From c6e4e52754e512debe257d4d010f228eb5747e87 Mon Sep 17 00:00:00 2001 From: Leif Ibsen Date: Tue, 17 Sep 2019 16:09:30 +0200 Subject: [PATCH] 1.0.0 --- .gitignore | 4 + Package.swift | 28 + README.md | 35 + Sources/BigInt/BigInt.swift | 1617 +++++ Sources/BigInt/BitSieve.swift | 149 + Sources/BigInt/ExpMod.swift | 262 + Sources/BigInt/Karatsuba.swift | 76 + Sources/BigInt/Limbs.swift | 709 ++ Sources/BigInt/ToomCook.swift | 119 + Tests/BigIntTests/AdditionTest.swift | 80 + Tests/BigIntTests/BitTest.swift | 84 + Tests/BigIntTests/ConstructorTest.swift | 117 + Tests/BigIntTests/DivModTest.swift | 74 + Tests/BigIntTests/ExpModTest.swift | 71 + Tests/BigIntTests/GcdTest.swift | 61 + Tests/BigIntTests/KaratsubaTest.swift | 42 + Tests/BigIntTests/LimbTest.swift | 33 + Tests/BigIntTests/ModInverseTest.swift | 40 + Tests/BigIntTests/MulSquareTest.swift | 50 + Tests/BigIntTests/MultiplicationTest.swift | 89 + Tests/BigIntTests/PerformanceTest.swift | 97 + Tests/BigIntTests/PrimeTest.swift | 47 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new file mode 100644 index 0000000..02c0875 --- /dev/null +++ b/.gitignore @@ -0,0 +1,4 @@ +.DS_Store +/.build +/Packages +/*.xcodeproj diff --git a/Package.swift b/Package.swift new file mode 100644 index 0000000..465560a --- /dev/null +++ b/Package.swift @@ -0,0 +1,28 @@ +// swift-tools-version:5.0 +// The swift-tools-version declares the minimum version of Swift required to build this package. + +import PackageDescription + +let package = Package( + name: "BigInt", + products: [ + // Products define the executables and libraries produced by a package, and make them visible to other packages. + .library( + name: "BigInt", + targets: ["BigInt"]), + ], + dependencies: [ + // Dependencies declare other packages that this package depends on. + // .package(url: /* package url */, from: "1.0.0"), + ], + targets: [ + // Targets are the basic building blocks of a package. A target can define a module or a test suite. + // Targets can depend on other targets in this package, and on products in packages which this package depends on. + .target( + name: "BigInt", + dependencies: []), + .testTarget( + name: "BigIntTests", + dependencies: ["BigInt"]), + ] +) diff --git a/README.md b/README.md new file mode 100644 index 0000000..5f0ce93 --- /dev/null +++ b/README.md @@ -0,0 +1,35 @@ +

Description

+ +The BigInt package provides arbitrary-precision integer arithmetic in Swift. +It contains functionality comparable to that of the Java BigInteger class. + +BigInt requires Swift 5.0. + + +

Usage

+Using Swift Package Manager, in your projects Package.swift file add a dependency like
+``` +dependencies: [ +// Dependencies declare other packages that this package depends on. +.package(url: "https://github.com/leif-ibsen/BigInt", from: "1.0.0"), +], +``` + +

References

+ +Algorithms from the following books have been used in the implementation. +There are references in the source code where appropriate. + + + + +

Acknowledgement

+ +The BitSieve class used in the implementation is a translation to Swift of the corresponding class from Java BigInteger. +The GCD algorithm and the Karatsuba and ToomCook multiplication algorithms are modelled after the corresponding algorithms in Java BigInteger. + diff --git a/Sources/BigInt/BigInt.swift b/Sources/BigInt/BigInt.swift new file mode 100755 index 0000000..1e34959 --- /dev/null +++ b/Sources/BigInt/BigInt.swift @@ -0,0 +1,1617 @@ +// +// BigInt.swift +// BigInt +// +// Created by Leif Ibsen on 24/12/2018. +// Copyright © 2018 Leif Ibsen. All rights reserved. +// + +/// Unsigned 8 bit value +public typealias Byte = UInt8 +/// Array of unsigned 8 bit values +public typealias Bytes = [UInt8] + +/// Unsigned 64 bit value +public typealias Limb = UInt64 +/// Array of unsigned 64 bit values +public typealias Limbs = [UInt64] + +precedencegroup ExponentiationPrecedence { + associativity: left + higherThan: MultiplicationPrecedence + lowerThan: BitwiseShiftPrecedence +} + +infix operator ** : ExponentiationPrecedence + +/// A signed integer of unbounded size. +/// A BInt value is represented with magnitude and sign. +/// The magnitude is an array of unsigned 64 bit integers (a.k.a. Limbs). +/// The sign is a boolean value, *true* means value < 0, *false* means value >= 0 +/// The representation is little-endian, least significant Limb has index 0. +/// The representation is minimal, there is no leading zero Limbs. +/// The exception is that the value 0 is represented as a single 64 bit zero Limb and sign *false* +public struct BInt: CustomStringConvertible, Equatable, Hashable { + + static let digits: [Character] = [ + "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", + "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n", "o", "p", "q", "r", "s", "t", "u", "v", "w", "x", "y", "z", + "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X", "Y", "Z"] + + + // MARK: - Constants + + /// BInt(0) + public static let ZERO = BInt(0) + /// BInt(1) + public static let ONE = BInt(1) + /// BInt(2) + public static let TWO = BInt(2) + /// BInt(3) + public static let THREE = BInt(3) + /// BInt(10) + public static let TEN = BInt(10) + + + // MARK: - Initializers + + /// Constructs a BInt from magnitude and sign + /// + /// - Parameters: + /// - magnitude: magnitude of value + /// - isNegative: *true* means negative value, *false* means 0 or positive value, default is *false* + public init(_ magnitude: Limbs, _ isNegative : Bool = false) { + self.magnitude = magnitude + self.magnitude.normalize() + self.isNegative = isNegative + if self.isZero { + self.isNegative = false + } + } + + /// Constructs a BInt from an Int value + /// + /// - Parameter x: Int value + public init(_ x: Int) { + if x == Int.min { + self.init([0x8000000000000000], true) + } else if x < 0 { + self.init([Limb(-x)], true) + } else { + self.init([Limb(x)], false) + } + } + + /// Constructs a BInt from a String value and radix + /// + /// - Parameters: + /// - x: String value to be converted + /// - radix: Radix of x, from 2 to 36 inclusive, default is 10 + /// + /// Examples: + /// * BInt("90abcdef", radix = 16) + /// * BInt("111110010", radix = 2) + /// * BInt("1cdefghijk44", radix = 26) + public init?(_ x: String, radix: Int = 10) { + if radix < 2 || radix > 36 { + return nil + } + var number = x + var sign: Bool + if x.hasPrefix("-") { + sign = true + number.remove(at: number.startIndex) + } else { + sign = false + } + var magnitude = [Limb(0)] + var base = [Limb(1)] + let lrdx = [Limb(radix)] + for s in number.reversed() { + if let digit = BInt.digits.firstIndex(of: s) { + let d = digit < 36 ? digit : digit - 26 + if d >= radix { + return nil + } + magnitude.add(base.times([Limb(d)])) + base.multiply(lrdx) + } else { + return nil + } + } + self.init(magnitude, sign) + } + + /// Constructs a random BInt with a specified number of bits + /// + /// - Precondition: bitWidth is positive + /// - Parameter bitWidth: Number of bits + public init(bitWidth: Int) { + precondition(bitWidth > 0, "Bitwidth must be positive") + let (q, r) = bitWidth.quotientAndRemainder(dividingBy: 64) + var limbs = Limbs(repeating: 0, count: r == 0 ? q : q + 1) + BInt.randomLimbs(&limbs) + if r > 0 { + limbs[limbs.count - 1] <<= 64 - r + limbs[limbs.count - 1] >>= 64 - r + } + self.init(limbs) + } + + /// Constructs a BInt from a big-endian magnitude byte array representation + /// + /// - Precondition: Byte array is not empty + /// - Parameter x: Magnitude big-endian byte array + /// + /// Examples: + /// * The byte array [1, 0, 0] represents BInt value 65536 + /// * The byte array [128, 0] represents BInt value 32768 + /// * The byte array [255, 127] represents BInt value 65407 + public init(magnitude x: Bytes) { + precondition(!x.isEmpty, "Empty byte array") + var bb = x + if bb[0] > 127 { + bb.insert(0, at: 0) + } + self.init(signed: bb) + } + + /// Constructs a BInt from a big-endian 2's complement byte array representation + /// + /// - Precondition: Byte array is not empty + /// - Parameter x: 2's complement big-endian byte array + /// + /// Examples: + /// * The byte array [1, 0, 0] represents BInt value 65536 + /// * The byte array [128, 0] represents BInt value -32768 + /// * The byte array [255, 127] represents BInt value -129 + public init(signed x: Bytes) { + precondition(!x.isEmpty, "Empty byte array") + self.isNegative = x[0] > 127 + var bb = x + if self.isNegative { + while bb.count > 1 && bb[0] == 255 { + bb.remove(at: 0) + } + } else { + while bb.count > 1 && bb[0] == 0 { + bb.remove(at: 0) + } + } + if self.isNegative { + var carry = true + var bbi = bb.count + for _ in 0 ..< bb.count { + bbi -= 1 + bb[bbi] = ~bb[bbi] + if carry { + if bb[bbi] == 255 { + bb[bbi] = 0 + } else { + bb[bbi] += 1 + carry = false + } + } + } + if carry { + bb.insert(1, at: 0) + } + } + let chunks = bb.count / 8 + let remaining = bb.count - chunks * 8 + self.magnitude = Limbs(repeating: 0, count: chunks + (remaining == 0 ? 0 : 1)) + var bi = 0 + var li = self.magnitude.count + if remaining > 0 { + li -= 1 + } + for _ in 0 ..< remaining { + self.magnitude[li] <<= 8 + self.magnitude[li] |= Limb(bb[bi]) + bi += 1 + } + for _ in 0 ..< chunks { + li -= 1 + for _ in 0 ..< 8 { + self.magnitude[li] <<= 8 + self.magnitude[li] |= Limb(bb[bi]) + bi += 1 + } + } + } + + + // MARK: Stored properties + + /// The sign, *true* if *self* < 0, *false* otherwise + public internal(set) var isNegative: Bool + + /// The magnitude limb array + public internal(set) var magnitude: Limbs + + + // MARK: Computed properties + + /// The number of bits in the magnitude of *self*. 0 if *self* = 0 + public var bitWidth: Int { + return self.magnitude.bitWidth + } + + /// Base 10 string value of *self* + public var description: String { + return self.asString() + } + + /// Is *true* if *self* is even, *false* if *self* is odd + public var isEven: Bool { + return self.magnitude[0] & 1 == 0 + } + + /// Is *false* if *self* = 0, *true* otherwise + public var isNotZero: Bool { + return self.magnitude.count > 1 || self.magnitude[0] > 0 + } + + /// Is *true* if *self* is odd, *false* if *self* is even + public var isOdd: Bool { + return self.magnitude[0] & 1 == 1 + } + + /// Is *true* if *self* = 1, *false* otherwise + public var isOne: Bool { + return self.magnitude.count == 1 && self.magnitude[0] == 1 && !self.isNegative + } + + /// Is *true* if *self* > 0, *false* otherwise + public var isPositive: Bool { + return !self.isNegative && self.isNotZero + } + + /// Is *true* if *self* = 0, *false* otherwise + public var isZero: Bool { + return self.magnitude.count == 1 && self.magnitude[0] == 0 + } + + /// The number of leading zero bits in the magnitude of *self*. 0 if *self* = 0 + public var leadingZeroBitCount: Int { + return self.isZero ? 0 : self.magnitude.last!.leadingZeroBitCount + } + + /// Is 0 if *self* = 0, 1 if *self* > 0, and -1 if *self* < 0 + public var signum: Int { + return self.isZero ? 0 : (self.isNegative ? -1 : 1) + } + + /// The number of trailing zero bits in the magnitude of *self*. 0 if *self* = 0 + public var trailingZeroBitCount: Int { + return self.magnitude.trailingZeroBitCount() + } + + mutating func setSign(_ sign: Bool) { + self.isNegative = self.isZero ? false : sign + } + + + // MARK: Conversion functions to Int, String, and Bytes + + /// *self* as an Int + /// + /// - Returns: *self* as an Int or *nil* if *self* is not representable as an Int + public func asInt() -> Int? { + if self.magnitude.count > 1 { + return nil + } + let mag0 = self.magnitude[0] + if self.isNegative { + return mag0 > 0x8000000000000000 ? nil : (mag0 == 0x8000000000000000 ? Int.min : -Int(mag0)) + } else { + return mag0 < 0x8000000000000000 ? Int(mag0) : nil + } + } + + static let zeros = "000000000000000000000000000000000000000000000000000000000000000" + static let limbDigits = [0, 0, + 63, 40, 31, 27, 24, 22, 21, 20, 19, 18, 17, 17, 16, 16, 15, 15, 15, + 15, 14, 14, 14, 14, 13, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12] + static let limbRadix = [BInt(0), BInt(0), + BInt([0x8000000000000000]), + BInt([0xa8b8b452291fe821]), + BInt([0x4000000000000000]), + BInt([0x6765c793fa10079d]), + BInt([0x41c21cb8e1000000]), + BInt([0x3642798750226111]), + BInt([0x8000000000000000]), + BInt([0xa8b8b452291fe821]), + BInt([0x8ac7230489e80000]), + BInt([0x4d28cb56c33fa539]), + BInt([0x1eca170c00000000]), + BInt([0x780c7372621bd74d]), + BInt([0x1e39a5057d810000]), + BInt([0x5b27ac993df97701]), + BInt([0x1000000000000000]), + BInt([0x27b95e997e21d9f1]), + BInt([0x5da0e1e53c5c8000]), + BInt([0xd2ae3299c1c4aedb]), + BInt([0x16bcc41e90000000]), + BInt([0x2d04b7fdd9c0ef49]), + BInt([0x5658597bcaa24000]), + BInt([0xa0e2073737609371]), + BInt([0xc29e98000000000]), + BInt([0x14adf4b7320334b9]), + BInt([0x226ed36478bfa000]), + BInt([0x383d9170b85ff80b]), + BInt([0x5a3c23e39c000000]), + BInt([0x8e65137388122bcd]), + BInt([0xdd41bb36d259e000]), + BInt([0xaee5720ee830681]), + BInt([0x1000000000000000]), + BInt([0x172588ad4f5f0981]), + BInt([0x211e44f7d02c1000]), + BInt([0x2ee56725f06e5c71]), + BInt([0x41c21cb8e1000000])] + + /// Byte array representation of magnitude value + /// + /// - Returns: Minimal big-endian magnitude byte array representation + /// + /// Examples: + /// * BInt(1).asMagnitudeBytes() = [1] + /// * BInt(-1).asMagnitudeBytes() = [1] + public func asMagnitudeBytes() -> Bytes { + var bb = (self.isNegative ? -self : self).asSignedBytes() + while bb.count > 1 && bb[0] == 0 { + bb.remove(at: 0) + } + return bb + } + + /// Byte array representation of 2's complement value + /// + /// - Returns: Minimal big-endian 2's complement byte array representation + /// + /// Examples: + /// * BInt(1).asSignedBytes() = [1] + /// * BInt(-1).asSignedBytes() = [255] + public func asSignedBytes() -> Bytes { + var xl = self.magnitude + if self.isNegative { + var carry = true + for i in 0 ..< xl.count { + xl[i] = ~xl[i] + if carry { + if xl[i] == Limb.max { + xl[i] = 0 + } else { + xl[i] += 1 + carry = false + } + } + } + } + var bb = Bytes(repeating: 0, count: xl.count * 8) + var bbi = bb.count + for i in 0 ..< xl.count { + var l = xl[i] + for _ in 0 ..< 8 { + bbi -= 1 + bb[bbi] = Byte(l & 0xff) + l >>= 8 + } + } + if self.isNegative { + if bb[0] < 128 { + bb.insert(255, at: 0) + } + while bb.count > 1 && bb[0] == 255 && bb[1] > 127 { + bb.remove(at: 0) + } + } else { + if bb[0] > 127 { + bb.insert(0, at: 0) + } + while bb.count > 1 && bb[0] == 0 && bb[1] < 128 { + bb.remove(at: 0) + } + } + return bb + } + + /// *self* as a String with a given radix + /// + /// - Precondition: Radix between 2 and 36 inclusive + /// - Parameters: + /// - radix: Radix from 2 to 36 inclusive + /// - uppercase: *true* to use uppercase letters, *false* to use lowercase letters, default is *false* + /// - Returns: *self* as a String in the given radix + public func asString(radix: Int = 10, uppercase: Bool = false) -> String { + precondition(radix >= 2 && radix <= 36, "Wrong radix \(radix)") + if self.isZero { + return "0" + } + let d = BInt.limbRadix[radix] + var digitGroups = [String]() + var tmp = BInt(self.magnitude) + while tmp.isNotZero { + let (q, r) = tmp.quotientAndRemainder(dividingBy: d) + digitGroups.append(r.isZero ? "0" : String(r.magnitude[0], radix: radix, uppercase: uppercase)) + tmp = q + } + var result = self.isNegative ? "-" : "" + result += digitGroups.last! + for i in (0 ..< digitGroups.count - 1).reversed() { + let leadingZeros = BInt.limbDigits[radix] - digitGroups[i].count + result += BInt.zeros.prefix(leadingZeros) + result += digitGroups[i] + } + return result + } + + static func toSignedBytesPair(_ x: BInt, _ y: BInt) -> (bx: Bytes, by: Bytes) { + var bx = x.asSignedBytes() + var by = y.asSignedBytes() + let x0 = bx[0] > Byte(0x7f) ? Byte(0xff) : 0 + let y0 = by[0] > Byte(0x7f) ? Byte(0xff) : 0 + while bx.count < by.count { + bx.insert(x0, at: 0) + } + while by.count < bx.count { + by.insert(y0, at: 0) + } + return (bx, by) + } + + + // MARK: Bit operation functions + + /// Bitwise **and** operator + /// + /// - Parameters: + /// - x: First value + /// - y: Second value + /// - Returns: BInt(signed: bx & by) where + /// - bx = x.asSignedBytes() + /// - by = y.asSignedBytes() + public static func &(x: BInt, y: BInt) -> BInt { + var (bx, by) = BInt.toSignedBytesPair(x, y) + for i in 0 ..< bx.count { + bx[i] &= by[i] + } + return BInt(signed: bx) + } + + /// x = x & y + /// + /// - Parameters: + /// - x: Left hand parameter + /// - y: Right hand parameter + public static func &=(x: inout BInt, y: BInt) { + x = x & y + } + + /// Bitwise **or** operator + /// + /// - Parameters: + /// - x: First value + /// - y: Second value + /// - Returns: BInt(signed: bx | by) where + /// - bx = x.asSignedBytes() + /// - by = y.asSignedBytes() + public static func |(x: BInt, y: BInt) -> BInt { + var (bx, by) = BInt.toSignedBytesPair(x, y) + for i in 0 ..< bx.count { + bx[i] |= by[i] + } + return BInt(signed: bx) + } + + /// x = x | y + /// + /// - Parameters: + /// - x: Left hand parameter + /// - y: Right hand parameter + public static func |=(x: inout BInt, y: BInt) { + x = x | y + } + + /// Bitwise **xor** operator + /// + /// - Parameters: + /// - x: First value + /// - y: Second value + /// - Returns: BInt(signed: bx ^ by) where + /// - bx = x.asSignedBytes() + /// - by = y.asSignedBytes() + public static func ^(x: BInt, y: BInt) -> BInt { + var (bx, by) = BInt.toSignedBytesPair(x, y) + for i in 0 ..< bx.count { + bx[i] ^= by[i] + } + return BInt(signed: bx) + } + + /// x = x ^ y + /// + /// - Parameters: + /// - x: Left hand parameter + /// - y: Right hand parameter + public static func ^=(x: inout BInt, y: BInt) { + x = x ^ y + } + + /// Bitwise **not** operator + /// + /// - Parameter x: BInt value + /// - Returns: -x - 1 + public static prefix func ~(x: BInt) -> BInt { + return -x - 1 + } + + /// Clear a specified bit + /// + /// - Parameter n: Bit number + public mutating func clearBit(_ n: Int) { + self.magnitude.setBitAt(n, to: false) + } + + /// Invert a specified bit + /// + /// - Parameter n: Bit number + public mutating func flipBit(_ n: Int) { + self.magnitude.setBitAt(n, to: !self.magnitude.getBitAt(n)) + } + + /// Set a specified bit + /// + /// - Parameter n: Bit number + public mutating func setBit(_ n: Int) { + self.magnitude.setBitAt(n, to: true) + } + + /// Test a specified bit + /// + /// - Parameter n: Bit number + /// - Returns: *true* if bit is set, *false* otherwise + public func testBit(_ n: Int) -> Bool { + return self.magnitude.getBitAt(n) + } + + + // MARK: Addition functions + + /// Prefix plus + /// + /// - Parameter x: BInt value + /// - Returns: x + public prefix static func +(x: BInt) -> BInt { + return x + } + + /// Addition + /// + /// - Parameters: + /// - x: First addend + /// - y: Second addend + /// - Returns: x + y + public static func +(x: BInt, y: BInt) -> BInt { + var sum = x + sum += y + return sum + } + + /// Addition + /// + /// - Parameters: + /// - x: First addend + /// - y: Second addend + /// - Returns: x + y + public static func +(x: Int, y: BInt) -> BInt { + var sum = y + sum += x + return sum + } + + /// Addition + /// + /// - Parameters: + /// - x: First addend + /// - y: Second addend + /// - Returns: x + y + public static func +(x: BInt, y: Int) -> BInt { + var sum = x + sum += y + return sum + } + + /// x = x + y + /// + /// - Parameters: + /// - x: Left hand addend + /// - y: Right hand addend + public static func +=(x: inout BInt, y: BInt) { + if x.isNegative == y.isNegative { + x.magnitude.add(y.magnitude) + } else { + let cmp = x.magnitude.difference(y.magnitude) + if cmp < 0 { + x.isNegative = !x.isNegative + } else if cmp == 0 { + x.isNegative = false + } + } + } + + /// x = x + y + /// + /// - Parameters: + /// - x: Left hand addend + /// - y: Right hand addend + public static func +=(x: inout BInt, y: Int) { + let absy = Limb(Swift.abs(y)) + if (y < 0 && x.isNegative) || (y >= 0 && !x.isNegative) { + x.magnitude.add(absy) + } else { + let cmp = x.magnitude.compare(absy) + x.magnitude.difference(absy) + if cmp < 0 { + x.isNegative = !x.isNegative + } else if cmp == 0 { + x.isNegative = false + } + } + } + + + // MARK: Negation functions + + /// Negates *self* + public mutating func negate() { + if self.isNotZero { + self.isNegative = !self.isNegative + } + } + + /// Negation + /// + /// - Parameter x: Operand + /// - Returns: -x + public static prefix func -(x: BInt) -> BInt { + var y = x + y.negate() + return y + } + + + // MARK: Subtraction functions + + /// Subtraction + /// + /// - Parameters: + /// - x: Minuend + /// - y: Subtrahend + /// - Returns: x - y + public static func -(x: BInt, y: BInt) -> BInt { + var diff = x + diff -= y + return diff + } + + /// Subtraction + /// + /// - Parameters: + /// - x: Minuend + /// - y: Subtrahend + /// - Returns: x - y + public static func -(x: Int, y: BInt) -> BInt { + var diff = y + diff -= x + return -diff + } + + /// Subtraction + /// + /// - Parameters: + /// - x: Minuend + /// - y: Subtrahend + /// - Returns: x - y + public static func -(x: BInt, y: Int) -> BInt { + var diff = x + diff -= y + return diff + } + + /// x = x - y + /// + /// - Parameters: + /// - x: Left hand minuend + /// - y: Right hand subtrahend + public static func -=(x: inout BInt, y: BInt) { + if x.isNegative == y.isNegative { + let cmp = x.magnitude.difference(y.magnitude) + if cmp < 0 { + x.isNegative = !x.isNegative + } else if cmp == 0 { + x.isNegative = false + } + } else { + x.magnitude.add(y.magnitude) + } + } + + /// x = x - y + /// + /// - Parameters: + /// - x: Left hand minuend + /// - y: Right hand subtrahend + public static func -=(x: inout BInt, y: Int) { + let absy = Limb(Swift.abs(y)) + if (y < 0 && !x.isNegative) || (y >= 0 && x.isNegative) { + x.magnitude.add(absy) + } else { + let cmp = x.magnitude.compare(absy) + x.magnitude.difference(absy) + if cmp < 0 { + x.isNegative = !x.isNegative + } else if cmp == 0 { + x.isNegative = false + } + } + } + + + // MARK: Multiplication functions + + /// Multiplication + /// + /// - Parameters: + /// - x: Multiplier + /// - y: Multiplicand + /// - Returns: x * y + public static func *(x: BInt, y: BInt) -> BInt { + var prod = x + prod *= y + return prod + } + + /// Multiplication + /// + /// - Parameters: + /// - x: Multiplier + /// - y: Multiplicand + /// - Returns: x * y + public static func *(x: Int, y: BInt) -> BInt { + var prod = y + prod *= x + return prod + } + + /// Multiplication + /// + /// - Parameters: + /// - x: Multiplier + /// - y: Multiplicand + /// - Returns: x * y + public static func *(x: BInt, y: Int) -> BInt { + var prod = x + prod *= y + return prod + } + + /// x = x * y + /// + /// - Parameters: + /// - x: Left hand multiplier + /// - y: Right hand multiplicand + public static func *=(x: inout BInt, y: BInt) { + x.magnitude.multiply(y.magnitude) + x.setSign(x.isNegative != y.isNegative) + } + + /// x = x * y + /// + /// - Parameters: + /// - x: Left hand multiplier + /// - y: Right hand multiplicand + public static func *=(x: inout BInt, y: Int) { + if y > 0 { + x.magnitude.multiply(Limb(y)) + } else if y < 0 { + x.magnitude.multiply(Limb(-y)) + x.setSign(!x.isNegative) + } else { + x = BInt.ZERO + } + } + + + // MARK: Division functions + + /// Division + /// + /// - Precondition: Divisor is not zero + /// - Parameter x: Divisor + /// - Returns: Quotient and remainder of *self* / x + public func quotientAndRemainder(dividingBy x: BInt) -> (quotient: BInt, remainder: BInt) { + var quotient = BInt.ZERO + var remainder = BInt.ZERO + self.quotientAndRemainder(dividingBy: x, "ient, &remainder) + return (quotient, remainder) + } + + /// Division + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Divisor + /// - quotient: Set to the quotient of *self* / x + /// - remainder: Set to the remainder of *self* / x + public func quotientAndRemainder(dividingBy x: BInt, _ quotient: inout BInt, _ remainder: inout BInt) { + self.magnitude.divMod(x.magnitude, "ient.magnitude, &remainder.magnitude) + quotient.setSign(self.isNegative != x.isNegative) + remainder.setSign(self.isNegative) + } + + /// Division + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + /// - Returns: x / y + public static func /(x: BInt, y: BInt) -> BInt { + return x.quotientAndRemainder(dividingBy: y).quotient + } + + /// Division + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + /// - Returns: x / y + public static func /(x: Int, y: BInt) -> BInt { + return BInt(x) / y + } + + /// Division + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + /// - Returns: x / y + public static func /(x: BInt, y: Int) -> BInt { + return x / BInt(y) + } + + /// x = x / y + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Left hand dividend + /// - y: Right hand divisor + public static func /=(x: inout BInt, y: BInt) { + x = x / y + } + + /// x = x / y + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Left hand dividend + /// - y: Right hand divisor + public static func /=(x: inout BInt, y: Int) { + x = x / y + } + + + // MARK: Remainder and modulus functions + + /// Remainder + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + /// - Returns: x % y + public static func %(x: BInt, y: BInt) -> BInt { + return x.quotientAndRemainder(dividingBy: y).remainder + } + + /// Remainder + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + /// - Returns: x % y + public static func %(x: Int, y: BInt) -> BInt { + return BInt(x) % y + } + + /// Remainder + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + /// - Returns: x % y + public static func %(x: BInt, y: Int) -> BInt { + return x % BInt(y) + } + + /// x = x % y + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + public static func %=(x: inout BInt,y: BInt) { + x = x % y + } + + /// x = x % y + /// + /// - Precondition: Divisor is not zero + /// - Parameters: + /// - x: Dividend + /// - y: Divisor + public static func %=(x: inout BInt, y: Int) { + x = x % y + } + + /// Modulus + /// + /// - Precondition: Divisor is not zero + /// - Parameter x: Divisor + /// - Returns: *self* *mod* x, a non-negative value + public func mod(_ x: BInt) -> BInt { + let r = self % x + if x.isNegative { + return r.isNegative ? r - x : r + } else { + return r.isNegative ? r + x : r + } + } + + /* + * Prime Numbers - algorithm 2.1.4 + * + * Return self modinverse m + */ + /// Inverse modulus + /// + /// - Precondition: *self* and modulus are coprime + /// - Parameter m: Modulus + /// - Returns: If *self* and m are coprime, x such that (*self* * x) mod m = 1 + public func modInverse(_ m: BInt) -> BInt { + var a = BInt.ONE + var b = BInt.ZERO + var g = self.mod(m) + var u = BInt.ZERO + var v = BInt.ONE + var w = m + while w.isPositive { + let q = g / w + (a, b, g, u, v, w) = (u, v, w, a - q * u, b - q * v, g - q * w) + } + precondition(g.isOne, "Modulus and self are not coprime") + return a.isNegative ? a + m : a + } + + + // MARK: Exponentiation functions + + /// Exponentiation + /// + /// - Precondition: Exponent is non-negative + /// - Parameters: + /// - a: Operand + /// - x: Non-negative exponent + /// - Returns: a^x + public static func **(a: BInt, x: Int) -> BInt { + precondition(x >= 0, "Exponent must be non-negative") + return x == 2 ? BInt(a.magnitude.squared()) : BInt(a.magnitude.raisedTo(x), a.isNegative && (x & 1 == 1)) + } + + /* + * Return (self ** x) mod m + * + * Use Barrett reduction algorithm for x.bitWidth < 512, else use Montgomery reduction algorithm + */ + /// Modular exponentiation + /// + /// - Precondition: Modulus is positive + /// - Parameters: + /// - x: The exponent + /// - m: The modulus, a positive number + /// - Returns: (*self*^x) mod m for positive x, ((*self*^-x) mod m) modInverse m for negative x + public func expMod(_ x: BInt, _ m: BInt) -> BInt { + precondition(m.isPositive, "Modulus must be positive") + if m.isOne { + return BInt.ZERO + } + let exponent = x.isNegative ? -x : x + var result: BInt + if exponent.magnitude.count < 9 { + result = BarrettModulus(self, m).expMod(exponent) + } else { + let trailing = m.trailingZeroBitCount + let pow2Modulus = BInt.ONE << trailing + let oddModulus = m >> trailing + if pow2Modulus.isOne { + result = OddModulus(self, oddModulus).expMod(exponent) + } else if oddModulus.isOne { + result = Pow2Modulus(self, pow2Modulus).expMod(exponent) + } else { + let pow2Mod = Pow2Modulus(self, pow2Modulus) + let oddMod = OddModulus(self, oddModulus) + let a1 = oddMod.expMod(exponent) + let a2 = pow2Mod.expMod(exponent) + let y1 = pow2Modulus.modInverse(oddModulus) + let y2 = oddModulus.modInverse(pow2Modulus) + result = (a1 * pow2Modulus * y1 + a2 * oddModulus * y2).mod(m) + } + } + if x.isNegative { + result = result.modInverse(m) + } + if self.isNegative { + return x.isEven || result.isZero ? result : m - result + } else { + return result + } + } + + + // MARK: Comparison functions + + /// Equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x = y, *false* otherwise + public static func ==(x: BInt, y: BInt) -> Bool { + return x.magnitude == y.magnitude && x.isNegative == y.isNegative + } + + /// Equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x = y, *false* otherwise + public static func ==(x: BInt, y: Int) -> Bool { + return x == BInt(y) + } + + /// Equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x = y, *false* otherwise + public static func ==(x: Int, y: BInt) -> Bool { + return BInt(x) == y + } + + /// Not equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x != y, *false* otherwise + public static func !=(x: BInt, y: BInt) -> Bool { + return x.magnitude != y.magnitude || x.isNegative != y.isNegative + } + + /// Not equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x != y, *false* otherwise + public static func !=(x: BInt, y: Int) -> Bool { + return x != BInt(y) + } + + /// Not equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x != y, *false* otherwise + public static func !=(x: Int, y: BInt) -> Bool { + return BInt(x) != y + } + + /// Less then + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x < y, *false* otherwise + public static func <(x: BInt, y: BInt) -> Bool { + if x.isNegative && y.isNegative { + return y.magnitude.lessThan(x.magnitude) + } else if !x.isNegative && y.isNegative { + return false + } else if x.isNegative && !y.isNegative { + return true + } else { + return x.magnitude.lessThan(y.magnitude) + } + } + + /// Less than + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x < y, *false* otherwise + public static func <(x: BInt, y: Int) -> Bool { + return x < BInt(y) + } + + /// Less than + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x < y, *false* otherwise + public static func <(x: Int, y: BInt) -> Bool { + return BInt(x) < y + } + + /// Greater than + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x > y, *false* otherwise + public static func >(x: BInt, y: BInt) -> Bool { + return y < x + } + + /// Greater than + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x > y, *false* otherwise + public static func >(x: Int, y: BInt) -> Bool { + return y < x + } + + /// Greater than + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x > y, *false* otherwise + public static func >(x: BInt, y: Int) -> Bool { + return y < x + } + + /// Less than or equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x <= y, *false* otherwise + public static func <=(x: BInt, y: BInt) -> Bool { + return !(y < x) + } + + /// Less than or equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x <= y, *false* otherwise + public static func <=(x: Int, y: BInt) -> Bool { + return !(y < x) + } + + /// Less than or equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x <= y, *false* otherwise + public static func <=(x: BInt, y: Int) -> Bool { + return !(y < x) + } + + /// Greater than or equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x >= y, *false* otherwise + public static func >=(x: BInt, y: BInt) -> Bool { + return !(x < y) + } + + /// Greater than or equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x >= y, *false* otherwise + public static func >=(x: Int, y: BInt) -> Bool { + return !(x < y) + } + + /// Greater than or equal + /// + /// - Parameters: + /// - x: First operand + /// - y: Second operand + /// - Returns: *true* if x >= y, *false* otherwise + public static func >=(x: BInt, y: Int) -> Bool { + return !(x < y) + } + + + // MARK: Shift functions + + /// Logical left shift + /// + /// - Parameters: + /// - x: Operand + /// - n: Shift count + /// - Returns: + /// - BInt(x.magnitude << n, x.isNegative) if n > 0 + /// - BInt(x.magnitude >> -n, x.isNegative) if n < 0 + /// - x if n = 0 + public static func <<(x: BInt, n: Int) -> BInt { + if n < 0 { + return x >> -n + } + return BInt(n == 1 ? x.magnitude.shifted1Left() : x.magnitude.shiftedLeft(n), x.isNegative) + } + + /// x = x << n + /// + /// - Parameters: + /// - x: Operand + /// - n: Shift count + public static func <<=(x: inout BInt, n: Int) { + if n < 0 { + x.magnitude.shiftRight(-n) + } else if n == 1 { + x.magnitude.shift1Left() + } else { + x.magnitude.shiftLeft(n) + } + } + + /// Logical right shift + /// + /// - Parameters: + /// - x: Operand + /// - n: Shift count + /// - Returns: + /// - BInt(x.magnitude >> n, x.isNegative) if n > 0 + /// - BInt(x.magnitude << -n, x.isNegative) if n < 0 + /// - x if n = 0 + public static func >>(x: BInt, n: Int) -> BInt { + if n < 0 { + return x << -n + } + return BInt(n == 1 ? x.magnitude.shifted1Right() : x.magnitude.shiftedRight(n), x.isNegative) + } + + /// x = x >> n + /// + /// - Parameters: + /// - x: Operand + /// - n: Shift count + public static func >>=(x: inout BInt, n: Int) { + if n < 0 { + x.magnitude.shiftLeft(-n) + } else if n == 1 { + x.magnitude.shift1Right() + } else { + x.magnitude.shiftRight(n) + } + if x.isZero { + x.isNegative = false + } + } + + + // MARK: Prime number functions + + static internal var random = SystemRandomNumberGenerator() + + static internal func randomBytes(_ bytes: inout Bytes) { + for i in 0 ..< bytes.count { + bytes[i] = BInt.random.next() + } + } + + static internal func randomLimbs(_ limbs: inout Limbs) { + for i in 0 ..< limbs.count { + limbs[i] = BInt.random.next() + } + } + + // Small prime product + static let SPP = BInt(3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41) + + static func smallPrime(_ bitLength: Int) -> BInt { + let multiple8 = bitLength & 0x7 == 0 + let length = multiple8 ? (bitLength + 7) >> 3 + 1 : (bitLength + 7) >> 3 + var bytes = Bytes(repeating: 0, count: length) + let highBit = Byte(1 << ((bitLength + 7) & 0x7)) // High bit of high int + let highMask = Byte((Int(highBit) << 1) - 1) // Bits to keep in high int + + while true { + BInt.randomBytes(&bytes) + if multiple8 { + bytes[0] = 0 + bytes[1] = (bytes[1] & highMask) | highBit + } else { + bytes[0] = (bytes[0] & highMask) | highBit + } + let x = BInt(signed: bytes) + if bitLength > 6 { + let r = x % SPP + if r % 3 == 0 || r % 5 == 0 || r % 7 == 0 || r % 11 == 0 || r % 13 == 0 || r % 17 == 0 || + r % 19 == 0 || r % 23 == 0 || r % 29 == 0 || r % 31 == 0 || r % 37 == 0 || r % 41 == 0 { + continue + } + } + if x.isProbablyPrime() { + return x + } + } + } + + static func largePrime(_ bitLength: Int, _ p: Int) -> BInt { + var x = BInt(bitWidth: bitLength) + x.setBit(bitLength - 1) + x.clearBit(0) + var bs = BitSieve(x, p) + var candidate = bs.retrieve() + while candidate == nil || candidate!.bitWidth != bitLength { + x += BInt(2 * bs.length) + if x.bitWidth != bitLength { + x = BInt(bitWidth: bitLength) + x.setBit(bitLength - 1) + } + x.clearBit(0) + bs = BitSieve(x, p) + candidate = bs.retrieve() + } + return candidate! + } + + /// Checks whether *self* is prime using the Miller-Rabin algorithm + /// + /// - Parameter p: If *true* is returned, *self* is prime with probability > 1-1/2^p + /// - Returns: *true* if *self* is probably prime, *false* if *self* is definitely not prime + public func isProbablyPrime(_ p: Int = 30) -> Bool { + if self < 2 { + return false + } + var rounds: Int + if self.bitWidth < 100 { + rounds = 50 + } else if self.bitWidth < 256 { + rounds = 27 + } else if bitWidth < 512 { + rounds = 15 + } else if bitWidth < 768 { + rounds = 8 + } else if bitWidth < 1024 { + rounds = 4 + } else { + rounds = 2 + } + rounds = min((p + 1) / 2, rounds) + let s1 = self - 1 + for _ in 0 ..< rounds { + if !self.pass(s1.randomLessThan() + 1) { + return false + } + } + return true + } + + func pass(_ a: BInt) -> Bool { + let s_1 = self - 1 + let k = s_1.trailingZeroBitCount + let m = s_1 >> k + var x = a.expMod(m, self) + if x == 1 { + return true + } + if k > 0 { + for _ in 0 ..< k - 1 { + if x == s_1 { + return true + } + x = (x * x) % self + } + } + return x == s_1 + } + + /// A probable prime number with a given bitwidth + /// + /// - Parameters: + /// - bitWidth: The bitWidth + /// - p: The returned number is prime with probability > 1-1/2^p, default value is 30 + /// - Returns: A prime number with the specified bitwidth and probability + public static func probablePrime(_ bitWidth: Int, _ p: Int = 30) -> BInt { + return bitWidth < 100 ? smallPrime(bitWidth) : largePrime(bitWidth, p) + } + + + // MARK: Miscellaneous functions + + /// Absolute value + /// + /// - Returns: The absolute value of *self* + public func abs() -> BInt { + return BInt(self.magnitude) + } + + /// Greatest common divisor + /// + /// - Parameter x: Operand + /// - Returns: Greatest common divisor of magnitude of *self* and magnitude of x + public func gcd(_ x: BInt) -> BInt { + return BInt(self.magnitude.gcd(x.magnitude)) + } + + /* + * Prime Numbers - algorithm 2.3.5 + */ + /// Jacobi symbol + /// + /// - Parameters: + /// - m: An integer value + /// - Returns: The Jacobi symbol of *self* and m, or 0 if it does not exist + public func jacobiSymbol(_ m: BInt) -> Int { + var m1 = m + var a = self % m1 + var t = 1 + while a.isNotZero { + var x: BInt + while a.isEven { + a >>= 1 + x = m1 % 8 + if x == 3 || x == 5 { + t = -t + } + } + x = a + a = m1 + m1 = x + if a % 4 == 3 && m1 % 4 == 3 { + t = -t + } + a %= m1 + } + return m1 == 1 ? t : 0 + } + + /// Random value + /// + /// - Precondition: *self* is positive + /// - Returns: A random value < absolute value of *self* + public func randomLessThan() -> BInt { + precondition(self.isPositive, "Must be positive") + return BInt(bitWidth: self.bitWidth) % self + } + + /* + * Prime Numbers - exercise 4.11 + */ + /// n'th root of a non-negative number + /// + /// - Precondition: + /// - *self* is non-negative + /// - n is positive + /// - Parameter n: The root + /// - Returns: Largest x such that x^n <= *self* + public func root(_ n: Int) -> BInt { + precondition(!self.isNegative, "\(n)'th root of negative number") + precondition(n > 0, "non-positive root") + if self.isZero { + return BInt.ZERO + } + let bn = BInt(n) + let bn1 = bn - 1 + var x = BInt.ONE << (self.bitWidth / n + 1) + while true { + let xx = x ** (n - 1) + let y = (self / xx + x * bn1) / bn + if y >= x { + return x + } + x = y + } + } + + /* + * Prime Numbers - algorithm 9.2.11 + */ + /// Square root of a non-negative number + /// + /// - Precondition: *self* is non-negative + /// - Returns: Largest x such that x^2 <= *self* + public func sqrt() -> BInt { + precondition(!self.isNegative, "Square root of negative number") + if self.isZero { + return BInt.ZERO + } + var x = BInt.ONE << (self.bitWidth / 2 + 1) + while true { + let y = (self / x + x) >> 1 + if y >= x { + return x + } + x = y + } + } + + /* + * Prime Numbers - algorithm 2.3.8 + */ + /// Square root modulo a prime number + /// + /// - Parameter p: An odd prime number + /// - Returns: x, such that x^2 mod p = *self*, or *nil* if no such x exists + public func sqrtMod(_ p: BInt) -> BInt? { + let A = self % p + switch (p % 8).asInt() { + case 3, 7: + return A.expMod((p + 1) >> 2, p) + + case 5: + var x = A.expMod((p + 3) >> 3, p) + if (x * x) % p != A % p { + x = x * BInt.TWO.expMod((p - 1) >> 2, p) % p + } + return x + + case 1: + let p_1 = p - 1 + var d: BInt? + let p_3 = p - 3 + while true { + d = p_3.randomLessThan() + 2 + if d!.jacobiSymbol(p) == -1 { + break + } + } + var s = 0 + var t = p_1 + while t.isEven { + s += 1 + t >>= 1 + } + let A1 = A.expMod(t, p) + let D = d!.expMod(t, p) + var m = BInt.ZERO + var exp = BInt.TWO ** (s - 1) + for i in 0 ..< s { + let DD = (D ** m.asInt()!) * A1 + if DD.expMod(exp, p) == p_1 { + m.setBit(i) + } + exp >>= 1 + } + return (A.expMod((t + 1) >> 1, p) * D.expMod(m >> 1, p)) % p + + default: + return nil + } + } + +} diff --git a/Sources/BigInt/BitSieve.swift b/Sources/BigInt/BitSieve.swift new file mode 100755 index 0000000..3cd2b33 --- /dev/null +++ b/Sources/BigInt/BitSieve.swift @@ -0,0 +1,149 @@ +// +// BitSieve.swift +// BigInt +// +// Created by Leif Ibsen on 15/11/2018. +// Copyright © 2018 Leif Ibsen. All rights reserved. +// + +/* + * BitSieve class from Java BigInteger translated to Swift + */ +class BitSieve { + + var bits: [UInt64] + let length: Int + let base: BInt + let prob: Int + + static let smallSieve = BitSieve() + + init() { + self.length = 150 * 64 + self.bits = [UInt64](repeating: 0, count: ((length - 1) >> 6) + 1) + self.base = BInt.ZERO + self.prob = 0 + + // Mark 1 as composite + set(0) + var nextIndex = 1 + var nextPrime = 3 + var more = true + + // Find primes and remove their multiples from sieve + repeat { + sieveSingle(self.length, nextIndex + nextPrime, nextPrime) + (nextIndex, more) = sieveSearch(self.length, nextIndex + 1) + nextPrime = nextIndex << 1 + 1 + } while more && (nextPrime < self.length) + } + + /** + * Construct a bit sieve of searchLen bits used for finding prime number + * candidates. The new sieve begins at the specified base, which must + * be even. + */ + init(_ base: BInt, _ prob: Int) { + /* + * Candidates are indicated by clear bits in the sieve. As a candidates + * nonprimality is calculated, a bit is set in the sieve to eliminate + * it. To reduce storage space and increase efficiency, no even numbers + * are represented in the sieve (each bit in the sieve represents an + * odd number). + */ + self.base = base + self.length = 16 * base.bitWidth + self.bits = [UInt64](repeating: 0, count: ((self.length - 1) >> 6) + 1) + self.prob = prob + var start = 0 + + var (step, more) = BitSieve.smallSieve.sieveSearch(BitSieve.smallSieve.length, start) + var convertedStep = step << 1 + 1 + + // Construct the large sieve at an even offset specified by base + repeat { + // Calculate base mod convertedStep + let (_, r) = self.base.magnitude.divMod1(UInt64(convertedStep)) + start = Int(r) + + // Take each multiple of step out of sieve + start = convertedStep - start + if start & 1 == 0 { + start += convertedStep + } + sieveSingle(self.length, (start - 1) >> 1, convertedStep) + + // Find next prime from small sieve + (step, more) = BitSieve.smallSieve.sieveSearch(BitSieve.smallSieve.length, step + 1) + convertedStep = step << 1 + 1 + } while more + } + + /** + * Get the value of the bit at the specified index. + */ + func get(_ bitIndex: Int) -> Bool { + return (self.bits[bitIndex >> 6] & Limbs.UMasks[bitIndex & 0x3f]) != 0 + } + + /** + * Set the bit at the specified index. + */ + func set(_ bitIndex: Int) { + self.bits[bitIndex >> 6] |= Limbs.UMasks[bitIndex & 0x3f] + } + + /** + * This method returns the index of the first clear bit in the search + * array that occurs at or after start. It will not search past the + * specified limit. It returns false if there is no such clear bit. + */ + func sieveSearch(_ limit: Int, _ start: Int) -> (Int, Bool) { + if start >= limit { + return (0, false) + } + var index = start + repeat { + if !get(index) { + return (index, true) + } + index += 1 + } while index < limit - 1 + return (0, false) + } + + /** + * Sieve a single set of multiples out of the sieve. Begin to remove + * multiples of the specified step starting at the specified start index, + * up to the specified limit. + */ + func sieveSingle(_ limit: Int, _ start: Int, _ step: Int) { + var x = start + while x < limit { + set(x) + x += step + } + } + + /** + * Test probable primes in the sieve and return successful candidates. + */ + func retrieve() -> BInt? { + // Examine the sieve one word at a time to find possible primes + var offset = 1 + for i in 0 ..< bits.count { + let nextWord = ~bits[i] + for j in 0 ..< 64 { + if nextWord & Limbs.UMasks[j] != 0 { + let candidate = self.base + offset + if candidate.isProbablyPrime(prob) { + return candidate + } + } + offset += 2 + } + } + return nil + } + +} diff --git a/Sources/BigInt/ExpMod.swift b/Sources/BigInt/ExpMod.swift new file mode 100755 index 0000000..e9b1b79 --- /dev/null +++ b/Sources/BigInt/ExpMod.swift @@ -0,0 +1,262 @@ +// +// ExpMod.swift +// BigInt +// +// Created by Leif Ibsen on 17/02/2019. +// Copyright © 2019 Leif Ibsen. All rights reserved. +// + +extension BInt { + + /* + * Montgomery helper class for (a ** x) mod m computation, from + * + * Montgomery Multiplication + * By Henry S. Warren, Jr. + * July 2012 + */ + class OddModulus { + + let modulus: Limbs + let base: Limbs + let Rsize: Int + var Rinv: Limbs = [1] + var mprime: Limbs = [0] + + init(_ a: BInt, _ modulus: BInt) { + self.modulus = modulus.magnitude + self.Rsize = self.modulus.count + self.base = a.magnitude.divMod(self.modulus).remainder + + // Compute Rinv and mprime such that R * Rinv - modulus * mprime = 1 + + for _ in 0 ..< Rsize * 64 { + if self.Rinv[0] & 1 == 0 { + self.Rinv.shift1Right() + self.mprime.shift1Right() + } else { + self.Rinv.add(self.modulus) + self.Rinv.shift1Right() + self.mprime.shift1Right() + self.mprime.setBitAt(Rsize * 64 - 1, to: true) + } + } + } + + func toMspace(_ x: Limbs) -> Limbs { + return (x.shiftedLeft(self.Rsize * 64)).divMod(self.modulus).remainder + } + + func fromMspace(_ x: Limbs) -> Limbs { + return (x.times(self.Rinv)).divMod(self.modulus).remainder + } + + func moduloR(_ x: inout Limbs) { + if x.count > self.Rsize { + x.removeLast(x.count - self.Rsize) + } + x.normalize() + } + + func divideR(_ x: inout Limbs) { + if x.count > self.Rsize { + x.removeFirst(self.Rsize) + } else { + x = [0] + } + } + + func reduce(_ t: inout Limbs) { + var u = t + moduloR(&u) + u.multiply(self.mprime) + moduloR(&u) + u.multiply(self.modulus) + u.add(t) + divideR(&u) + if !u.lessThan(self.modulus) { + u.difference(self.modulus) + } + t = u + } + + // Handbook of Applied Cryptography - sliding window algorithm 14.85 + func expMod(_ x: BInt) -> BInt { + let bw = x.bitWidth + let k = bw < 200 ? 4 : (bw < 700 ? 5 : 6) + var g = Array(repeating: Limbs(repeating: 0, count: 0), count: 1 << k) + g[0] = toMspace(self.base) + var g2 = g[0] + g2.square() + reduce(&g2) + for i in 1 ..< g.count { + g[i] = g[i - 1].times(g2) + reduce(&g[i]) + } + var result = toMspace([1]) + var i = x.bitWidth - 1 + while i >= 0 { + if x.testBit(i) { + var l = max(0, i - k + 1) + while !x.testBit(l) { + l += 1 + } + for _ in 0 ..< i - l + 1 { + result.square() + reduce(&result) + } + var ndx = 0 + for j in (l ... i).reversed() { + ndx <<= 1 + if x.testBit(j) { + ndx += 1 + } + } + ndx >>= 1 + result.multiply(g[ndx]) + reduce(&result) + i = l - 1 + } else { + result.square() + reduce(&result) + i -= 1 + } + } + return BInt(fromMspace(result)) + } + } + + /* + * Helper class for (a ** x) mod m computation for m a power of 2 + */ + class Pow2Modulus { + + var limbCount = 0 + var mask: Limb = 0 + var base: Limbs + + init(_ a: BInt, _ modulus: BInt) { + let trailing = modulus.trailingZeroBitCount + self.limbCount = trailing / Limb.bitWidth + if trailing % Limb.bitWidth != 0 { + self.limbCount += 1 + } + for _ in 0 ..< trailing % Limb.bitWidth { + self.mask = (self.mask << 1) + 1 + } + self.base = a.magnitude + reduce(&self.base) + } + + func reduce(_ t: inout Limbs) { + let k = t.count - self.limbCount + if k > 0 { + t.removeLast(k) + } + if t.count == self.limbCount { + t[t.count - 1] &= self.mask + } + t.normalize() + } + + func expMod(_ x: BInt) -> BInt { + var exponent = x.magnitude + var result: Limbs = [1] + while true { + if exponent[0] & 1 == 1 { + result.multiply(self.base) + reduce(&result) + } + exponent.shift1Right() + if exponent.equalTo(0) { + break + } + self.base.square() + reduce(&self.base) + } + return BInt(result) + } + + } + + /* + * Helper class for (a ** x) mod m computation for small exponents + */ + class BarrettModulus { + + var base: Limbs + let modulus: Limbs + let u: Limbs + let k1: Int + let km1: Int + + init(_ a: BInt, _ modulus: BInt) { + self.modulus = modulus.magnitude + self.k1 = self.modulus.count + 1 + self.km1 = self.modulus.count - 1 + self.base = a.magnitude + var uu = Limbs(repeating: 0, count: 2 * self.modulus.count + 1) + uu[2 * self.modulus.count] = 1 + self.u = uu.divMod(self.modulus).quotient + self.base = self.base.divMod(self.modulus).remainder + } + + func moduloK1(_ x: inout Limbs) { + if x.count > self.k1 { + x.removeLast(x.count - self.k1) + } + x.normalize() + } + + func divideK(_ k: Int, _ x: inout Limbs) { + if x.count > k { + x.removeFirst(k) + } else { + x = [0] + } + } + + // Handbook of Applied Cryptography - Barrett reduction algorithm 14.42 + func reduce(_ t: inout Limbs) { + if self.modulus.compare(t) > 0 { + return + } + var q = t + divideK(self.km1, &q) + q.multiply(self.u) + divideK(self.k1, &q) + moduloK1(&t) + q.multiply(self.modulus) + moduloK1(&q) + if t.compare(q) < 0 { + var bk = Limbs(repeating: 0, count: self.k1) + bk[self.k1 - 1] = 1 + t.add(bk) + } + t.difference(q) + while t.compare(self.modulus) >= 0 { + t.difference(self.modulus) + } + } + + func expMod(_ x: BInt) -> BInt { + var exponent = x.magnitude + var result: Limbs = [1] + while true { + if exponent[0] & 1 == 1 { + result.multiply(self.base) + reduce(&result) + } + exponent.shift1Right() + if exponent.equalTo(0) { + break + } + self.base.square() + reduce(&self.base) + } + return BInt(result) + } + + } + +} diff --git a/Sources/BigInt/Karatsuba.swift b/Sources/BigInt/Karatsuba.swift new file mode 100755 index 0000000..47a2aec --- /dev/null +++ b/Sources/BigInt/Karatsuba.swift @@ -0,0 +1,76 @@ +// +// Karatsuba.swift +// BigInt +// +// Created by Leif Ibsen on 14/02/2019. +// Copyright © 2019 Leif Ibsen. All rights reserved. +// + +/* + * Karatsuba multiplication modelled after the Karatsuba algorithm in Java BigInteger + */ +extension Array where Element == Limb { + + func getUpper(_ k: Int) -> Limbs { + var w: Limbs + if self.count <= k { + w = [0] + } else { + w = Limbs(repeating: 0, count: self.count - k) + for i in 0 ..< w.count { + w[i] = self[k + i] + } + } + return w + } + + func getLower(_ k: Int) -> Limbs { + var w = Limbs(repeating: 0, count: k < self.count ? k : self.count) + for i in 0 ..< w.count { + w[i] = self[i] + } + return w + } + + func karatsubaTimes(_ x: Limbs) -> Limbs { + let k = (Swift.max(self.count, x.count) + 1) >> 1 + var x0 = x.getLower(k) + let x1 = x.getUpper(k) + var s0 = self.getLower(k) + let s1 = self.getUpper(k) + var w0 = x0 + w0.multiply(s0) + var w1 = x1 + w1.multiply(s1) + let xcmp = x0.difference(x1) + let scmp = s0.difference(s1) + x0.multiply(s0) + w0.add(w0, k) + w0.add(w1, k << 1) + w0.add(w1, k) + if xcmp * scmp < 0 { + w0.add(x0, k) + } else { + w0.subtract(x0, k) + } + return w0 + } + + func karatsubaSquared() -> Limbs { + let k = (self.count + 1) >> 1 + var s0 = self.getLower(k) + let s1 = self.getUpper(k) + var w0 = s0 + w0.square() + var w1 = s1 + w1.square() + s0.difference(s1) + s0.square() + w0.add(w0, k) + w0.add(w1, k << 1) + w0.add(w1, k) + w0.subtract(s0, k) + return w0 + } + +} diff --git a/Sources/BigInt/Limbs.swift b/Sources/BigInt/Limbs.swift new file mode 100755 index 0000000..2ac9847 --- /dev/null +++ b/Sources/BigInt/Limbs.swift @@ -0,0 +1,709 @@ +// +// Limbs.swift +// BigInt +// +// Created by Leif Ibsen on 24/12/2018. +// Copyright © 2018 Leif Ibsen. All rights reserved. +// + +extension Array where Element == Limb { + + static let UMasks: Limbs = [ + 0x1,0x2,0x4,0x8, + 0x10,0x20,0x40,0x80, + 0x100,0x200,0x400,0x800, + 0x1000,0x2000,0x4000,0x8000, + 0x10000,0x20000,0x40000,0x80000, + 0x100000,0x200000,0x400000,0x800000, + 0x1000000,0x2000000,0x4000000,0x8000000, + 0x10000000,0x20000000,0x40000000,0x80000000, + 0x100000000,0x200000000,0x400000000,0x800000000, + 0x1000000000,0x2000000000,0x4000000000,0x8000000000, + 0x10000000000,0x20000000000,0x40000000000,0x80000000000, + 0x100000000000,0x200000000000,0x400000000000,0x800000000000, + 0x1000000000000,0x2000000000000,0x4000000000000,0x8000000000000, + 0x10000000000000,0x20000000000000,0x40000000000000,0x80000000000000, + 0x100000000000000,0x200000000000000,0x400000000000000,0x800000000000000, + 0x1000000000000000,0x2000000000000000,0x4000000000000000,0x8000000000000000] + + mutating func normalize() { + let sc = self.count + if sc == 0 { + self.append(0) + } else if sc > 1 { + var i = sc - 1 + while self[i] == 0 && i > 0 { + i -= 1 + } + self.removeSubrange(i + 1 ..< sc) + } + } + + mutating func ensureSize(_ size: Int) { + self.reserveCapacity(size) + while self.count < size { + self.append(0) + } + } + + func trailingZeroBitCount() -> Int { + if self == [0] { + return 0 + } + var i = 0 + while self[i] == 0 { + i += 1 + } + return i * 64 + self[i].trailingZeroBitCount + } + + /* + * Bit operations + */ + + var bitWidth: Int { + var lastBits = 0 + var last = self.last! + while last != 0 { + last >>= 1 + lastBits += 1 + } + return (self.count - 1) * 64 + lastBits + } + + func getBitAt(_ i: Int) -> Bool { + let limbIndex = i >> 6 + if limbIndex >= self.count { + return false + } + return (self[limbIndex] & Limbs.UMasks[i & 0x3f]) != 0 + } + + mutating func setBitAt(_ i: Int, to bit: Bool) { + let limbIndex = i >> 6 + if limbIndex >= self.count && !bit { + return + } + self.ensureSize(limbIndex + 1) + if bit { + self[limbIndex] |= Limbs.UMasks[i & 0x3f] + } else { + self[limbIndex] &= ~Limbs.UMasks[i & 0x3f] + } + } + + /* + * Comparing + */ + + // return -1 if self < x, +1 if self > x, and 0 if self = x + func compare(_ x: Limbs) -> Int { + let scount = self.count + let xcount = x.count + if scount < xcount { + return -1 + } else if scount > xcount { + return 1 + } + var i = xcount - 1 + while i >= 0 { + if self[i] < x[i] { + return -1 + } else if self[i] > x[i] { + return 1 + } + i -= 1 + } + return 0 + } + + // return -1 if self < x, +1 if self > x, and 0 if self = x + func compare(_ x: Limb) -> Int { + if self.count > 1 { + return 1 + } + return self[0] == x ? 0 : (self[0] < x ? -1 : 1) + } + + // return self < x + func lessThan(_ x: Limbs) -> Bool { + return self.compare(x) < 0 + } + + // return self > x + func greaterThan(_ x: Limbs) -> Bool { + return self.compare(x) > 0 + } + + // return self = x + func equalTo(_ x: Limb) -> Bool { + return self.count == 1 && self[0] == x + } + + /* + * Shifting + */ + + // self = self << 1 + mutating func shift1Left() { + var b = self[0] & 0x8000000000000000 != 0 + self[0] <<= 1 + for i in 1 ..< self.count { + let b1 = self[i] & 0x8000000000000000 != 0 + self[i] <<= 1 + if b { + self[i] |= 1 + } + b = b1 + } + if b { + self.append(1) + } + } + + // self = self << shifts + mutating func shiftLeft(_ shifts: Int) { + let limbShifts = shifts >> 6 + let bitShifts = shifts & 0x3f + if bitShifts > 0 { + let b = self[self.count - 1] >> (64 - bitShifts) + for i in (0 ..< self.count).reversed() { + self[i] <<= bitShifts + if i > 0 { + self[i] |= self[i - 1] >> (64 - bitShifts) + } + } + self.append(b) + } + for _ in 0 ..< limbShifts { + self.insert(0, at: 0) + } + self.normalize() + } + + // return self << 1 + func shifted1Left() -> Limbs { + var res = self + res.shift1Left() + return res + } + + // return self << shifts + func shiftedLeft(_ shifts: Int) -> Limbs { + var res = self + res.shiftLeft(shifts) + return res + } + + // self = self >> 1 + mutating func shift1Right() { + for i in 0 ..< self.count { + if i > 0 && self[i] & 1 == 1 { + self[i - 1] |= 0x8000000000000000 + } + self[i] >>= 1 + } + self.normalize() + } + + // self = self >> shifts + mutating func shiftRight(_ shifts: Int) { + let limbShifts = Swift.min(shifts >> 6, self.count) + self.removeFirst(limbShifts) + let bitShifts = shifts & 0x3f + if bitShifts > 0 { + for i in 0 ..< self.count { + if i > 0 { + self[i - 1] |= self[i] << (64 - bitShifts) + } + self[i] >>= bitShifts + } + } + self.normalize() + } + + // return self >> 1 + func shifted1Right() -> Limbs { + var res = self + res.shift1Right() + return res + } + + // return self >> shifts + func shiftedRight(_ shifts: Int) -> Limbs { + var res = self + res.shiftRight(shifts) + return res + } + + /* + * Addition + */ + + // self[offset ..< self.count] = self[offset ..< self.count] + x + mutating func add(_ x: Limbs, _ offset: Int = 0, _ uselastcarry: Bool = true) { + if x.equalTo(0) { + return + } + self.ensureSize(x.count + offset) + var carry = false + for i in 0 ..< x.count { + let io = i + offset + if carry { + self[io] = self[io] &+ 1 + if self[io] == 0 { + self[io] = x[i] + // carry still lives + } else { + (self[io], carry) = self[io].addingReportingOverflow(x[i]) + } + } else { + (self[io], carry) = self[io].addingReportingOverflow(x[i]) + } + } + var i = x.count + offset + while carry && i < self.count { + self[i] = self[i] &+ 1 + carry = self[i] == 0 + i += 1 + } + if carry && uselastcarry { + self.append(1) + } + } + + // self = self + [x] + mutating func add(_ x: Limb) { + var carry: Bool + (self[0], carry) = self[0].addingReportingOverflow(x) + var i = 1 + while carry && i < self.count { + self[i] = self[i] &+ 1 + carry = self[i] == 0 + i += 1 + } + if carry { + self.append(1) + } + } + + /* + * Subtraction + */ + + // self[offset ..< self.count] = self[offset ..< self.count] - x, return borrow + mutating func subtract(_ x: Limbs, _ offset: Int) -> Bool { + self.ensureSize(x.count + offset) + var borrow = false + for i in 0 ..< x.count { + let io = i + offset + if borrow { + if self[io] == 0 { + self[io] = 0xffffffffffffffff - x[i] + // borrow still lives + } else { + self[io] -= 1 + (self[io], borrow) = self[io].subtractingReportingOverflow(x[i]) + } + } else { + (self[io], borrow) = self[io].subtractingReportingOverflow(x[i]) + } + } + var i = x.count + offset + while borrow && i < self.count { + self[i] = self[i] &- 1 + borrow = self[i] == 0xffffffffffffffff + i += 1 + } + return borrow + } + + // self = abs(self - x), return self.compare(x) + mutating func difference(_ x: Limbs) -> Int { + var xx = x + let cmp = self.compare(xx) + if cmp < 0 { + swap(&self, &xx) + } + var borrow = false + for i in 0 ..< xx.count { + if borrow { + if self[i] == 0 { + self[i] = 0xffffffffffffffff - xx[i] + // borrow still lives + } else { + self[i] -= 1 + (self[i], borrow) = self[i].subtractingReportingOverflow(xx[i]) + } + } else { + (self[i], borrow) = self[i].subtractingReportingOverflow(xx[i]) + } + } + var i = xx.count + while borrow && i < self.count { + self[i] = self[i] &- 1 + borrow = self[i] == 0xffffffffffffffff + i += 1 + } + self.normalize() + return cmp + } + + // self = abs(self - [x]), return self.compare([x]) + mutating func difference(_ x: Limb) -> Int { + var xx = [x] + let cmp = self.compare(xx) + if cmp < 0 { + swap(&self, &xx) + } + var borrow: Bool + (self[0], borrow) = self[0].subtractingReportingOverflow(xx[0]) + var i = 1 + while borrow && i < self.count { + self[i] = self[i] &- 1 + borrow = self[i] == 0xffffffffffffffff + i += 1 + } + self.normalize() + return cmp + } + + /* + * Multiplication + */ + + // Threshold for Karatsuba multplication + static let KA_THR = 100 + // Threshold for Toom Cook multplication + static let TC_THR = 200 + + // self = self * x + // Seminumerical Algorithms chapter 4.3.1 - algorithm M + mutating func multiply(_ x: Limbs) { + let m = self.count + var w: Limbs + if m > Limbs.KA_THR && x.count > Limbs.KA_THR { + if m > Limbs.TC_THR || x.count > Limbs.TC_THR { + w = self.toomCookTimes(x) + } else { + w = self.karatsubaTimes(x) + } + } else { + let n = x.count + w = Limbs(repeating: 0, count: m + n) + var carry: Limb + var ovfl1, ovfl2: Bool + for i in 0 ..< m { + carry = 0 + for j in 0 ..< n { + let (hi, lo) = self[i].multipliedFullWidth(by: x[j]) + (w[i + j], ovfl1) = w[i + j].addingReportingOverflow(lo) + (w[i + j], ovfl2) = w[i + j].addingReportingOverflow(carry) + carry = hi &+ ((ovfl1 ? 1 : 0) + (ovfl2 ? 1 : 0)) + } + w[i + n] = carry + } + } + w.normalize() + self = w + } + + // self = self * x + mutating func multiply(_ x: Limb) { + let m = self.count + var w = Limbs(repeating: 0, count: m + 1) + var carry: Limb + var ovfl1, ovfl2: Bool + for i in 0 ..< m { + carry = 0 + let (hi, lo) = self[i].multipliedFullWidth(by: x) + (w[i], ovfl1) = w[i].addingReportingOverflow(lo) + (w[i], ovfl2) = w[i].addingReportingOverflow(carry) + carry = hi &+ ((ovfl1 ? 1 : 0) + (ovfl2 ? 1 : 0)) + w[i + 1] = carry + } + w.normalize() + self = w + } + + func times(_ x: Limbs) -> Limbs { + var w = self + w.multiply(x) + return w + } + + // self = self * self + mutating func square() { + let m = self.count + var w: Limbs + if m > Limbs.TC_THR { + w = self.toomCookSquared() + } else if m > Limbs.KA_THR { + w = self.karatsubaSquared() + } else { + w = Limbs(repeating: 0, count: m << 1) + // Compute diagonal elements and divide by 2 + (w[1], w[0]) = self[0].multipliedFullWidth(by: self[0]) + w[0] >>= 1 + if w[1] & 1 == 1 { + w[0] |= 0x8000000000000000 + } + w[1] >>= 1 + var j = 2 + for i in 1 ..< m { + (w[j + 1], w[j]) = self[i].multipliedFullWidth(by: self[i]) + if w[j] & 1 == 1 { + w[j - 1] |= 0x8000000000000000 + } + w[j] >>= 1 + if w[j + 1] & 1 == 1 { + w[j] |= 0x8000000000000000 + } + w[j + 1] >>= 1 + j += 2 + } + // Add off-diagonal products + var carry: Limb + var ovfl1, ovfl2: Bool + for i in 0 ..< m { + carry = 0 + for j in i + 1 ..< m { + let (hi, lo) = self[i].multipliedFullWidth(by: self[j]) + (w[i + j], ovfl1) = w[i + j].addingReportingOverflow(lo) + (w[i + j], ovfl2) = w[i + j].addingReportingOverflow(carry) + carry = hi &+ ((ovfl1 ? 1 : 0) + (ovfl2 ? 1 : 0)) + } + (w[i + m], ovfl1) = w[i + m].addingReportingOverflow(carry) + if ovfl1 { + w[i + m + 1] += 1 + } + } + // Multiply by 2 and adjust last bit + w.shift1Left() + w[0] |= (self[0] & 1) + } + w.normalize() + self = w + } + + func squared() -> Limbs { + var w = self + w.square() + return w + } + + /* + * Division and modulus + */ + + // Hacker's Delight - algorithm 9.2 + // (hi || lo) / d => (q, r) + static func div128(_ hi: Limb, _ lo: Limb, _ d: Limb) -> (q: Limb, r: Limb) { + precondition(d > 0, "Division by zero") + var hi = hi + var lo = lo + for _ in 0 ..< 64 { + let t: Limb = hi & 0x8000000000000000 == 0 ? 0 : 0xffffffffffffffff + hi = (hi << 1) | (lo >> 63) + lo <<= 1 + if (hi | t) >= d { + hi = hi &- d + lo = lo &+ 1 + } + } + return (lo, hi) + } + + // Seminumerical Algorithms chapter 4.3.1 - exercise 16 + func divMod1(_ v: Limb) -> (quotient: Limbs, remainder: Limb) { + if self.equalTo(0) { + return ([0], 0) + } + var w = Limbs(repeating: 0, count: self.count) + var r = Limb(0) + for j in (0 ..< self.count).reversed() { + (w[j], r) = Limbs.div128(r, self[j], v) + } + w.normalize() + return (w, r) + } + + func divMod(_ v: Limbs) -> (quotient: Limbs, remainder: Limbs) { + var quotient = Limbs() + var remainder = Limbs() + self.divMod(v, "ient, &remainder) + return (quotient, remainder) + } + + // Seminumerical Algorithms chapter 4.3.1 - algorithm D + func divMod(_ v: Limbs, _ quotient: inout Limbs, _ remainder: inout Limbs) { + if self.lessThan(v) { + quotient = [0] + remainder = self + } else if v.count == 1 { + let (q, r) = self.divMod1(v[0]) + quotient = q + remainder = [r] + } else { + remainder = self + var v = v + let n = v.count + let m = remainder.count + let d = v[n - 1].leadingZeroBitCount + v.shiftLeft(d) + remainder.shiftLeft(d) + remainder.append(0) + var qhat = Limb(0) + var rhat = Limb(0) + var k = m - n + quotient = Limbs(repeating: 0, count: k + 1) + var ovfl: Bool + repeat { + if v[n - 1] == remainder[k + n] { + qhat = 0xffffffffffffffff + (rhat, ovfl) = remainder[k + n].addingReportingOverflow(remainder[k + n - 1]) + } else { + (qhat, rhat) = Limbs.div128(remainder[k + n], remainder[k + n - 1], v[n - 1]) + ovfl = false + } + while !ovfl { + let (hi, lo) = qhat.multipliedFullWidth(by: v[n - 2]) + if hi < rhat || (hi == rhat && lo <= remainder[k + n - 2]) { + break + } + qhat -= 1 + (rhat, ovfl) = rhat.addingReportingOverflow(v[n - 1]) + } + if qhat != 0 { + let borrow = remainder.subtract(v.times([qhat]), k) + if borrow { + qhat -= 1 + remainder.add(v, k, false) + } + } + quotient[k] = qhat + k -= 1 + } while k >= 0 + remainder.shiftRight(d) + quotient.normalize() + } + } + + /* + * gcd algorithms modelled after the gcd algorithms in Java BigInteger + */ + func gcd(_ x: Limbs) -> Limbs { + if self == [0] { + return x + } + var u = self + var v = x + while v != [0] { + if v.count == u.count { + return u.binaryGcd(v) + } + let r = u.divMod(v).remainder + u = v + v = r + } + return u + } + + func binaryGcd(_ x: Limbs) -> Limbs { + if self == [0] { + return x + } + var u = self + var v = x + let k = Swift.min(u.trailingZeroBitCount(), v.trailingZeroBitCount()) + u.shiftRight(k) + v.shiftRight(k) + var t: Limbs + var tSign: Bool + if u[0] & 1 == 1 { + t = v + tSign = true + } else { + t = u + tSign = false + } + while t != [0] { + t.shiftRight(t.trailingZeroBitCount()) + if tSign { + v = t + } else { + u = t + } + if u.count == 1 && v.count == 1 { + return Limbs.binaryGcd(u[0], v[0]).shiftedLeft(k) + } + let cmp = u.compare(v) + if cmp < 0 { + v.difference(u) + t = v + tSign = true + } else if cmp > 0 { + u.difference(v) + t = u + tSign = false + } else { + break + } + } + return u.shiftedLeft(k) + } + + /* + * gcd of single Limb values + */ + static func binaryGcd(_ a: Limb, _ b: Limb) -> Limbs { + if b == 0 { + return [a] + } + if a == 0 { + return [b] + } + var a = a + var b = b + let atz = a.trailingZeroBitCount + let btz = b.trailingZeroBitCount + a >>= atz + b >>= btz + let t = Swift.min(atz, btz) + while a != b { + if a > b { + a -= b + a >>= a.trailingZeroBitCount + } else { + b -= a + b >>= b.trailingZeroBitCount + } + } + return [a << t] + } + + /* + * Exponentiation + */ + + func raisedTo(_ x: Int) -> Limbs { + if x == 0 { + return [1] + } + if x == 1 { + return self + } + var base = self + var exponent = x + var y: Limbs = [1] + while exponent > 1 { + if exponent & 1 != 0 { + y.multiply(base) + } + base.square() + exponent >>= 1 + } + return base.times(y) + } +} + diff --git a/Sources/BigInt/ToomCook.swift b/Sources/BigInt/ToomCook.swift new file mode 100755 index 0000000..0fb9521 --- /dev/null +++ b/Sources/BigInt/ToomCook.swift @@ -0,0 +1,119 @@ +// +// ToomCook.swift +// BigInt +// +// Created by Leif Ibsen on 07/02/2019. +// Copyright © 2019 Leif Ibsen. All rights reserved. +// + +/* + * ToomCook multiplication modelled after the ToomCook algorithm in Java BigInteger + */ +extension Array where Element == Limb { + + static func divideBy3(_ d: BInt) -> BInt { + var quotient = Limbs(repeating: 0, count: d.magnitude.count) + var remainder = Limb(0) + for i in 0 ..< quotient.count { + let w = remainder > d.magnitude[i] ? remainder - d.magnitude[i] : d.magnitude[i] - remainder + let x = w &* 0xaaaaaaaaaaaaaaab + quotient[i] = x + remainder = x < 0x5555555555555556 ? 0 : (x < 0xaaaaaaaaaaaaaaab ? 1 : 2) + } + return BInt(quotient, d.isNegative) + } + + func getSlice(_ n: Int, _ k: Int) -> BInt { + var w: Limbs + if n * k >= self.count { + w = [] + } else if (n + 1) * k < self.count { + w = Limbs(repeating: 0, count: k) + } else { + w = Limbs(repeating: 0, count: self.count - n * k) + } + for i in 0 ..< w.count { + w[i] = self[n * k + i] + } + return BInt(w) + } + + func toomCookTimes(_ x: Limbs) -> Limbs { + let k = (Swift.max(self.count, x.count) + 2) / 3 + let x0 = x.getSlice(0, k) + let x1 = x.getSlice(1, k) + let x2 = x.getSlice(2, k) + let s0 = self.getSlice(0, k) + let s1 = self.getSlice(1, k) + let s2 = self.getSlice(2, k) + + let pp = x0 + x2 + let p1 = pp + x1 + let pm1 = pp - x1 + let pm2 = ((pm1 + x2) << 1) - x0 + let qq = s0 + s2 + let q1 = qq + s1 + let qm1 = qq - s1 + let qm2 = ((qm1 + s2) << 1) - s0 + + var r0 = x0 * s0 + let r1 = p1 * q1 + let rm1 = pm1 * qm1 + let rm2 = pm2 * qm2 + let rinf = x2 * s2 + var rr3 = Limbs.divideBy3(rm2 - r1) + var rr1 = (r1 - rm1) >> 1 + var rr2 = rm1 - r0 + + rr3 = (rr2 - rr3) >> 1 + (rinf << 1) + rr2 += rr1 + rr2 -= rinf + rr1 -= rr3 + + var offset = k << 2 + r0.magnitude.add(rinf.magnitude, offset) + offset -= k + r0.magnitude.add(rr3.magnitude, offset) + offset -= k + r0.magnitude.add(rr2.magnitude, offset) + r0.magnitude.add(rr1.magnitude, k) + return r0.magnitude + } + + func toomCookSquared() -> Limbs { + let k = (self.count + 2) / 3 + let s0 = self.getSlice(0, k) + let s1 = self.getSlice(1, k) + let s2 = self.getSlice(2, k) + + let qq = s0 + s2 + let q1 = qq + s1 + let qm1 = qq - s1 + let qm2 = ((qm1 + s2) << 1) - s0 + + var r0 = s0 ** 2 + let r1 = q1 ** 2 + let rm1 = qm1 ** 2 + let rm2 = qm2 ** 2 + let rinf = s2 ** 2 + + var rr3 = Limbs.divideBy3(rm2 - r1) + var rr1 = (r1 - rm1) >> 1 + var rr2 = rm1 - r0 + + rr3 = (rr2 - rr3) >> 1 + (rinf << 1) + rr2 += rr1 + rr2 -= rinf + rr1 -= rr3 + + var offset = k << 2 + r0.magnitude.add(rinf.magnitude, offset) + offset -= k + r0.magnitude.add(rr3.magnitude, offset) + offset -= k + r0.magnitude.add(rr2.magnitude, offset) + r0.magnitude.add(rr1.magnitude, k) + return r0.magnitude + } + +} diff --git a/Tests/BigIntTests/AdditionTest.swift b/Tests/BigIntTests/AdditionTest.swift new file mode 100755 index 0000000..0a0558c --- /dev/null +++ b/Tests/BigIntTests/AdditionTest.swift @@ -0,0 +1,80 @@ +// +// AdditionTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 17/01/2019. +// + +import XCTest + +class AdditionTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + XCTAssertEqual(BInt(7) + BInt(4), BInt(11)) + XCTAssertEqual(BInt(7) + BInt(-4), BInt(3)) + XCTAssertEqual(BInt(-7) + BInt(4), BInt(-3)) + XCTAssertEqual(BInt(-7) + BInt(-4), BInt(-11)) + XCTAssertEqual(BInt(-7) + BInt(0), BInt(-7)) + XCTAssertEqual(BInt(7) + BInt(0), BInt(7)) + XCTAssertEqual(BInt(7) + 4, BInt(11)) + XCTAssertEqual(BInt(7) + (-4), BInt(3)) + XCTAssertEqual(BInt(-7) + 4, BInt(-3)) + XCTAssertEqual(BInt(-7) + (-4), BInt(-11)) + XCTAssertEqual(BInt(-7) + 0, BInt(-7)) + XCTAssertEqual(BInt(7) + 0, BInt(7)) + XCTAssertEqual(7 + BInt(4), BInt(11)) + XCTAssertEqual(7 + BInt(-4), BInt(3)) + XCTAssertEqual((-7) + BInt(4), BInt(-3)) + XCTAssertEqual((-7) + BInt(-4), BInt(-11)) + XCTAssertEqual((-7) + BInt(0), BInt(-7)) + XCTAssertEqual(7 + BInt(0), BInt(7)) + } + + func test2() { + var x1 = BInt(7) + x1 += BInt(4) + XCTAssertEqual(x1, BInt(11)) + var x2 = BInt(7) + x2 += BInt(-4) + XCTAssertEqual(x2, BInt(3)) + var x3 = BInt(-7) + x3 += BInt(4) + XCTAssertEqual(x3, BInt(-3)) + var x4 = BInt(-7) + x4 += BInt(-4) + XCTAssertEqual(x4, BInt(-11)) + } + + func test3() { + for i in 0 ..< 1000 { + let a = i % 2 == 0 ? BInt(bitWidth: 1000) : -BInt(bitWidth: 1000) + let b = BInt(bitWidth: 800) + let n = BInt(bitWidth: 500) + let ab = a + b + let an = a * n + XCTAssertEqual(ab - a, b) + XCTAssertEqual(ab - b, a) + XCTAssertEqual(an - a, a * (n - 1)) + } + } + + func test4() { + for i in 0 ..< 1000 { + let a = i % 2 == 0 ? BInt(bitWidth: 1000) : -BInt(bitWidth: 1000) + var b = a + for _ in 0 ..< 100 { + b += a + } + XCTAssertEqual(b, a * 101) + } + } + +} diff --git a/Tests/BigIntTests/BitTest.swift b/Tests/BigIntTests/BitTest.swift new file mode 100755 index 0000000..0593eda --- /dev/null +++ b/Tests/BigIntTests/BitTest.swift @@ -0,0 +1,84 @@ +// +// BitTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 31/12/2018. +// + +import XCTest + +class BitTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func doTest(_ x1: BInt) { + var x2 = x1 + x2.negate() + XCTAssertEqual(x1, -x2) + } + + func test1() { + doTest(BInt(0)) + doTest(BInt(1)) + doTest(BInt(-1)) + doTest(BInt(bitWidth: 200)) + } + + func test2() { + let a = BInt(bitWidth: 300) + let b = BInt(bitWidth: 300) + let c = BInt(bitWidth: 300) + XCTAssertEqual(b ^ b ^ b, b) + XCTAssertEqual(a & (b | c), (a & b) | (a & c)) + } + + func test3() { + let a = BInt(bitWidth: 300) + var b = a + for i in 0 ..< 300 { + b.flipBit(i) + } + for i in 0 ..< 300 { + XCTAssertEqual(a.testBit(i), !b.testBit(i)) + } + } + + func test4() { + let a = BInt(bitWidth: 300) + XCTAssertEqual(a | BInt.ZERO, a) + XCTAssertEqual(a & BInt.ZERO, BInt.ZERO) + XCTAssertEqual(a & ~BInt.ZERO, a) + XCTAssertEqual(a ^ BInt.ZERO, a) + XCTAssertEqual(a ^ ~BInt.ZERO, ~a) + } + + func test5() { + let b3 = BInt(3) + let bm3 = BInt(-3) + let b7 = BInt(7) + let bm7 = BInt(-7) + XCTAssertEqual(b3 & b7, BInt(3)) + XCTAssertEqual(b3 & bm7, BInt(1)) + XCTAssertEqual(bm3 & b7, BInt(5)) + XCTAssertEqual(bm3 & bm7, BInt(-7)) + XCTAssertEqual(b3 | b7, BInt(7)) + XCTAssertEqual(b3 | bm7, BInt(-5)) + XCTAssertEqual(bm3 | b7, BInt(-1)) + XCTAssertEqual(bm3 | bm7, BInt(-3)) + XCTAssertEqual(b3 ^ b7, BInt(4)) + XCTAssertEqual(b3 ^ bm7, BInt(-6)) + XCTAssertEqual(bm3 ^ b7, BInt(-6)) + XCTAssertEqual(bm3 ^ bm7, BInt(4)) + XCTAssertEqual(~b3, BInt(-4)) + XCTAssertEqual(~bm3, BInt(2)) + XCTAssertEqual(~b7, BInt(-8)) + XCTAssertEqual(~bm7, BInt(6)) + } + +} diff --git a/Tests/BigIntTests/ConstructorTest.swift b/Tests/BigIntTests/ConstructorTest.swift new file mode 100755 index 0000000..568cff4 --- /dev/null +++ b/Tests/BigIntTests/ConstructorTest.swift @@ -0,0 +1,117 @@ +// +// ConstructorTest.swift +// XBigIntegerTests +// +// Created by Leif Ibsen on 11/12/2018. +// + +import XCTest + +class ConstructorTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + let x1 = BInt("123", radix: 10)! + XCTAssertEqual(x1.magnitude.count, 1) + XCTAssertFalse((x1.isNegative)) + let x2 = BInt("-0", radix: 10)! + XCTAssertEqual(x2.magnitude.count, 1) + XCTAssertFalse((x2.isNegative)) + let b: Bytes = [0, 0, 0, 123] + let x3 = BInt(signed: b) + XCTAssertEqual(x1, x3) + let x4 = BInt(magnitude: b) + XCTAssertEqual(x1, x4) + let x5 = BInt(bitWidth: 100) + XCTAssertTrue(x5.bitWidth <= 100) + let x6 = BInt("-123")! + XCTAssertEqual(x6.magnitude[0], 123) + let x71 = BInt(bitWidth: 100) + let x72 = BInt(signed: x71.asSignedBytes()) + XCTAssertEqual(x71, x72) + let x8 = BInt.ONE << 317 + XCTAssertEqual(x8.trailingZeroBitCount, 317) + let x9 = BInt(1) + XCTAssertFalse(x9.isEven) + XCTAssertTrue(x9.isOdd) + XCTAssertTrue(x9.isPositive) + XCTAssertFalse(x9.isNegative) + XCTAssertFalse(x9.isZero) + XCTAssertTrue(x9.isNotZero) + let x10 = BInt(-1) + XCTAssertFalse(x10.isEven) + XCTAssertTrue(x10.isOdd) + XCTAssertFalse(x10.isPositive) + XCTAssertTrue(x10.isNegative) + XCTAssertFalse(x10.isZero) + XCTAssertTrue(x10.isNotZero) + let x11 = BInt(0) + XCTAssertTrue(x11.isEven) + XCTAssertFalse(x11.isOdd) + XCTAssertFalse(x11.isPositive) + XCTAssertFalse(x11.isNegative) + XCTAssertTrue(x11.isZero) + XCTAssertFalse(x11.isNotZero) + let x12 = BInt(bitWidth: 1) + XCTAssertTrue(x12 == BInt.ONE || x12 == BInt.ZERO) + let x13 = BInt("12345670", radix: 8) + XCTAssertEqual(x13, BInt("2739128")) + let x14 = BInt("12345678", radix: 8) + XCTAssertEqual(x14, nil) + } + + func test2() { + let x0 = BInt(0) + let x1 = BInt(1) + let xm1 = BInt(-1) + XCTAssertEqual(x0.magnitude.count, 1) + XCTAssertEqual(x1.magnitude.count, 1) + XCTAssertEqual(xm1.magnitude.count, 1) + XCTAssertFalse(x0.isNegative) + XCTAssertFalse(x1.isNegative) + XCTAssertTrue(xm1.isNegative) + XCTAssertTrue(x0.isZero) + XCTAssertTrue(x1.isPositive) + XCTAssertTrue(xm1.isNegative) + } + + func test3() { + let x0 = BInt([0x8000000000000000], true) + XCTAssertEqual(x0.asInt(), Int.min) + let x1 = BInt(signed: [0]) + XCTAssertEqual(x1, BInt.ZERO) + let x2 = BInt(signed: [0, 255]) + XCTAssertEqual(x2, BInt(255)) + let x3 = BInt(signed: [255]) + XCTAssertEqual(x3, BInt(-1)) + let x4 = BInt(signed: [255, 0]) + XCTAssertEqual(x4, BInt(-256)) + let x5 = BInt(signed: [255, 255]) + XCTAssertEqual(x5, BInt(-1)) + } + + func test4() { + let x1 = BInt(magnitude: [0]) + XCTAssertEqual(x1, BInt.ZERO) + let x2 = BInt(magnitude: [0, 255]) + XCTAssertEqual(x2, BInt(255)) + let x3 = BInt(magnitude: [255]) + XCTAssertEqual(x3, BInt(255)) + let x4 = BInt(magnitude: [255, 0]) + XCTAssertEqual(x4, BInt(65280)) + let x5 = BInt(magnitude: [255, 255]) + XCTAssertEqual(x5, BInt(65535)) + let x6 = BInt(magnitude: [0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff]) + XCTAssertEqual(x6, BInt("fffffffffffffffffffffffffffffffeffffffffffffffff", radix: 16)!) + let x7 = BInt(signed: [0x00, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff]) + XCTAssertEqual(x7, BInt("fffffffffffffffffffffffffffffffeffffffffffffffff", radix: 16)!) + } + +} diff --git a/Tests/BigIntTests/DivModTest.swift b/Tests/BigIntTests/DivModTest.swift new file mode 100755 index 0000000..b92a738 --- /dev/null +++ b/Tests/BigIntTests/DivModTest.swift @@ -0,0 +1,74 @@ +// +// DivModTest.swift +// XBigIntegerTests +// +// Created by Leif Ibsen on 11/12/2018. +// + +import XCTest + +class DivModTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func doTest1(_ bw1: Int, _ bw2: Int) { + let x1 = BInt(bitWidth: bw1) + let x2 = BInt(bitWidth: bw2) + BInt.ONE + doTest2(x1, x2) + } + + func doTest2(_ x1: BInt, _ x2: BInt) { + let r1 = x1 % x2 + let q1 = x1 / x2 + XCTAssertEqual(x1, x2 * q1 + r1) + let (q2, r2) = x1.quotientAndRemainder(dividingBy: x2) + XCTAssertEqual(q1, q2) + XCTAssertEqual(r1, r2) + var q3 = BInt.ZERO + var r3 = BInt.ZERO + x1.quotientAndRemainder(dividingBy: x2, &q3, &r3) + XCTAssertEqual(q1, q3) + XCTAssertEqual(r1, r3) + } + + func test1() { + doTest1(30, 20) + doTest1(30, 120) + doTest1(130, 20) + doTest1(130, 120) + } + + func test2() { + XCTAssertEqual(BInt(7) % BInt(4), BInt(3)) + XCTAssertEqual(BInt(-7) % BInt(4), BInt(-3)) + XCTAssertEqual(BInt(7) % BInt(-4), BInt(3)) + XCTAssertEqual(BInt(-7) % BInt(-4), BInt(-3)) + XCTAssertEqual(BInt(7).mod(BInt(4)), BInt(3)) + XCTAssertEqual(BInt(-7).mod(BInt(4)), BInt(1)) + XCTAssertEqual(BInt(7).mod(BInt(-4)), BInt(3)) + XCTAssertEqual(BInt(-7).mod(BInt(-4)), BInt(1)) + doTest2(BInt(7), BInt(4)) + doTest2(BInt(-7), BInt(4)) + doTest2(BInt(7), BInt(-4)) + doTest2(BInt(-7), BInt(-4)) + doTest2(BInt(Limbs(repeating: UInt64.max, count: 50)), BInt(Limbs(repeating: UInt64.max, count: 35))) + } + + func test3() { + XCTAssertEqual(BInt(0) / BInt(7), BInt.ZERO) + XCTAssertEqual(-BInt(0) / BInt(7), BInt.ZERO) + XCTAssertEqual(BInt(0) / BInt(-7), BInt.ZERO) + XCTAssertEqual(-BInt(0) / BInt(-7), BInt.ZERO) + XCTAssertEqual(BInt(7) % BInt(7), BInt.ZERO) + XCTAssertEqual(BInt(-7) % BInt(7), BInt.ZERO) + XCTAssertEqual(BInt(7) % BInt(-7), BInt.ZERO) + XCTAssertEqual(BInt(-7) % BInt(-7), BInt.ZERO) + } + +} diff --git a/Tests/BigIntTests/ExpModTest.swift b/Tests/BigIntTests/ExpModTest.swift new file mode 100755 index 0000000..28409de --- /dev/null +++ b/Tests/BigIntTests/ExpModTest.swift @@ -0,0 +1,71 @@ +// +// ExpModTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 04/02/2019. +// + +import XCTest + +class ExpModTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + XCTAssertEqual(BInt(3).expMod(BInt(7), BInt(1)), BInt(0)) + XCTAssertEqual(BInt(2).expMod(BInt(10), BInt(1000)), BInt(24)) + } + + func test2() { + let a1 = BInt("43271512091896741409394736939900206368860872098952555939")! + let x1 = BInt("42267688843543983783885633938053687583065584419295560555")! + let m1 = BInt("65559338391610243479024015552681806795487318988418855174")! + let mp1 = BInt("14708028296754883426561209612579760556999800024666797837")! + XCTAssertEqual(a1.expMod(x1, m1), mp1) + XCTAssertEqual(a1.expMod(-x1, m1), mp1.modInverse(m1)) + XCTAssertEqual((-a1).expMod(x1, m1), m1 - mp1) + XCTAssertEqual((-a1).expMod(-x1, m1), (m1 - mp1).modInverse(m1)) + let a2 = BInt("9879959622001881798420480569351120749752891168795071469741009590796905186183625061410538508653929799901162907503196502223071902180994253404412067954774342232969326053454779870840130810532326017165678692636647404921424922403748460111140358572478743271512091896741409394736939900206368860872098952555939")! + let x2 = BInt("4149842346989426807721754542711895351513161856205879655378968612408032038996656528445703209766166328006052965811745650295935314936056815509075542554308409615827636639641110901357012471113482422183741588797481891328204616583065067700486989814417842267688843543983783885633938053687583065584419295560555")! + let m2 = BInt("10524302966485349118258764179820205386685991992586369700154893101599927732040662774460446149003080427232451962311367600902738242964142492968383265627950930467854069828393051189273332522792516344807937835132537814794042705435787095095919023768140765559338391610243479024015552681806795487318988418855174")! + let mp2 = BInt("1122200972247120546333043544356752277213829455746835691791369400547247327000588079094470008395425401827849174141816902264513461023891163414270218627387835660351885298195540293936217720560009323739879363833688454142584347044964970051959361123182603999640496505409063560777205949747764493609693376510835")! + XCTAssertEqual(a2.expMod(x2, m2), mp2) + XCTAssertEqual(a2.expMod(-x2, m2), mp2.modInverse(m2)) + XCTAssertEqual((-a2).expMod(x2, m2), m2 - mp2) + XCTAssertEqual((-a2).expMod(-x2, m2), (m2 - mp2).modInverse(m2)) + let a3 = BInt("2988348162058574136915891421498819466320163312926952423791023078876139")! + let x3 = BInt("2351399303373464486466122544523690094744975233415544072992656881240319")! + let m3 = BInt("10000000000000000000000000000000000000000")! + let mp3 = BInt("1527229998585248450016808958343740453059")! + XCTAssertEqual(a3.expMod(x3, m3), mp3) + XCTAssertEqual(a3.expMod(-x3, m3), mp3.modInverse(m3)) + XCTAssertEqual((-a3).expMod(x3, m3), m3 - mp3) + XCTAssertEqual((-a3).expMod(-x3, m3), (m3 - mp3).modInverse(m3)) + } + + func test3() { + XCTAssertEqual(BInt(-2).expMod(BInt(0), BInt(11)), BInt(1)) + XCTAssertEqual(BInt(2).expMod(BInt(0), BInt(11)), BInt(1)) + XCTAssertEqual(BInt(1).expMod(BInt(2), BInt(11)), BInt(1)) + XCTAssertEqual(BInt(1).expMod(BInt(-2), BInt(11)), BInt(1)) + XCTAssertEqual(BInt(-2).expMod(BInt(-3), BInt(11)), BInt(4)) + XCTAssertEqual(BInt(-11).expMod(BInt(2), BInt(11)), BInt(0)) + XCTAssertEqual(BInt(2).expMod(BInt(5), BInt(11)), BInt(10)) + XCTAssertEqual(BInt(2).expMod(BInt(-5), BInt(11)), BInt(10)) + XCTAssertEqual(BInt(2).expMod(BInt(4), BInt(11)), BInt(5)) + XCTAssertEqual(BInt(2).expMod(BInt(-4), BInt(11)), BInt(9)) + XCTAssertEqual(BInt(-2).expMod(BInt(5), BInt(11)), BInt(1)) + XCTAssertEqual(BInt(-2).expMod(BInt(-5), BInt(11)), BInt(1)) + XCTAssertEqual(BInt(-2).expMod(BInt(4), BInt(11)), BInt(5)) + XCTAssertEqual(BInt(-2).expMod(BInt(-4), BInt(11)), BInt(9)) + XCTAssertEqual(BInt(0).expMod(BInt(5), BInt(11)), BInt(0)) + XCTAssertEqual(BInt(0).expMod(BInt(4), BInt(11)), BInt(0)) + } + +} diff --git a/Tests/BigIntTests/GcdTest.swift b/Tests/BigIntTests/GcdTest.swift new file mode 100755 index 0000000..84f1237 --- /dev/null +++ b/Tests/BigIntTests/GcdTest.swift @@ -0,0 +1,61 @@ +// +// gcdTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 15/01/2019. +// + +import XCTest + +class GcdTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + XCTAssertEqual(BInt.ZERO.gcd(BInt.ZERO), BInt.ZERO) + XCTAssertEqual(BInt.ONE.gcd(BInt.ZERO), BInt.ONE) + XCTAssertEqual(BInt.ZERO.gcd(BInt.ONE), BInt.ONE) + let x = BInt(bitWidth: 100) + XCTAssertEqual(x.gcd(BInt.ONE), BInt.ONE) + XCTAssertEqual(x.gcd(BInt.ZERO), x) + XCTAssertEqual(x.gcd(x), x) + XCTAssertEqual(BInt(4).gcd(BInt(272)), BInt(4)) + } + + func test2() { + var i = 1 + for _ in 0 ..< 3 { + i *= 10 + let x = BInt(bitWidth: i) + let y = BInt(bitWidth: 2 * i) + let gcd = x.gcd(y) + let (qx, rx) = x.quotientAndRemainder(dividingBy: gcd) + let (qy, ry) = y.quotientAndRemainder(dividingBy: gcd) + XCTAssertEqual(rx, BInt.ZERO) + XCTAssertEqual(ry, BInt.ZERO) + XCTAssertEqual(qx.gcd(qy), BInt.ONE) + XCTAssertEqual(qy.gcd(qx), BInt.ONE) + } + } + + func test3() { + for _ in 0 ..< 1000 { + let x1 = BInt(bitWidth: 10) + XCTAssertEqual(x1.gcd(x1), x1) + XCTAssertEqual(x1.gcd(x1 + 1), BInt.ONE) + let x2 = BInt(bitWidth: 60) + XCTAssertEqual(x2.gcd(x2), x2) + XCTAssertEqual(x2.gcd(x2 + 1), BInt.ONE) + let x3 = BInt(bitWidth: 1000) + XCTAssertEqual(x3.gcd(x3), x3) + XCTAssertEqual(x3.gcd(x3 + 1), BInt.ONE) + } + } + +} diff --git a/Tests/BigIntTests/KaratsubaTest.swift b/Tests/BigIntTests/KaratsubaTest.swift new file mode 100755 index 0000000..87b5ef3 --- /dev/null +++ b/Tests/BigIntTests/KaratsubaTest.swift @@ -0,0 +1,42 @@ +// +// KaratsubaTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 15/02/2019. +// + +import XCTest + +class KaratsubaTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + for _ in 0 ..< 10 { + let a = BInt(bitWidth: (Limbs.KA_THR + Limbs.TC_THR) / 2 * 64) + let b = BInt(bitWidth: (Limbs.KA_THR + Limbs.TC_THR) / 2 * 64) + let p = a * b + let (q, r) = p.quotientAndRemainder(dividingBy: a) + XCTAssertEqual(q, b) + XCTAssertEqual(r, BInt.ZERO) + } + } + + func test2() { + for _ in 0 ..< 10 { + let a = BInt(bitWidth: (Limbs.KA_THR + Limbs.TC_THR) / 2 * 64) + let p = a ** 2 + let (q, r) = p.quotientAndRemainder(dividingBy: a) + XCTAssertEqual(q, a) + XCTAssertEqual(r, BInt.ZERO) + } + } + + +} diff --git a/Tests/BigIntTests/LimbTest.swift b/Tests/BigIntTests/LimbTest.swift new file mode 100755 index 0000000..3f8d7c8 --- /dev/null +++ b/Tests/BigIntTests/LimbTest.swift @@ -0,0 +1,33 @@ +// +// LimbTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 24/12/2018. +// + +import XCTest + +class LimbTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + var x1: Limbs = [0, 0, 0] + XCTAssertEqual(x1.count, 3) + x1.ensureSize(5) + XCTAssertEqual(x1.count, 5) + x1.normalize() + XCTAssertEqual(x1.count, 1) + x1 = [0, 0, 1] + XCTAssertEqual(x1.bitWidth, 129) + x1.setBitAt(1, to: true) + XCTAssertEqual(BInt(x1), (BInt(1) << 128) + 2) + } + +} diff --git a/Tests/BigIntTests/ModInverseTest.swift b/Tests/BigIntTests/ModInverseTest.swift new file mode 100755 index 0000000..1316c66 --- /dev/null +++ b/Tests/BigIntTests/ModInverseTest.swift @@ -0,0 +1,40 @@ +// +// ModInverseTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 02/01/2019. +// + +import XCTest + +class ModInverseTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func doTest(_ n: Int, _ m: Int) { + let p1 = BInt.probablePrime(m) + var x1 = BInt(bitWidth: n) + if x1 == 0 { + x1 = BInt(1) + } + if x1 % p1 == BInt.ONE { + XCTAssert((x1 * x1.modInverse(p1)).mod(p1) == BInt.ONE) + } + } + + func test1() { + doTest(3, 4) + doTest(30, 40) + doTest(300, 400) + doTest(4, 3) + doTest(40, 30) + doTest(400, 300) + } + +} diff --git a/Tests/BigIntTests/MulSquareTest.swift b/Tests/BigIntTests/MulSquareTest.swift new file mode 100755 index 0000000..1650ca2 --- /dev/null +++ b/Tests/BigIntTests/MulSquareTest.swift @@ -0,0 +1,50 @@ +// +// MulSquareTest.swift +// BigIntegerTests +// +// Created by Leif Ibsen on 19/12/2018. +// + +import XCTest + +class MulSquareTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + for i in 1 ... 10 { + let a1 = (BInt(1) << (i * 10)).randomLessThan() + let a2 = (BInt(1) << (i * 100)).randomLessThan() + let a3 = (BInt(1) << (i * 1000)).randomLessThan() + let a4 = (BInt(1) << (i * 10000)).randomLessThan() + let b1 = -a1 + let b2 = -a2 + let b3 = -a3 + let b4 = -a4 + XCTAssertEqual(a1 ** 2, a1 * a1) + XCTAssertEqual(a2 ** 2, a2 * a2) + XCTAssertEqual(a3 ** 2, a3 * a3) + XCTAssertEqual(a4 ** 2, a4 * a4) + XCTAssertEqual(b1 ** 2, b1 * b1) + XCTAssertEqual(b2 ** 2, b2 * b2) + XCTAssertEqual(b3 ** 2, b3 * b3) + XCTAssertEqual(b4 ** 2, b4 * b4) + } + } + + func test2() { + var b1 = BInt(1) + let a1 = BInt(bitWidth: 100) + for i in 0 ..< 10 { + XCTAssertEqual(a1 ** i, b1) + b1 *= a1 + } + } + +} diff --git a/Tests/BigIntTests/MultiplicationTest.swift b/Tests/BigIntTests/MultiplicationTest.swift new file mode 100755 index 0000000..e90da5c --- /dev/null +++ b/Tests/BigIntTests/MultiplicationTest.swift @@ -0,0 +1,89 @@ +// +// MultiplicationTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 19/01/2019. +// + +import XCTest + +class MultiplicationTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func doTest1(_ i: Int) { + let a = BInt(bitWidth: 200 * i) + let b = BInt(bitWidth: 100 * i) + let c = a * b + let (q, r) = c.quotientAndRemainder(dividingBy: a) + XCTAssertEqual(r, BInt(0)) + XCTAssertEqual(q, b) + let (q1, r1) = c.quotientAndRemainder(dividingBy: b) + XCTAssertEqual(r1, BInt(0)) + XCTAssertEqual(q1, a) + var am = a.magnitude + am.square() + XCTAssertEqual(a * a, BInt(am)) + } + + func test1() { + for i in 1 ..< 10 { + doTest1(i) + } + doTest1(1000) + } + + func test2() { + XCTAssertEqual(BInt.ONE * BInt.ZERO, BInt.ZERO) + XCTAssertEqual(BInt(-1) * BInt.ZERO, BInt.ZERO) + XCTAssertEqual(BInt.ZERO * BInt.ONE, BInt.ZERO) + XCTAssertEqual(BInt.ZERO * BInt(-1), BInt.ZERO) + XCTAssertEqual(BInt.ONE * 0, BInt.ZERO) + XCTAssertEqual(BInt(-1) * 0, BInt.ZERO) + XCTAssertEqual(0 * BInt.ONE, BInt.ZERO) + XCTAssertEqual(0 * BInt(-1), BInt.ZERO) + } + + func test3() { + XCTAssertEqual(BInt(7) * BInt(4), BInt(28)) + XCTAssertEqual(BInt(7) * BInt(-4), BInt(-28)) + XCTAssertEqual(BInt(-7) * BInt(4), BInt(-28)) + XCTAssertEqual(BInt(-7) * BInt(-4), BInt(28)) + XCTAssertEqual(BInt(-7) * BInt(0), BInt(0)) + XCTAssertEqual(BInt(7) * BInt(0), BInt(0)) + XCTAssertEqual(BInt(7) * 4, BInt(28)) + XCTAssertEqual(BInt(7) * (-4), BInt(-28)) + XCTAssertEqual(BInt(-7) * 4, BInt(-28)) + XCTAssertEqual(BInt(-7) * (-4), BInt(28)) + XCTAssertEqual(BInt(-7) * 0, BInt(0)) + XCTAssertEqual(BInt(7) * 0, BInt(0)) + XCTAssertEqual(7 * BInt(4), BInt(28)) + XCTAssertEqual(7 * BInt(-4), BInt(-28)) + XCTAssertEqual((-7) * BInt(4), BInt(-28)) + XCTAssertEqual((-7) * BInt(-4), BInt(28)) + XCTAssertEqual((-7) * BInt(0), BInt(0)) + XCTAssertEqual(7 * BInt(0), BInt(0)) + } + + func test5() { + var x1 = BInt(7) + x1 *= BInt(4) + XCTAssertEqual(x1, BInt(28)) + var x2 = BInt(7) + x2 *= BInt(-4) + XCTAssertEqual(x2, BInt(-28)) + var x3 = BInt(-7) + x3 *= BInt(4) + XCTAssertEqual(x3, BInt(-28)) + var x4 = BInt(-7) + x4 *= BInt(-4) + XCTAssertEqual(x4, BInt(28)) + } + +} diff --git a/Tests/BigIntTests/PerformanceTest.swift b/Tests/BigIntTests/PerformanceTest.swift new file mode 100755 index 0000000..b464a63 --- /dev/null +++ b/Tests/BigIntTests/PerformanceTest.swift @@ -0,0 +1,97 @@ +// +// PerformanceTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 17/01/2019. +// + +import XCTest + +class PerformanceTest: XCTestCase { + + let a1000 = BInt("3187705437890850041662973758105262878153514794996698172406519277876060364087986868049465132757493318066301987043192958841748826350731448419937544810921786918975580180410200630645469411588934094075222404396990984350815153163569041641732160380739556436955287671287935796642478260435292021117614349253825")! + let b1000 = BInt("9159373012373110951130589007821321098436345855865428979299172149373720601254669552044211236974571462005332583657082428026625366060511329189733296464187785766230514564038057370938741745651937465362625449921195096442684523511715110908407508139315000469851121118117438147266381183636498494901233452870695")! + let c2000 = BInt("1190583332681083129323588684910845359379915367459759242106618067261956856381281184752008892106576666833853411939711280970145570546868549934865719229538926506588929417873149597614787608112658086250354719939407543740242931571462165384138560315454455247539461818779966171917173966217706187439870264672508450210272481951994459523586160979759782950984370978171111340529321052541588344733968902238813379990628157732181128074253104347868153860527298911917508606081710893794973605227829729403843750412766366804402629686458092685235454222856584200220355212623917637542398554907364450159627359316156463617143173")! + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + + func testAddition() { + let N = 100000 + var x = BInt.ZERO + self.measure { + for _ in 0 ..< N { + x = a1000 + b1000 + } + } + } + + func testSubtraction() { + let N = 100000 + var x = BInt.ZERO + self.measure { + for _ in 0 ..< N { + x = a1000 - b1000 + } + } + } + + func testMultiplication() { + let N = 1000 + var x = BInt.ZERO + self.measure { + for _ in 0 ..< N { + x = a1000 * b1000 + } + } + } + + func testSquaring() { + let N = 1000 + var a = Limbs(repeating: 0, count: 0) + self.measure { + for _ in 0 ..< N { + a = a1000.magnitude + a.square() + } + } + } + + func testDivision() { + let N = 1000 + var q = BInt.ZERO + var r = BInt.ZERO + self.measure { + for _ in 0 ..< N { + (q, r) = c2000.quotientAndRemainder(dividingBy: a1000) + } + } + } + + func testGcd() { + let N = 10 + var x = BInt.ZERO + self.measure { + for _ in 0 ..< N { + x = a1000.gcd(b1000) + } + } + } + + func testExpMod() { + let N = 10 + var x = BInt.ZERO + self.measure { + for _ in 0 ..< N { + x = a1000.expMod(b1000, c2000) + } + } + } + +} diff --git a/Tests/BigIntTests/PrimeTest.swift b/Tests/BigIntTests/PrimeTest.swift new file mode 100755 index 0000000..d031054 --- /dev/null +++ b/Tests/BigIntTests/PrimeTest.swift @@ -0,0 +1,47 @@ +// +// PrimeTest.swift +// BigIntegerTests +// +// Created by Leif Ibsen on 15/12/2018. +// + +import XCTest + +class PrimeTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func doMersenne(_ x: Int) { + XCTAssertTrue((BInt(1) << x - 1).isProbablyPrime()) + XCTAssertTrue(!(BInt(1) << x + 1).isProbablyPrime()) + } + func test1() { + XCTAssertTrue(BInt(100).isProbablyPrime(0)) + XCTAssertTrue(!BInt(100).isProbablyPrime()) + for i in 1 ... 5 { + XCTAssertTrue(BInt.probablePrime(100 * i).isProbablyPrime()) + } + } + + func test2() { + doMersenne(3) + doMersenne(5) + doMersenne(7) + doMersenne(13) + doMersenne(17) + doMersenne(19) + doMersenne(31) + doMersenne(61) + doMersenne(89) + doMersenne(107) + doMersenne(127) + doMersenne(521) + } + +} diff --git a/Tests/BigIntTests/RootTest.swift b/Tests/BigIntTests/RootTest.swift new file mode 100644 index 0000000..834e290 --- /dev/null +++ b/Tests/BigIntTests/RootTest.swift @@ -0,0 +1,45 @@ +// +// RootTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 11/09/2019. +// + +import XCTest + +class RootTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + for _ in 0 ..< 1000 { + let x = BInt(bitWidth: 1000) + let y = x.sqrt() + XCTAssert(y * y <= x) + XCTAssert((y + 1) * (y + 1) > x) + for n in 1 ... 10 { + let y = x.root(n) + XCTAssert(y ** n <= x) + XCTAssert((y + 1) ** n > x) + } + } + } + + func test2() { + for _ in 0 ..< 100 { + let p = BInt.probablePrime(100) + for _ in 0 ..< 100 { + let x = BInt(bitWidth: 300) + let s = x.sqrtMod(p) + XCTAssert(x.jacobiSymbol(p) != 1 || (s! ** 2).mod(p) == x.mod(p)) + } + } + } + +} diff --git a/Tests/BigIntTests/ShiftTest.swift b/Tests/BigIntTests/ShiftTest.swift new file mode 100755 index 0000000..ce444bb --- /dev/null +++ b/Tests/BigIntTests/ShiftTest.swift @@ -0,0 +1,33 @@ +// +// ShiftTest.swift +// XBigIntegerTests +// +// Created by Leif Ibsen on 11/12/2018. +// + +import XCTest + +class ShiftTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + let x1 = BInt(bitWidth: 200) + XCTAssertEqual(x1, (x1 << 3) >> 3) + XCTAssertEqual(x1 << -3, x1 >> 3) + XCTAssertEqual(x1 << 3, x1 >> -3) + let x2 = BInt("1234567890")! + XCTAssertEqual(x2, (x2 << 200) / (BInt.ONE << 200)) + var x3 = x2 + x3 <<= 1 + XCTAssertEqual(x3, BInt("2469135780")!) + } + + +} diff --git a/Tests/BigIntTests/SubtractionTest.swift b/Tests/BigIntTests/SubtractionTest.swift new file mode 100755 index 0000000..5e55174 --- /dev/null +++ b/Tests/BigIntTests/SubtractionTest.swift @@ -0,0 +1,58 @@ +// +// SubtractionTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 19/01/2019. +// + +import XCTest + +class SubtractionTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + XCTAssertEqual(BInt(7) - BInt(4), BInt(3)) + XCTAssertEqual(BInt(7) - BInt(-4), BInt(11)) + XCTAssertEqual(BInt(-7) - BInt(4), BInt(-11)) + XCTAssertEqual(BInt(-7) - BInt(-4), BInt(-3)) + XCTAssertEqual(BInt(-7) - BInt(0), BInt(-7)) + XCTAssertEqual(BInt(7) - BInt(0), BInt(7)) + XCTAssertEqual(BInt(7) - 4, BInt(3)) + XCTAssertEqual(BInt(7) - (-4), BInt(11)) + XCTAssertEqual(BInt(-7) - 4, BInt(-11)) + XCTAssertEqual(BInt(-7) - (-4), BInt(-3)) + XCTAssertEqual(BInt(-7) - 0, BInt(-7)) + XCTAssertEqual(BInt(7) - 0, BInt(7)) + XCTAssertEqual(7 - BInt(4), BInt(3)) + XCTAssertEqual(7 - BInt(-4), BInt(11)) + XCTAssertEqual((-7) - BInt(4), BInt(-11)) + XCTAssertEqual((-7) - BInt(-4), BInt(-3)) + XCTAssertEqual((-7) - BInt(0), BInt(-7)) + XCTAssertEqual(7 - BInt(0), BInt(7)) + } + + func test2() { + var x1 = BInt(7) + x1 -= BInt(4) + XCTAssertEqual(x1, BInt(3)) + var x2 = BInt(7) + x2 -= BInt(-4) + XCTAssertEqual(x2, BInt(11)) + var x3 = BInt(-7) + x3 -= BInt(4) + XCTAssertEqual(x3, BInt(-11)) + var x4 = BInt(-7) + x4 -= BInt(-4) + XCTAssertEqual(x4, BInt(-3)) + + } + + +} diff --git a/Tests/BigIntTests/ToByteArrayTest.swift b/Tests/BigIntTests/ToByteArrayTest.swift new file mode 100644 index 0000000..69935f3 --- /dev/null +++ b/Tests/BigIntTests/ToByteArrayTest.swift @@ -0,0 +1,40 @@ +// +// ToByteArrayTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 05/09/2019. +// + +import XCTest + +class ToByteArrayTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + for _ in 0 ..< 10000 { + let x = BInt(bitWidth: 1000) + let x1 = BInt(magnitude: x.asMagnitudeBytes()) + XCTAssertEqual(x, x1) + let x2 = BInt(signed: x.asSignedBytes()) + XCTAssertEqual(x, x2) + } + } + + func test2() { + for _ in 0 ..< 10000 { + let x = -BInt(bitWidth: 1000) + let x1 = -BInt(magnitude: x.asMagnitudeBytes()) + XCTAssertEqual(x, x1) + let x2 = BInt(signed: x.asSignedBytes()) + XCTAssertEqual(x, x2) + } + } + +} diff --git a/Tests/BigIntTests/ToStringTest.swift b/Tests/BigIntTests/ToStringTest.swift new file mode 100755 index 0000000..f591a9a --- /dev/null +++ b/Tests/BigIntTests/ToStringTest.swift @@ -0,0 +1,48 @@ +// +// ToStringTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 04/02/2019. +// + +import XCTest + +class ToStringTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func testToString1() { + var x = BInt(1) + var s = "1" + for _ in 0 ..< 100 { + x *= 10 + s += "0" + XCTAssertEqual(s, x.asString(radix: 10)) + } + } + + func testToString2() { + let x: Limbs = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] + let b = BInt(x, false) + for r in 2 ... 36 { + let b1 = BInt(b.asString(radix: r), radix: r) + XCTAssertEqual(b, b1) + } + } + + func testToString3() { + let x: Limbs = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] + let b = BInt(x, true) + for r in 2 ... 36 { + let b1 = BInt(b.asString(radix: r), radix: r) + XCTAssertEqual(b, b1) + } + } + +} diff --git a/Tests/BigIntTests/ToomCookTest.swift b/Tests/BigIntTests/ToomCookTest.swift new file mode 100755 index 0000000..cc95034 --- /dev/null +++ b/Tests/BigIntTests/ToomCookTest.swift @@ -0,0 +1,41 @@ +// +// ToomCookTest.swift +// BigIntTests +// +// Created by Leif Ibsen on 09/02/2019. +// + +import XCTest + +class ToomCookTest: XCTestCase { + + override func setUp() { + // Put setup code here. This method is called before the invocation of each test method in the class. + } + + override func tearDown() { + // Put teardown code here. This method is called after the invocation of each test method in the class. + } + + func test1() { + for _ in 0 ..< 10 { + let a = BInt(bitWidth: Limbs.TC_THR * 5 * 64) + let b = BInt(bitWidth: Limbs.TC_THR * 2 * 64) + let p = a * b + let (q, r) = p.quotientAndRemainder(dividingBy: a) + XCTAssertEqual(q, b) + XCTAssertEqual(r, BInt.ZERO) + } + } + + func test2() { + for _ in 0 ..< 10 { + let a = BInt(bitWidth: Limbs.TC_THR * 10 * 64) + let p = a ** 2 + let (q, r) = p.quotientAndRemainder(dividingBy: a) + XCTAssertEqual(q, a) + XCTAssertEqual(r, BInt.ZERO) + } + } + +} diff --git a/Tests/BigIntTests/XCTestManifests.swift b/Tests/BigIntTests/XCTestManifests.swift new file mode 100755 index 0000000..ab63890 --- /dev/null +++ b/Tests/BigIntTests/XCTestManifests.swift @@ -0,0 +1,9 @@ +import XCTest + +#if !os(macOS) +public func allTests() -> [XCTestCaseEntry] { + return [ + testCase(BigIntegerTests.allTests), + ] +} +#endif \ No newline at end of file diff --git a/Tests/LinuxMain.swift b/Tests/LinuxMain.swift new file mode 100644 index 0000000..ff9f8c0 --- /dev/null +++ b/Tests/LinuxMain.swift @@ -0,0 +1,7 @@ +import XCTest + +import BigIntTests + +var tests = [XCTestCaseEntry]() +tests += BigIntTests.allTests() +XCTMain(tests) diff --git a/docs/Structs.html b/docs/Structs.html new file mode 100644 index 0000000..97523ff --- /dev/null +++ b/docs/Structs.html @@ -0,0 +1,113 @@ + + + + Structures Reference + + + + + + + + + + +
+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+

Structures

+

The following structures are available globally.

+ +
+
+
+
    +
  • +
    + + + + BInt + +
    +
    +
    +
    +
    +
    +

    A signed integer of unbounded size. +A BInt value is represented with magnitude and sign. +The magnitude is an array of unsigned 64 bit integers (a.k.a. Limbs). +The sign is a boolean value, true means value < 0, false means value >= 0 +The representation is little-endian, least significant Limb has index 0. +The representation is minimal, there is no leading zero Limbs. +The exception is that the value 0 is represented as a single 64 bit zero Limb and sign false

    + + See more +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public struct BInt : CustomStringConvertible, Equatable, Hashable
    + +
    +
    +
    +
    +
    • +
+
+
+
+ +
+
+ + + diff --git a/docs/Structs/BInt.html b/docs/Structs/BInt.html new file mode 100644 index 0000000..e12deab --- /dev/null +++ b/docs/Structs/BInt.html @@ -0,0 +1,5734 @@ + + + + BInt Structure Reference + + + + + + + + + + +
+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+

BInt

+
+
+ 
public struct BInt : CustomStringConvertible, Equatable, Hashable
+ +
+
+

A signed integer of unbounded size. +A BInt value is represented with magnitude and sign. +The magnitude is an array of unsigned 64 bit integers (a.k.a. Limbs). +The sign is a boolean value, true means value < 0, false means value >= 0 +The representation is little-endian, least significant Limb has index 0. +The representation is minimal, there is no leading zero Limbs. +The exception is that the value 0 is represented as a single 64 bit zero Limb and sign false

+ +
+
+
+
+ + + +

Constants

+ +
+
    +
  • +
    + + + + ZERO + +
    +
    +
    +
    +
    +
    +

    BInt(0)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let ZERO: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + ONE + +
    +
    +
    +
    +
    +
    +

    BInt(1)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let ONE: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + TWO + +
    +
    +
    +
    +
    +
    +

    BInt(2)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let TWO: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + THREE + +
    +
    +
    +
    +
    +
    +

    BInt(3)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let THREE: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + TEN + +
    +
    +
    +
    +
    +
    +

    BInt(10)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let TEN: BInt
    + +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Initializers

+ +
+
    +
  • +
    + + + + init(_:_:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from magnitude and sign

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(_ magnitude: Limbs, _ isNegative: Bool = false)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + magnitude + + +
    +

    magnitude of value

    +
    +
    + + isNegative + + +
    +

    true means negative value, false means 0 or positive value, default is false

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(_:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from an Int value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(_ x: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Int value

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(_:radix:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from a String value and radix

    + +

    Examples:

    + +
      +
    • BInt(90abcdef, radix = 16)
      • +
    • BInt(111110010, radix = 2)
      • +
    • BInt(1cdefghijk44, radix = 26)
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init?(_ x: String, radix: Int = 10)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    String value to be converted

    +
    +
    + + radix + + +
    +

    Radix of x, from 2 to 36 inclusive, default is 10

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(bitWidth:) + +
    +
    +
    +
    +
    +
    +

    Constructs a random BInt with a specified number of bits

    +
    +

    Precondition

    + bitWidth is positive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(bitWidth: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + bitWidth + + +
    +

    Number of bits

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(magnitude:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from a big-endian magnitude byte array representation

    +
    +

    Precondition

    + Byte array is not empty + +
    + +

    Examples:

    + +
      +
    • The byte array [1, 0, 0] represents BInt value 65536
      • +
    • The byte array [128, 0] represents BInt value 32768
      • +
    • The byte array [255, 127] represents BInt value 65407
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(magnitude x: Bytes)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Magnitude big-endian byte array

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(signed:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from a big-endian 2’s complement byte array representation

    +
    +

    Precondition

    + Byte array is not empty + +
    + +

    Examples:

    + +
      +
    • The byte array [1, 0, 0] represents BInt value 65536
      • +
    • The byte array [128, 0] represents BInt value -32768
      • +
    • The byte array [255, 127] represents BInt value -129
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(signed x: Bytes)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    2’s complement big-endian byte array

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Stored properties

+ +
+
    +
  • +
    + + + + isNegative + +
    +
    +
    +
    +
    +
    +

    The sign, true if self < 0, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public internal(set) var isNegative: Bool
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + magnitude + +
    +
    +
    +
    +
    +
    +

    The magnitude limb array

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public internal(set) var magnitude: Limbs
    + +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Computed properties

+ +
+
    +
  • +
    + + + + bitWidth + +
    +
    +
    +
    +
    +
    +

    The number of bits in the magnitude of self. 0 if self = 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var bitWidth: Int { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + description + +
    +
    +
    +
    +
    +
    +

    Base 10 string value of self

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var description: String { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isEven + +
    +
    +
    +
    +
    +
    +

    Is true if self is even, false if self is odd

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isEven: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isNotZero + +
    +
    +
    +
    +
    +
    +

    Is false if self = 0, true otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isNotZero: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isOdd + +
    +
    +
    +
    +
    +
    +

    Is true if self is odd, false if self is even

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isOdd: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isOne + +
    +
    +
    +
    +
    +
    +

    Is true if self = 1, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isOne: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isPositive + +
    +
    +
    +
    +
    +
    +

    Is true if self > 0, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isPositive: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isZero + +
    +
    +
    +
    +
    +
    +

    Is true if self = 0, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isZero: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + leadingZeroBitCount + +
    +
    +
    +
    +
    +
    +

    The number of leading zero bits in the magnitude of self. 0 if self = 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var leadingZeroBitCount: Int { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + signum + +
    +
    +
    +
    +
    +
    +

    Is 0 if self = 0, 1 if self > 0, and -1 if self < 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var signum: Int { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + trailingZeroBitCount + +
    +
    +
    +
    +
    +
    +

    The number of trailing zero bits in the magnitude of self. 0 if self = 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var trailingZeroBitCount: Int { get }
    + +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Conversion functions to Int, String, and Bytes

+ +
+
    +
  • +
    + + + + asInt() + +
    +
    +
    +
    +
    +
    +

    self as an Int

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asInt() -> Int?
    + +
    +
    +
    +

    Return Value

    +

    self as an Int or nil if self is not representable as an Int

    +
    +
    +
    +
    • +
  • +
    + + + + asMagnitudeBytes() + +
    +
    +
    +
    +
    +
    +

    Byte array representation of magnitude value

    + +

    Examples:

    + +
      +
    • BInt(1).asMagnitudeBytes() = [1]
      • +
    • BInt(-1).asMagnitudeBytes() = [1]
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asMagnitudeBytes() -> Bytes
    + +
    +
    +
    +

    Return Value

    +

    Minimal big-endian magnitude byte array representation

    +
    +
    +
    +
    • +
  • +
    + + + + asSignedBytes() + +
    +
    +
    +
    +
    +
    +

    Byte array representation of 2’s complement value

    + +

    Examples:

    + +
      +
    • BInt(1).asSignedBytes() = [1]
      • +
    • BInt(-1).asSignedBytes() = [255]
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asSignedBytes() -> Bytes
    + +
    +
    +
    +

    Return Value

    +

    Minimal big-endian 2’s complement byte array representation

    +
    +
    +
    +
    • +
  • + +
    +
    +
    +
    +
    +

    self as a String with a given radix

    +
    +

    Precondition

    + Radix between 2 and 36 inclusive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asString(radix: Int = 10, uppercase: Bool = false) -> String
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + radix + + +
    +

    Radix from 2 to 36 inclusive

    +
    +
    + + uppercase + + +
    +

    true to use uppercase letters, false to use lowercase letters, default is false

    +
    +
    +
    +
    +

    Return Value

    +

    self as a String in the given radix

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Bit operation functions

+ +
+
    +
  • +
    + + + + &(_:_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise and operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func & (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First value

    +
    +
    + + y + + +
    +

    Second value

    +
    +
    +
    +
    +

    Return Value

    +

    BInt(signed: bx & by) where + +

      +
    • bx = x.asSignedBytes()
      • +
    • by = y.asSignedBytes()
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + &=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x & y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func &= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand parameter

    +
    +
    + + y + + +
    +

    Right hand parameter

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + |(_:_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise or operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func | (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First value

    +
    +
    + + y + + +
    +

    Second value

    +
    +
    +
    +
    +

    Return Value

    +

    BInt(signed: bx | by) where + +

      +
    • bx = x.asSignedBytes()
      • +
    • by = y.asSignedBytes()
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + |=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x | y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func |= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand parameter

    +
    +
    + + y + + +
    +

    Right hand parameter

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + ^(_:_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise xor operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func ^ (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First value

    +
    +
    + + y + + +
    +

    Second value

    +
    +
    +
    +
    +

    Return Value

    +

    BInt(signed: bx ^ by) where + +

      +
    • bx = x.asSignedBytes()
      • +
    • by = y.asSignedBytes()
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + ^=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x ^ y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func ^= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand parameter

    +
    +
    + + y + + +
    +

    Right hand parameter

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + ~(_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise not operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public prefix static func ~ (x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    BInt value

    +
    +
    +
    +
    +

    Return Value

    +

    -x - 1

    +
    +
    +
    +
    • +
  • +
    + + + + clearBit(_:) + +
    +
    +
    +
    +
    +
    +

    Clear a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func clearBit(_ n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + flipBit(_:) + +
    +
    +
    +
    +
    +
    +

    Invert a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func flipBit(_ n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + setBit(_:) + +
    +
    +
    +
    +
    +
    +

    Set a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func setBit(_ n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + testBit(_:) + +
    +
    +
    +
    +
    +
    +

    Test a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func testBit(_ n: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +

    Return Value

    +

    true if bit is set, false otherwise

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Addition functions

+ +
+
    +
  • +
    + + + + +(_:) + +
    +
    +
    +
    +
    +
    +

    Prefix plus

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public prefix static func + (x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    BInt value

    +
    +
    +
    +
    +

    Return Value

    +

    x

    +
    +
    +
    +
    • +
  • +
    + + + + +(_:_:) + +
    +
    +
    +
    +
    +
    +

    Addition

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func + (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First addend

    +
    +
    + + y + + +
    +

    Second addend

    +
    +
    +
    +
    +

    Return Value

    +

    x + y

    +
    +
    +
    +
    • +
  • +
    + + + + +(_:_:) + +
    +
    +
    +
    +
    +
    +

    Addition

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func + (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First addend

    +
    +
    + + y + + +
    +

    Second addend

    +
    +
    +
    +
    +

    Return Value

    +

    x + y

    +
    +
    +
    +
    • +
  • +
    + + + + +(_:_:) + +
    +
    +
    +
    +
    +
    +

    Addition

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func + (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First addend

    +
    +
    + + y + + +
    +

    Second addend

    +
    +
    +
    +
    +

    Return Value

    +

    x + y

    +
    +
    +
    +
    • +
  • +
    + + + + +=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x + y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func += (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand addend

    +
    +
    + + y + + +
    +

    Right hand addend

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + +=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x + y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func += (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand addend

    +
    +
    + + y + + +
    +

    Right hand addend

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Negation functions

+ +
+
    +
  • +
    + + + + negate() + +
    +
    +
    +
    +
    +
    +

    Negates self

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func negate()
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + -(_:) + +
    +
    +
    +
    +
    +
    +

    Negation

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public prefix static func - (x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    +
    +
    +

    Return Value

    +

    -x

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Subtraction functions

+ +
+
    +
  • +
    + + + + -(_:_:) + +
    +
    +
    +
    +
    +
    +

    Subtraction

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func - (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Minuend

    +
    +
    + + y + + +
    +

    Subtrahend

    +
    +
    +
    +
    +

    Return Value

    +

    x - y

    +
    +
    +
    +
    • +
  • +
    + + + + -(_:_:) + +
    +
    +
    +
    +
    +
    +

    Subtraction

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func - (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Minuend

    +
    +
    + + y + + +
    +

    Subtrahend

    +
    +
    +
    +
    +

    Return Value

    +

    x - y

    +
    +
    +
    +
    • +
  • +
    + + + + -(_:_:) + +
    +
    +
    +
    +
    +
    +

    Subtraction

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func - (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Minuend

    +
    +
    + + y + + +
    +

    Subtrahend

    +
    +
    +
    +
    +

    Return Value

    +

    x - y

    +
    +
    +
    +
    • +
  • +
    + + + + -=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x - y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func -= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand minuend

    +
    +
    + + y + + +
    +

    Right hand subtrahend

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + -=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x - y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func -= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand minuend

    +
    +
    + + y + + +
    +

    Right hand subtrahend

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Multiplication functions

+ +
+
    +
  • +
    + + + + *(_:_:) + +
    +
    +
    +
    +
    +
    +

    Multiplication

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func * (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Multiplier

    +
    +
    + + y + + +
    +

    Multiplicand

    +
    +
    +
    +
    +

    Return Value

    +

    x * y

    +
    +
    +
    +
    • +
  • +
    + + + + *(_:_:) + +
    +
    +
    +
    +
    +
    +

    Multiplication

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func * (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Multiplier

    +
    +
    + + y + + +
    +

    Multiplicand

    +
    +
    +
    +
    +

    Return Value

    +

    x * y

    +
    +
    +
    +
    • +
  • +
    + + + + *(_:_:) + +
    +
    +
    +
    +
    +
    +

    Multiplication

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func * (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Multiplier

    +
    +
    + + y + + +
    +

    Multiplicand

    +
    +
    +
    +
    +

    Return Value

    +

    x * y

    +
    +
    +
    +
    • +
  • +
    + + + + *=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x * y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func *= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand multiplier

    +
    +
    + + y + + +
    +

    Right hand multiplicand

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + *=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x * y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func *= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand multiplier

    +
    +
    + + y + + +
    +

    Right hand multiplicand

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Division functions

+ +
+
    +
  • + +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func quotientAndRemainder(dividingBy x: BInt) -> (quotient: BInt, remainder: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    Quotient and remainder of self / x

    +
    +
    +
    +
    • +
  • + +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func quotientAndRemainder(dividingBy x: BInt, _ quotient: inout BInt, _ remainder: inout BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + + + + + +
    + + x + + +
    +

    Divisor

    +
    +
    + + quotient + + +
    +

    Set to the quotient of self / x

    +
    +
    + + remainder + + +
    +

    Set to the remainder of self / x

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + /(_:_:) + +
    +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func / (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x / y

    +
    +
    +
    +
    • +
  • +
    + + + + /(_:_:) + +
    +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func / (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x / y

    +
    +
    +
    +
    • +
  • +
    + + + + /(_:_:) + +
    +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func / (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x / y

    +
    +
    +
    +
    • +
  • +
    + + + + /=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x / y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func /= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand dividend

    +
    +
    + + y + + +
    +

    Right hand divisor

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + /=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x / y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func /= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand dividend

    +
    +
    + + y + + +
    +

    Right hand divisor

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Remainder and modulus functions

+ +
+
    +
  • +
    + + + + %(_:_:) + +
    +
    +
    +
    +
    +
    +

    Remainder

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func % (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x % y

    +
    +
    +
    +
    • +
  • +
    + + + + %(_:_:) + +
    +
    +
    +
    +
    +
    +

    Remainder

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func % (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x % y

    +
    +
    +
    +
    • +
  • +
    + + + + %(_:_:) + +
    +
    +
    +
    +
    +
    +

    Remainder

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func % (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x % y

    +
    +
    +
    +
    • +
  • +
    + + + + %=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x % y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func %= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + %=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x % y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func %= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + mod(_:) + +
    +
    +
    +
    +
    +
    +

    Modulus

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func mod(_ x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    self mod x, a non-negative value

    +
    +
    +
    +
    • +
  • +
    + + + + modInverse(_:) + +
    +
    +
    +
    +
    +
    +

    Inverse modulus

    +
    +

    Precondition

    + self and modulus are coprime + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func modInverse(_ m: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + m + + +
    +

    Modulus

    +
    +
    +
    +
    +

    Return Value

    +

    If self and m are coprime, x such that (self * x) mod m = 1

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Exponentiation functions

+ +
+
    +
  • +
    + + + + **(_:_:) + +
    +
    +
    +
    +
    +
    +

    Exponentiation

    +
    +

    Precondition

    + Exponent is non-negative + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func ** (a: BInt, x: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + a + + +
    +

    Operand

    +
    +
    + + x + + +
    +

    Non-negative exponent

    +
    +
    +
    +
    +

    Return Value

    +

    a^x

    +
    +
    +
    +
    • +
  • +
    + + + + expMod(_:_:) + +
    +
    +
    +
    +
    +
    +

    Modular exponentiation

    +
    +

    Precondition

    + Modulus is positive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func expMod(_ x: BInt, _ m: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    The exponent

    +
    +
    + + m + + +
    +

    The modulus, a positive number

    +
    +
    +
    +
    +

    Return Value

    +

    (self^x) mod m for positive x, ((self^-x) mod m) modInverse m for negative x

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Comparison functions

+ +
+
    +
  • +
    + + + + ==(_:_:) + +
    +
    +
    +
    +
    +
    +

    Equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func == (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x = y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + ==(_:_:) + +
    +
    +
    +
    +
    +
    +

    Equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func == (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x = y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + ==(_:_:) + +
    +
    +
    +
    +
    +
    +

    Equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func == (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x = y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + !=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Not equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func != (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x != y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + !=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Not equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func != (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x != y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + !=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Not equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func != (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x != y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less then

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func < (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x < y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func < (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x < y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func < (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x < y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func > (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x > y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func > (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x > y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func > (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x > y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <= (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x <= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <= (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x <= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <= (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x <= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >= (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x >= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >= (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x >= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >= (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x >= y, false otherwise

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Shift functions

+ +
+
    +
  • +
    + + + + <<(_:_:) + +
    +
    +
    +
    +
    +
    +

    Logical left shift

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func << (x: BInt, n: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +

    Return Value

    +

      +
    • BInt(x.magnitude << n, x.isNegative) if n > 0
      • +
    • BInt(x.magnitude >> -n, x.isNegative) if n < 0
      • +
    • x if n = 0
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + <<=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x << n

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <<= (x: inout BInt, n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + >>(_:_:) + +
    +
    +
    +
    +
    +
    +

    Logical right shift

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >> (x: BInt, n: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +

    Return Value

    +

      +
    • BInt(x.magnitude >> n, x.isNegative) if n > 0
      • +
    • BInt(x.magnitude << -n, x.isNegative) if n < 0
      • +
    • x if n = 0
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + >>=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x >> n

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >>= (x: inout BInt, n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Prime number functions

+ +
+
    +
  • +
    + + + + isProbablyPrime(_:) + +
    +
    +
    +
    +
    +
    +

    Checks whether self is prime using the Miller-Rabin algorithm

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func isProbablyPrime(_ p: Int = 30) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + p + + +
    +

    If true is returned, self is prime with probability > 1-½^p

    +
    +
    +
    +
    +

    Return Value

    +

    true if self is probably prime, false if self is definitely not prime

    +
    +
    +
    +
    • +
  • +
    + + + + probablePrime(_:_:) + +
    +
    +
    +
    +
    +
    +

    A probable prime number with a given bitwidth

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func probablePrime(_ bitWidth: Int, _ p: Int = 30) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + bitWidth + + +
    +

    The bitWidth

    +
    +
    + + p + + +
    +

    The returned number is prime with probability > 1-½^p, default value is 30

    +
    +
    +
    +
    +

    Return Value

    +

    A prime number with the specified bitwidth and probability

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Miscellaneous functions

+ +
+
    +
  • +
    + + + + abs() + +
    +
    +
    +
    +
    +
    +

    Absolute value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func abs() -> BInt
    + +
    +
    +
    +

    Return Value

    +

    The absolute value of self

    +
    +
    +
    +
    • +
  • +
    + + + + gcd(_:) + +
    +
    +
    +
    +
    +
    +

    Greatest common divisor

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func gcd(_ x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    +
    +
    +

    Return Value

    +

    Greatest common divisor of magnitude of self and magnitude of x

    +
    +
    +
    +
    • +
  • +
    + + + + jacobiSymbol(_:) + +
    +
    +
    +
    +
    +
    +

    Jacobi symbol

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func jacobiSymbol(_ m: BInt) -> Int
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + m + + +
    +

    An integer value

    +
    +
    +
    +
    +

    Return Value

    +

    The Jacobi symbol of self and m, or 0 if it does not exist

    +
    +
    +
    +
    • +
  • +
    + + + + randomLessThan() + +
    +
    +
    +
    +
    +
    +

    Random value

    +
    +

    Precondition

    + self is positive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func randomLessThan() -> BInt
    + +
    +
    +
    +

    Return Value

    +

    A random value < absolute value of self

    +
    +
    +
    +
    • +
  • +
    + + + + root(_:) + +
    +
    +
    +
    +
    +
    +

    n'th root of a non-negative number

    +
    +

    Precondition

    +
      +
    • self is non-negative
      • +
    • n is positive
      • +
    + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func root(_ n: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    The root

    +
    +
    +
    +
    +

    Return Value

    +

    Largest x such that x^n <= self

    +
    +
    +
    +
    • +
  • +
    + + + + sqrt() + +
    +
    +
    +
    +
    +
    +

    Square root of a non-negative number

    +
    +

    Precondition

    + self is non-negative + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func sqrt() -> BInt
    + +
    +
    +
    +

    Return Value

    +

    Largest x such that x^2 <= self

    +
    +
    +
    +
    • +
  • +
    + + + + sqrtMod(_:) + +
    +
    +
    +
    +
    +
    +

    Square root modulo a prime number

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func sqrtMod(_ p: BInt) -> BInt?
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + p + + +
    +

    An odd prime number

    +
    +
    +
    +
    +

    Return Value

    +

    x, such that x^2 mod p = self, or nil if no such x exists

    +
    +
    +
    +
    • +
+
+
+
+ +
+
+ + + diff --git a/docs/Typealiases.html b/docs/Typealiases.html new file mode 100644 index 0000000..2a06fbe --- /dev/null +++ b/docs/Typealiases.html @@ -0,0 +1,187 @@ + + + + Type Aliases Reference + + + + + + + + + + +
+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+

Type Aliases

+

The following type aliases are available globally.

+ +
+
+
+
    +
  • +
    + + + + Byte + +
    +
    +
    +
    +
    +
    +

    Unsigned 8 bit value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Byte = UInt8
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + Bytes + +
    +
    +
    +
    +
    +
    +

    Array of unsigned 8 bit values

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Bytes = [UInt8]
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + Limb + +
    +
    +
    +
    +
    +
    +

    Unsigned 64 bit value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Limb = UInt64
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + Limbs + +
    +
    +
    +
    +
    +
    +

    Array of unsigned 64 bit values

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Limbs = [UInt64]
    + +
    +
    +
    +
    +
    • +
+
+
+
+ +
+
+ + + diff --git a/docs/badge.svg b/docs/badge.svg new file mode 100644 index 0000000..a096fec --- /dev/null +++ b/docs/badge.svg @@ -0,0 +1,28 @@ + + + + + + + + + + + + + + + + documentation + + + documentation + + + 100% + + + 100% + + + diff --git a/docs/css/highlight.css b/docs/css/highlight.css new file mode 100644 index 0000000..d0db0e1 --- /dev/null +++ b/docs/css/highlight.css @@ -0,0 +1,200 @@ +/* Credit to https://gist.github.com/wataru420/2048287 */ +.highlight { + /* Comment */ + /* Error */ + /* Keyword */ + /* Operator */ + /* Comment.Multiline */ + /* Comment.Preproc */ + /* Comment.Single */ + /* Comment.Special */ + /* Generic.Deleted */ + /* Generic.Deleted.Specific */ + /* Generic.Emph */ + /* Generic.Error */ + /* Generic.Heading */ + /* Generic.Inserted */ + /* Generic.Inserted.Specific */ + /* Generic.Output */ + /* Generic.Prompt */ + /* Generic.Strong */ + /* Generic.Subheading */ + /* Generic.Traceback */ + /* Keyword.Constant */ + /* Keyword.Declaration */ + /* Keyword.Pseudo */ + /* Keyword.Reserved */ + /* Keyword.Type */ + /* Literal.Number */ + /* Literal.String */ + /* Name.Attribute */ + /* Name.Builtin */ + /* Name.Class */ + /* Name.Constant */ + /* Name.Entity */ + /* Name.Exception */ + /* Name.Function */ + /* Name.Namespace */ + /* Name.Tag */ + /* Name.Variable */ + /* Operator.Word */ + /* Text.Whitespace */ + /* Literal.Number.Float */ + /* Literal.Number.Hex */ + /* Literal.Number.Integer */ + /* Literal.Number.Oct */ + /* Literal.String.Backtick */ + /* Literal.String.Char */ + /* Literal.String.Doc */ + /* Literal.String.Double */ + /* Literal.String.Escape */ + /* Literal.String.Heredoc */ + /* Literal.String.Interpol */ + /* Literal.String.Other */ + /* Literal.String.Regex */ + /* Literal.String.Single */ + /* Literal.String.Symbol */ + /* Name.Builtin.Pseudo */ + /* Name.Variable.Class */ + /* Name.Variable.Global */ + /* Name.Variable.Instance */ + /* Literal.Number.Integer.Long */ } + .highlight .c { + color: #999988; + font-style: italic; } + .highlight .err { + color: #a61717; + background-color: #e3d2d2; } + .highlight .k { + color: #000000; + font-weight: bold; } + .highlight .o { + color: #000000; + font-weight: bold; } + .highlight .cm { + color: #999988; + font-style: italic; } + .highlight .cp { + color: #999999; + font-weight: bold; } + .highlight .c1 { + color: #999988; + font-style: italic; } + .highlight .cs { + color: #999999; + font-weight: bold; + font-style: italic; } + .highlight .gd { + color: #000000; + background-color: #ffdddd; } + .highlight .gd .x { + color: #000000; + background-color: #ffaaaa; } + .highlight .ge { + color: #000000; + font-style: italic; } + .highlight .gr { + color: #aa0000; } + .highlight .gh { + color: #999999; } + .highlight .gi { + color: #000000; + background-color: #ddffdd; } + .highlight .gi .x { + color: #000000; + background-color: #aaffaa; } + .highlight .go { + color: #888888; } + .highlight .gp { + color: #555555; } + .highlight .gs { + font-weight: bold; } + .highlight .gu { + color: #aaaaaa; } + .highlight .gt { + color: #aa0000; } + .highlight .kc { + color: #000000; + font-weight: bold; } + .highlight .kd { + color: #000000; + font-weight: bold; } + .highlight .kp { + color: #000000; + font-weight: bold; } + .highlight .kr { + color: #000000; + font-weight: bold; } + .highlight .kt { + color: #445588; } + .highlight .m { + color: #009999; } + .highlight .s { + color: #d14; } + .highlight .na { + color: #008080; } + .highlight .nb { + color: #0086B3; } + .highlight .nc { + color: #445588; + font-weight: bold; } + .highlight .no { + color: #008080; } + .highlight .ni { + color: #800080; } + .highlight .ne { + color: #990000; + font-weight: bold; } + .highlight .nf { + color: #990000; } + .highlight .nn { + color: #555555; } + .highlight .nt { + color: #000080; } + .highlight .nv { + color: #008080; } + .highlight .ow { + color: #000000; + font-weight: bold; } + .highlight .w { + color: #bbbbbb; } + .highlight .mf { + color: #009999; } + .highlight .mh { + color: #009999; } + .highlight .mi { + color: #009999; } + .highlight .mo { + color: #009999; } + .highlight .sb { + color: #d14; } + .highlight .sc { + color: #d14; } + .highlight .sd { + color: #d14; } + .highlight .s2 { + color: #d14; } + .highlight .se { + color: #d14; } + .highlight .sh { + color: #d14; } + .highlight .si { + color: #d14; } + .highlight .sx { + color: #d14; } + .highlight .sr { + color: #009926; } + .highlight .s1 { + color: #d14; } + .highlight .ss { + color: #990073; } + .highlight .bp { + color: #999999; } + .highlight .vc { + color: #008080; } + .highlight .vg { + color: #008080; } + .highlight .vi { + color: #008080; } + .highlight .il { + color: #009999; } diff --git a/docs/css/jazzy.css b/docs/css/jazzy.css new file mode 100644 index 0000000..0541d71 --- /dev/null +++ b/docs/css/jazzy.css @@ -0,0 +1,343 @@ +html, body, div, span, h1, h3, h4, p, a, code, em, img, ul, li, table, tbody, tr, td { + background: transparent; + border: 0; + margin: 0; + outline: 0; + padding: 0; + vertical-align: baseline; } + +body { + background-color: #f2f2f2; + font-family: Helvetica, freesans, Arial, sans-serif; + font-size: 14px; + -webkit-font-smoothing: subpixel-antialiased; + word-wrap: break-word; } + +h1, h2, h3 { + margin-top: 0.8em; + margin-bottom: 0.3em; + font-weight: 100; + color: black; } + +h1 { + font-size: 2.5em; } + +h2 { + font-size: 2em; + border-bottom: 1px solid #e2e2e2; } + +h4 { + font-size: 13px; + line-height: 1.5; + margin-top: 21px; } + +h5 { + font-size: 1.1em; } + +h6 { + font-size: 1.1em; + color: #777; } + +.section-name { + color: gray; + display: block; + font-family: Helvetica; + font-size: 22px; + font-weight: 100; + margin-bottom: 15px; } + +pre, code { + font: 0.95em Menlo, monospace; + color: #777; + word-wrap: normal; } + +p code, li code { + background-color: #eee; + padding: 2px 4px; + border-radius: 4px; } + +a { + color: #0088cc; + text-decoration: none; } + +ul { + padding-left: 15px; } + +li { + line-height: 1.8em; } + +img { + max-width: 100%; } + +blockquote { + margin-left: 0; + padding: 0 10px; + border-left: 4px solid #ccc; } + +.content-wrapper { + margin: 0 auto; + width: 980px; } + +header { + font-size: 0.85em; + line-height: 26px; + background-color: #414141; + position: fixed; + width: 100%; + z-index: 1; } + header img { + padding-right: 6px; + vertical-align: -4px; + height: 16px; } + header a { + color: #fff; } + header p { + float: left; + color: #999; } + header .header-right { + float: right; + margin-left: 16px; } + +#breadcrumbs { + background-color: #f2f2f2; + height: 27px; + padding-top: 17px; + position: fixed; + width: 100%; + z-index: 1; + margin-top: 26px; } + #breadcrumbs #carat { + height: 10px; + margin: 0 5px; } + +.sidebar { + background-color: #f9f9f9; + border: 1px solid #e2e2e2; + overflow-y: auto; + overflow-x: hidden; + position: fixed; + top: 70px; + bottom: 0; + width: 230px; + word-wrap: normal; } + +.nav-groups { + list-style-type: none; + background: #fff; + padding-left: 0; } + +.nav-group-name { + border-bottom: 1px solid #e2e2e2; + font-size: 1.1em; + font-weight: 100; + padding: 15px 0 15px 20px; } + .nav-group-name > a { + color: #333; } + +.nav-group-tasks { + margin-top: 5px; } + +.nav-group-task { + font-size: 0.9em; + list-style-type: none; + white-space: nowrap; } + .nav-group-task a { + color: #888; } + +.main-content { + background-color: #fff; + border: 1px solid #e2e2e2; + margin-left: 246px; + position: absolute; + overflow: hidden; + padding-bottom: 20px; + top: 70px; + width: 734px; } + .main-content p, .main-content a, .main-content code, .main-content em, .main-content ul, .main-content table, .main-content blockquote { + margin-bottom: 1em; } + .main-content p { + line-height: 1.8em; } + .main-content section .section:first-child { + margin-top: 0; + padding-top: 0; } + .main-content section .task-group-section .task-group:first-of-type { + padding-top: 10px; } + .main-content section .task-group-section .task-group:first-of-type .section-name { + padding-top: 15px; } + .main-content section .heading:before { + content: ""; + display: block; + padding-top: 70px; + margin: -70px 0 0; } + +.section { + padding: 0 25px; } + +.highlight { + background-color: #eee; + padding: 10px 12px; + border: 1px solid #e2e2e2; + border-radius: 4px; + overflow-x: auto; } + +.declaration .highlight { + overflow-x: initial; + padding: 0 40px 40px 0; + margin-bottom: -25px; + background-color: transparent; + border: none; } + +.section-name { + margin: 0; + margin-left: 18px; } + +.task-group-section { + padding-left: 6px; + border-top: 1px solid #e2e2e2; } + +.task-group { + padding-top: 0px; } + +.task-name-container a[name]:before { + content: ""; + display: block; + padding-top: 70px; + margin: -70px 0 0; } + +.item { + padding-top: 8px; + width: 100%; + list-style-type: none; } + .item a[name]:before { + content: ""; + display: block; + padding-top: 70px; + margin: -70px 0 0; } + .item code { + background-color: transparent; + padding: 0; } + .item .token, .item .direct-link { + padding-left: 3px; + margin-left: 15px; + font-size: 11.9px; + transition: all 300ms; } + .item .token-open { + margin-left: 0px; } + .item .discouraged { + text-decoration: line-through; } + .item .declaration-note { + font-size: .85em; + color: gray; + font-style: italic; } + +.pointer-container { + border-bottom: 1px solid #e2e2e2; + left: -23px; + padding-bottom: 13px; + position: relative; + width: 110%; } + +.pointer { + background: #f9f9f9; + border-left: 1px solid #e2e2e2; + border-top: 1px solid #e2e2e2; + height: 12px; + left: 21px; + top: -7px; + -webkit-transform: rotate(45deg); + -moz-transform: rotate(45deg); + -o-transform: rotate(45deg); + transform: rotate(45deg); + position: absolute; + width: 12px; } + +.height-container { + display: none; + left: -25px; + padding: 0 25px; + position: relative; + width: 100%; + overflow: hidden; } + .height-container .section { + background: #f9f9f9; + border-bottom: 1px solid #e2e2e2; + left: -25px; + position: relative; + width: 100%; + padding-top: 10px; + padding-bottom: 5px; } + +.aside, .language { + padding: 6px 12px; + margin: 12px 0; + border-left: 5px solid #dddddd; + overflow-y: hidden; } + .aside .aside-title, .language .aside-title { + font-size: 9px; + letter-spacing: 2px; + text-transform: uppercase; + padding-bottom: 0; + margin: 0; + color: #aaa; + -webkit-user-select: none; } + .aside p:last-child, .language p:last-child { + margin-bottom: 0; } + +.language { + border-left: 5px solid #cde9f4; } + .language .aside-title { + color: #4b8afb; } + +.aside-warning, .aside-deprecated, .aside-unavailable { + border-left: 5px solid #ff6666; } + .aside-warning .aside-title, .aside-deprecated .aside-title, .aside-unavailable .aside-title { + color: #ff0000; } + +.graybox { + border-collapse: collapse; + width: 100%; } + .graybox p { + margin: 0; + word-break: break-word; + min-width: 50px; } + .graybox td { + border: 1px solid #e2e2e2; + padding: 5px 25px 5px 10px; + vertical-align: middle; } + .graybox tr td:first-of-type { + text-align: right; + padding: 7px; + vertical-align: top; + word-break: normal; + width: 40px; } + +.slightly-smaller { + font-size: 0.9em; } + +#footer { + position: relative; + top: 10px; + bottom: 0px; + margin-left: 25px; } + #footer p { + margin: 0; + color: #aaa; + font-size: 0.8em; } + +html.dash header, html.dash #breadcrumbs, html.dash .sidebar { + display: none; } +html.dash .main-content { + width: 980px; + margin-left: 0; + border: none; + width: 100%; + top: 0; + padding-bottom: 0; } +html.dash .height-container { + display: block; } +html.dash .item .token { + margin-left: 0; } +html.dash .content-wrapper { + width: auto; } +html.dash #footer { + position: static; } diff --git a/docs/docsets/.docset/Contents/Info.plist b/docs/docsets/.docset/Contents/Info.plist new file mode 100644 index 0000000..61863ec --- /dev/null +++ b/docs/docsets/.docset/Contents/Info.plist @@ -0,0 +1,20 @@ + + + + + CFBundleIdentifier + com.jazzy. + CFBundleName + + DocSetPlatformFamily + + isDashDocset + + dashIndexFilePath + index.html + isJavaScriptEnabled + + DashDocSetFamily + dashtoc + + diff --git a/docs/docsets/.docset/Contents/Resources/Documents/Structs.html b/docs/docsets/.docset/Contents/Resources/Documents/Structs.html new file mode 100644 index 0000000..97523ff --- /dev/null +++ b/docs/docsets/.docset/Contents/Resources/Documents/Structs.html @@ -0,0 +1,113 @@ + + + + Structures Reference + + + + + + + + + + +
+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+

Structures

+

The following structures are available globally.

+ +
+
+
+
    +
  • +
    + + + + BInt + +
    +
    +
    +
    +
    +
    +

    A signed integer of unbounded size. +A BInt value is represented with magnitude and sign. +The magnitude is an array of unsigned 64 bit integers (a.k.a. Limbs). +The sign is a boolean value, true means value < 0, false means value >= 0 +The representation is little-endian, least significant Limb has index 0. +The representation is minimal, there is no leading zero Limbs. +The exception is that the value 0 is represented as a single 64 bit zero Limb and sign false

    + + See more +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public struct BInt : CustomStringConvertible, Equatable, Hashable
    + +
    +
    +
    +
    +
    • +
+
+
+
+ +
+
+ + + diff --git a/docs/docsets/.docset/Contents/Resources/Documents/Structs/BInt.html b/docs/docsets/.docset/Contents/Resources/Documents/Structs/BInt.html new file mode 100644 index 0000000..e12deab --- /dev/null +++ b/docs/docsets/.docset/Contents/Resources/Documents/Structs/BInt.html @@ -0,0 +1,5734 @@ + + + + BInt Structure Reference + + + + + + + + + + +
+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+

BInt

+
+
+ 
public struct BInt : CustomStringConvertible, Equatable, Hashable
+ +
+
+

A signed integer of unbounded size. +A BInt value is represented with magnitude and sign. +The magnitude is an array of unsigned 64 bit integers (a.k.a. Limbs). +The sign is a boolean value, true means value < 0, false means value >= 0 +The representation is little-endian, least significant Limb has index 0. +The representation is minimal, there is no leading zero Limbs. +The exception is that the value 0 is represented as a single 64 bit zero Limb and sign false

+ +
+
+
+
+ + + +

Constants

+ +
+
    +
  • +
    + + + + ZERO + +
    +
    +
    +
    +
    +
    +

    BInt(0)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let ZERO: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + ONE + +
    +
    +
    +
    +
    +
    +

    BInt(1)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let ONE: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + TWO + +
    +
    +
    +
    +
    +
    +

    BInt(2)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let TWO: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + THREE + +
    +
    +
    +
    +
    +
    +

    BInt(3)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let THREE: BInt
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + TEN + +
    +
    +
    +
    +
    +
    +

    BInt(10)

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static let TEN: BInt
    + +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Initializers

+ +
+
    +
  • +
    + + + + init(_:_:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from magnitude and sign

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(_ magnitude: Limbs, _ isNegative: Bool = false)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + magnitude + + +
    +

    magnitude of value

    +
    +
    + + isNegative + + +
    +

    true means negative value, false means 0 or positive value, default is false

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(_:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from an Int value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(_ x: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Int value

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(_:radix:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from a String value and radix

    + +

    Examples:

    + +
      +
    • BInt(90abcdef, radix = 16)
      • +
    • BInt(111110010, radix = 2)
      • +
    • BInt(1cdefghijk44, radix = 26)
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init?(_ x: String, radix: Int = 10)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    String value to be converted

    +
    +
    + + radix + + +
    +

    Radix of x, from 2 to 36 inclusive, default is 10

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(bitWidth:) + +
    +
    +
    +
    +
    +
    +

    Constructs a random BInt with a specified number of bits

    +
    +

    Precondition

    + bitWidth is positive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(bitWidth: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + bitWidth + + +
    +

    Number of bits

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(magnitude:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from a big-endian magnitude byte array representation

    +
    +

    Precondition

    + Byte array is not empty + +
    + +

    Examples:

    + +
      +
    • The byte array [1, 0, 0] represents BInt value 65536
      • +
    • The byte array [128, 0] represents BInt value 32768
      • +
    • The byte array [255, 127] represents BInt value 65407
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(magnitude x: Bytes)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Magnitude big-endian byte array

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + init(signed:) + +
    +
    +
    +
    +
    +
    +

    Constructs a BInt from a big-endian 2’s complement byte array representation

    +
    +

    Precondition

    + Byte array is not empty + +
    + +

    Examples:

    + +
      +
    • The byte array [1, 0, 0] represents BInt value 65536
      • +
    • The byte array [128, 0] represents BInt value -32768
      • +
    • The byte array [255, 127] represents BInt value -129
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public init(signed x: Bytes)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    2’s complement big-endian byte array

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Stored properties

+ +
+
    +
  • +
    + + + + isNegative + +
    +
    +
    +
    +
    +
    +

    The sign, true if self < 0, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public internal(set) var isNegative: Bool
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + magnitude + +
    +
    +
    +
    +
    +
    +

    The magnitude limb array

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public internal(set) var magnitude: Limbs
    + +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Computed properties

+ +
+
    +
  • +
    + + + + bitWidth + +
    +
    +
    +
    +
    +
    +

    The number of bits in the magnitude of self. 0 if self = 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var bitWidth: Int { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + description + +
    +
    +
    +
    +
    +
    +

    Base 10 string value of self

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var description: String { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isEven + +
    +
    +
    +
    +
    +
    +

    Is true if self is even, false if self is odd

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isEven: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isNotZero + +
    +
    +
    +
    +
    +
    +

    Is false if self = 0, true otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isNotZero: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isOdd + +
    +
    +
    +
    +
    +
    +

    Is true if self is odd, false if self is even

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isOdd: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isOne + +
    +
    +
    +
    +
    +
    +

    Is true if self = 1, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isOne: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isPositive + +
    +
    +
    +
    +
    +
    +

    Is true if self > 0, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isPositive: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + isZero + +
    +
    +
    +
    +
    +
    +

    Is true if self = 0, false otherwise

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var isZero: Bool { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + leadingZeroBitCount + +
    +
    +
    +
    +
    +
    +

    The number of leading zero bits in the magnitude of self. 0 if self = 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var leadingZeroBitCount: Int { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + signum + +
    +
    +
    +
    +
    +
    +

    Is 0 if self = 0, 1 if self > 0, and -1 if self < 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var signum: Int { get }
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + trailingZeroBitCount + +
    +
    +
    +
    +
    +
    +

    The number of trailing zero bits in the magnitude of self. 0 if self = 0

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public var trailingZeroBitCount: Int { get }
    + +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Conversion functions to Int, String, and Bytes

+ +
+
    +
  • +
    + + + + asInt() + +
    +
    +
    +
    +
    +
    +

    self as an Int

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asInt() -> Int?
    + +
    +
    +
    +

    Return Value

    +

    self as an Int or nil if self is not representable as an Int

    +
    +
    +
    +
    • +
  • +
    + + + + asMagnitudeBytes() + +
    +
    +
    +
    +
    +
    +

    Byte array representation of magnitude value

    + +

    Examples:

    + +
      +
    • BInt(1).asMagnitudeBytes() = [1]
      • +
    • BInt(-1).asMagnitudeBytes() = [1]
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asMagnitudeBytes() -> Bytes
    + +
    +
    +
    +

    Return Value

    +

    Minimal big-endian magnitude byte array representation

    +
    +
    +
    +
    • +
  • +
    + + + + asSignedBytes() + +
    +
    +
    +
    +
    +
    +

    Byte array representation of 2’s complement value

    + +

    Examples:

    + +
      +
    • BInt(1).asSignedBytes() = [1]
      • +
    • BInt(-1).asSignedBytes() = [255]
      • +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asSignedBytes() -> Bytes
    + +
    +
    +
    +

    Return Value

    +

    Minimal big-endian 2’s complement byte array representation

    +
    +
    +
    +
    • +
  • + +
    +
    +
    +
    +
    +

    self as a String with a given radix

    +
    +

    Precondition

    + Radix between 2 and 36 inclusive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func asString(radix: Int = 10, uppercase: Bool = false) -> String
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + radix + + +
    +

    Radix from 2 to 36 inclusive

    +
    +
    + + uppercase + + +
    +

    true to use uppercase letters, false to use lowercase letters, default is false

    +
    +
    +
    +
    +

    Return Value

    +

    self as a String in the given radix

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Bit operation functions

+ +
+
    +
  • +
    + + + + &(_:_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise and operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func & (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First value

    +
    +
    + + y + + +
    +

    Second value

    +
    +
    +
    +
    +

    Return Value

    +

    BInt(signed: bx & by) where + +

      +
    • bx = x.asSignedBytes()
      • +
    • by = y.asSignedBytes()
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + &=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x & y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func &= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand parameter

    +
    +
    + + y + + +
    +

    Right hand parameter

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + |(_:_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise or operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func | (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First value

    +
    +
    + + y + + +
    +

    Second value

    +
    +
    +
    +
    +

    Return Value

    +

    BInt(signed: bx | by) where + +

      +
    • bx = x.asSignedBytes()
      • +
    • by = y.asSignedBytes()
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + |=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x | y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func |= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand parameter

    +
    +
    + + y + + +
    +

    Right hand parameter

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + ^(_:_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise xor operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func ^ (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First value

    +
    +
    + + y + + +
    +

    Second value

    +
    +
    +
    +
    +

    Return Value

    +

    BInt(signed: bx ^ by) where + +

      +
    • bx = x.asSignedBytes()
      • +
    • by = y.asSignedBytes()
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + ^=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x ^ y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func ^= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand parameter

    +
    +
    + + y + + +
    +

    Right hand parameter

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + ~(_:) + +
    +
    +
    +
    +
    +
    +

    Bitwise not operator

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public prefix static func ~ (x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    BInt value

    +
    +
    +
    +
    +

    Return Value

    +

    -x - 1

    +
    +
    +
    +
    • +
  • +
    + + + + clearBit(_:) + +
    +
    +
    +
    +
    +
    +

    Clear a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func clearBit(_ n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + flipBit(_:) + +
    +
    +
    +
    +
    +
    +

    Invert a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func flipBit(_ n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + setBit(_:) + +
    +
    +
    +
    +
    +
    +

    Set a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func setBit(_ n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + testBit(_:) + +
    +
    +
    +
    +
    +
    +

    Test a specified bit

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func testBit(_ n: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    Bit number

    +
    +
    +
    +
    +

    Return Value

    +

    true if bit is set, false otherwise

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Addition functions

+ +
+
    +
  • +
    + + + + +(_:) + +
    +
    +
    +
    +
    +
    +

    Prefix plus

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public prefix static func + (x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    BInt value

    +
    +
    +
    +
    +

    Return Value

    +

    x

    +
    +
    +
    +
    • +
  • +
    + + + + +(_:_:) + +
    +
    +
    +
    +
    +
    +

    Addition

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func + (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First addend

    +
    +
    + + y + + +
    +

    Second addend

    +
    +
    +
    +
    +

    Return Value

    +

    x + y

    +
    +
    +
    +
    • +
  • +
    + + + + +(_:_:) + +
    +
    +
    +
    +
    +
    +

    Addition

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func + (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First addend

    +
    +
    + + y + + +
    +

    Second addend

    +
    +
    +
    +
    +

    Return Value

    +

    x + y

    +
    +
    +
    +
    • +
  • +
    + + + + +(_:_:) + +
    +
    +
    +
    +
    +
    +

    Addition

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func + (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First addend

    +
    +
    + + y + + +
    +

    Second addend

    +
    +
    +
    +
    +

    Return Value

    +

    x + y

    +
    +
    +
    +
    • +
  • +
    + + + + +=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x + y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func += (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand addend

    +
    +
    + + y + + +
    +

    Right hand addend

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + +=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x + y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func += (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand addend

    +
    +
    + + y + + +
    +

    Right hand addend

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Negation functions

+ +
+
    +
  • +
    + + + + negate() + +
    +
    +
    +
    +
    +
    +

    Negates self

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public mutating func negate()
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + -(_:) + +
    +
    +
    +
    +
    +
    +

    Negation

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public prefix static func - (x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    +
    +
    +

    Return Value

    +

    -x

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Subtraction functions

+ +
+
    +
  • +
    + + + + -(_:_:) + +
    +
    +
    +
    +
    +
    +

    Subtraction

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func - (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Minuend

    +
    +
    + + y + + +
    +

    Subtrahend

    +
    +
    +
    +
    +

    Return Value

    +

    x - y

    +
    +
    +
    +
    • +
  • +
    + + + + -(_:_:) + +
    +
    +
    +
    +
    +
    +

    Subtraction

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func - (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Minuend

    +
    +
    + + y + + +
    +

    Subtrahend

    +
    +
    +
    +
    +

    Return Value

    +

    x - y

    +
    +
    +
    +
    • +
  • +
    + + + + -(_:_:) + +
    +
    +
    +
    +
    +
    +

    Subtraction

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func - (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Minuend

    +
    +
    + + y + + +
    +

    Subtrahend

    +
    +
    +
    +
    +

    Return Value

    +

    x - y

    +
    +
    +
    +
    • +
  • +
    + + + + -=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x - y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func -= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand minuend

    +
    +
    + + y + + +
    +

    Right hand subtrahend

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + -=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x - y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func -= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand minuend

    +
    +
    + + y + + +
    +

    Right hand subtrahend

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Multiplication functions

+ +
+
    +
  • +
    + + + + *(_:_:) + +
    +
    +
    +
    +
    +
    +

    Multiplication

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func * (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Multiplier

    +
    +
    + + y + + +
    +

    Multiplicand

    +
    +
    +
    +
    +

    Return Value

    +

    x * y

    +
    +
    +
    +
    • +
  • +
    + + + + *(_:_:) + +
    +
    +
    +
    +
    +
    +

    Multiplication

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func * (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Multiplier

    +
    +
    + + y + + +
    +

    Multiplicand

    +
    +
    +
    +
    +

    Return Value

    +

    x * y

    +
    +
    +
    +
    • +
  • +
    + + + + *(_:_:) + +
    +
    +
    +
    +
    +
    +

    Multiplication

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func * (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Multiplier

    +
    +
    + + y + + +
    +

    Multiplicand

    +
    +
    +
    +
    +

    Return Value

    +

    x * y

    +
    +
    +
    +
    • +
  • +
    + + + + *=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x * y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func *= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand multiplier

    +
    +
    + + y + + +
    +

    Right hand multiplicand

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + *=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x * y

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func *= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand multiplier

    +
    +
    + + y + + +
    +

    Right hand multiplicand

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Division functions

+ +
+
    +
  • + +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func quotientAndRemainder(dividingBy x: BInt) -> (quotient: BInt, remainder: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    Quotient and remainder of self / x

    +
    +
    +
    +
    • +
  • + +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func quotientAndRemainder(dividingBy x: BInt, _ quotient: inout BInt, _ remainder: inout BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + + + + + +
    + + x + + +
    +

    Divisor

    +
    +
    + + quotient + + +
    +

    Set to the quotient of self / x

    +
    +
    + + remainder + + +
    +

    Set to the remainder of self / x

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + /(_:_:) + +
    +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func / (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x / y

    +
    +
    +
    +
    • +
  • +
    + + + + /(_:_:) + +
    +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func / (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x / y

    +
    +
    +
    +
    • +
  • +
    + + + + /(_:_:) + +
    +
    +
    +
    +
    +
    +

    Division

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func / (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x / y

    +
    +
    +
    +
    • +
  • +
    + + + + /=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x / y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func /= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand dividend

    +
    +
    + + y + + +
    +

    Right hand divisor

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + /=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x / y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func /= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Left hand dividend

    +
    +
    + + y + + +
    +

    Right hand divisor

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Remainder and modulus functions

+ +
+
    +
  • +
    + + + + %(_:_:) + +
    +
    +
    +
    +
    +
    +

    Remainder

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func % (x: BInt, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x % y

    +
    +
    +
    +
    • +
  • +
    + + + + %(_:_:) + +
    +
    +
    +
    +
    +
    +

    Remainder

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func % (x: Int, y: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x % y

    +
    +
    +
    +
    • +
  • +
    + + + + %(_:_:) + +
    +
    +
    +
    +
    +
    +

    Remainder

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func % (x: BInt, y: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    x % y

    +
    +
    +
    +
    • +
  • +
    + + + + %=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x % y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func %= (x: inout BInt, y: BInt)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + %=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x % y

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func %= (x: inout BInt, y: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Dividend

    +
    +
    + + y + + +
    +

    Divisor

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + mod(_:) + +
    +
    +
    +
    +
    +
    +

    Modulus

    +
    +

    Precondition

    + Divisor is not zero + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func mod(_ x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Divisor

    +
    +
    +
    +
    +

    Return Value

    +

    self mod x, a non-negative value

    +
    +
    +
    +
    • +
  • +
    + + + + modInverse(_:) + +
    +
    +
    +
    +
    +
    +

    Inverse modulus

    +
    +

    Precondition

    + self and modulus are coprime + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func modInverse(_ m: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + m + + +
    +

    Modulus

    +
    +
    +
    +
    +

    Return Value

    +

    If self and m are coprime, x such that (self * x) mod m = 1

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Exponentiation functions

+ +
+
    +
  • +
    + + + + **(_:_:) + +
    +
    +
    +
    +
    +
    +

    Exponentiation

    +
    +

    Precondition

    + Exponent is non-negative + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func ** (a: BInt, x: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + a + + +
    +

    Operand

    +
    +
    + + x + + +
    +

    Non-negative exponent

    +
    +
    +
    +
    +

    Return Value

    +

    a^x

    +
    +
    +
    +
    • +
  • +
    + + + + expMod(_:_:) + +
    +
    +
    +
    +
    +
    +

    Modular exponentiation

    +
    +

    Precondition

    + Modulus is positive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func expMod(_ x: BInt, _ m: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    The exponent

    +
    +
    + + m + + +
    +

    The modulus, a positive number

    +
    +
    +
    +
    +

    Return Value

    +

    (self^x) mod m for positive x, ((self^-x) mod m) modInverse m for negative x

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Comparison functions

+ +
+
    +
  • +
    + + + + ==(_:_:) + +
    +
    +
    +
    +
    +
    +

    Equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func == (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x = y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + ==(_:_:) + +
    +
    +
    +
    +
    +
    +

    Equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func == (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x = y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + ==(_:_:) + +
    +
    +
    +
    +
    +
    +

    Equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func == (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x = y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + !=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Not equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func != (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x != y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + !=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Not equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func != (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x != y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + !=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Not equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func != (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x != y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less then

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func < (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x < y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func < (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x < y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func < (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x < y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func > (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x > y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func > (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x > y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func > (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x > y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <= (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x <= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <= (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x <= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + <=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Less than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <= (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x <= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >= (x: BInt, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x >= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >= (x: Int, y: BInt) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x >= y, false otherwise

    +
    +
    +
    +
    • +
  • +
    + + + + >=(_:_:) + +
    +
    +
    +
    +
    +
    +

    Greater than or equal

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >= (x: BInt, y: Int) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    First operand

    +
    +
    + + y + + +
    +

    Second operand

    +
    +
    +
    +
    +

    Return Value

    +

    true if x >= y, false otherwise

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Shift functions

+ +
+
    +
  • +
    + + + + <<(_:_:) + +
    +
    +
    +
    +
    +
    +

    Logical left shift

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func << (x: BInt, n: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +

    Return Value

    +

      +
    • BInt(x.magnitude << n, x.isNegative) if n > 0
      • +
    • BInt(x.magnitude >> -n, x.isNegative) if n < 0
      • +
    • x if n = 0
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + <<=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x << n

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func <<= (x: inout BInt, n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +
    +
    • +
  • +
    + + + + >>(_:_:) + +
    +
    +
    +
    +
    +
    +

    Logical right shift

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >> (x: BInt, n: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +

    Return Value

    +

      +
    • BInt(x.magnitude >> n, x.isNegative) if n > 0
      • +
    • BInt(x.magnitude << -n, x.isNegative) if n < 0
      • +
    • x if n = 0
      • +

    +
    +
    +
    +
    • +
  • +
    + + + + >>=(_:_:) + +
    +
    +
    +
    +
    +
    +

    x = x >> n

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func >>= (x: inout BInt, n: Int)
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    + + n + + +
    +

    Shift count

    +
    +
    +
    +
    +
    +
    • +
+
+
+
+ + + +

Prime number functions

+ +
+
    +
  • +
    + + + + isProbablyPrime(_:) + +
    +
    +
    +
    +
    +
    +

    Checks whether self is prime using the Miller-Rabin algorithm

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func isProbablyPrime(_ p: Int = 30) -> Bool
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + p + + +
    +

    If true is returned, self is prime with probability > 1-½^p

    +
    +
    +
    +
    +

    Return Value

    +

    true if self is probably prime, false if self is definitely not prime

    +
    +
    +
    +
    • +
  • +
    + + + + probablePrime(_:_:) + +
    +
    +
    +
    +
    +
    +

    A probable prime number with a given bitwidth

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public static func probablePrime(_ bitWidth: Int, _ p: Int = 30) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + + + + + +
    + + bitWidth + + +
    +

    The bitWidth

    +
    +
    + + p + + +
    +

    The returned number is prime with probability > 1-½^p, default value is 30

    +
    +
    +
    +
    +

    Return Value

    +

    A prime number with the specified bitwidth and probability

    +
    +
    +
    +
    • +
+
+
+
+ + + +

Miscellaneous functions

+ +
+
    +
  • +
    + + + + abs() + +
    +
    +
    +
    +
    +
    +

    Absolute value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func abs() -> BInt
    + +
    +
    +
    +

    Return Value

    +

    The absolute value of self

    +
    +
    +
    +
    • +
  • +
    + + + + gcd(_:) + +
    +
    +
    +
    +
    +
    +

    Greatest common divisor

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func gcd(_ x: BInt) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + x + + +
    +

    Operand

    +
    +
    +
    +
    +

    Return Value

    +

    Greatest common divisor of magnitude of self and magnitude of x

    +
    +
    +
    +
    • +
  • +
    + + + + jacobiSymbol(_:) + +
    +
    +
    +
    +
    +
    +

    Jacobi symbol

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func jacobiSymbol(_ m: BInt) -> Int
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + m + + +
    +

    An integer value

    +
    +
    +
    +
    +

    Return Value

    +

    The Jacobi symbol of self and m, or 0 if it does not exist

    +
    +
    +
    +
    • +
  • +
    + + + + randomLessThan() + +
    +
    +
    +
    +
    +
    +

    Random value

    +
    +

    Precondition

    + self is positive + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func randomLessThan() -> BInt
    + +
    +
    +
    +

    Return Value

    +

    A random value < absolute value of self

    +
    +
    +
    +
    • +
  • +
    + + + + root(_:) + +
    +
    +
    +
    +
    +
    +

    n'th root of a non-negative number

    +
    +

    Precondition

    +
      +
    • self is non-negative
      • +
    • n is positive
      • +
    + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func root(_ n: Int) -> BInt
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + n + + +
    +

    The root

    +
    +
    +
    +
    +

    Return Value

    +

    Largest x such that x^n <= self

    +
    +
    +
    +
    • +
  • +
    + + + + sqrt() + +
    +
    +
    +
    +
    +
    +

    Square root of a non-negative number

    +
    +

    Precondition

    + self is non-negative + +
    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func sqrt() -> BInt
    + +
    +
    +
    +

    Return Value

    +

    Largest x such that x^2 <= self

    +
    +
    +
    +
    • +
  • +
    + + + + sqrtMod(_:) + +
    +
    +
    +
    +
    +
    +

    Square root modulo a prime number

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public func sqrtMod(_ p: BInt) -> BInt?
    + +
    +
    +
    +

    Parameters

    + + + + + + + +
    + + p + + +
    +

    An odd prime number

    +
    +
    +
    +
    +

    Return Value

    +

    x, such that x^2 mod p = self, or nil if no such x exists

    +
    +
    +
    +
    • +
+
+
+
+ +
+
+ + + diff --git a/docs/docsets/.docset/Contents/Resources/Documents/Typealiases.html b/docs/docsets/.docset/Contents/Resources/Documents/Typealiases.html new file mode 100644 index 0000000..2a06fbe --- /dev/null +++ b/docs/docsets/.docset/Contents/Resources/Documents/Typealiases.html @@ -0,0 +1,187 @@ + + + + Type Aliases Reference + + + + + + + + + + +
+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+

Type Aliases

+

The following type aliases are available globally.

+ +
+
+
+
    +
  • +
    + + + + Byte + +
    +
    +
    +
    +
    +
    +

    Unsigned 8 bit value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Byte = UInt8
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + Bytes + +
    +
    +
    +
    +
    +
    +

    Array of unsigned 8 bit values

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Bytes = [UInt8]
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + Limb + +
    +
    +
    +
    +
    +
    +

    Unsigned 64 bit value

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Limb = UInt64
    + +
    +
    +
    +
    +
    • +
  • +
    + + + + Limbs + +
    +
    +
    +
    +
    +
    +

    Array of unsigned 64 bit values

    + +
    +
    +

    Declaration

    +
    +

    Swift

    + 
    public typealias Limbs = [UInt64]
    + +
    +
    +
    +
    +
    • +
+
+
+
+ +
+
+ + + diff --git a/docs/docsets/.docset/Contents/Resources/Documents/css/highlight.css b/docs/docsets/.docset/Contents/Resources/Documents/css/highlight.css new file mode 100644 index 0000000..d0db0e1 --- /dev/null +++ b/docs/docsets/.docset/Contents/Resources/Documents/css/highlight.css @@ -0,0 +1,200 @@ +/* Credit to https://gist.github.com/wataru420/2048287 */ +.highlight { + /* Comment */ + /* Error */ + /* Keyword */ + /* Operator */ + /* Comment.Multiline */ + /* Comment.Preproc */ + /* Comment.Single */ + /* Comment.Special */ + /* Generic.Deleted */ + /* Generic.Deleted.Specific */ + /* Generic.Emph */ + /* Generic.Error */ + /* Generic.Heading */ + /* Generic.Inserted */ + /* Generic.Inserted.Specific */ + /* Generic.Output */ + /* Generic.Prompt */ + /* Generic.Strong */ + /* Generic.Subheading */ + /* Generic.Traceback */ + /* Keyword.Constant */ + /* Keyword.Declaration */ + /* Keyword.Pseudo */ + /* Keyword.Reserved */ + /* Keyword.Type */ + /* Literal.Number */ + /* Literal.String */ + /* Name.Attribute */ + /* Name.Builtin */ + /* Name.Class */ + /* Name.Constant */ + /* Name.Entity */ + /* Name.Exception */ + /* Name.Function */ + /* Name.Namespace */ + /* Name.Tag */ + /* Name.Variable */ + /* Operator.Word */ + /* Text.Whitespace */ + /* Literal.Number.Float */ + /* Literal.Number.Hex */ + /* Literal.Number.Integer */ + /* Literal.Number.Oct */ + /* Literal.String.Backtick */ + /* Literal.String.Char */ + /* Literal.String.Doc */ + /* Literal.String.Double */ + /* Literal.String.Escape */ + /* Literal.String.Heredoc */ + /* Literal.String.Interpol */ + /* Literal.String.Other */ + /* Literal.String.Regex */ + /* Literal.String.Single */ + /* Literal.String.Symbol */ + /* Name.Builtin.Pseudo */ + /* Name.Variable.Class */ + /* Name.Variable.Global */ + /* Name.Variable.Instance */ + /* Literal.Number.Integer.Long */ } + .highlight .c { + color: #999988; + font-style: italic; } + .highlight .err { + color: #a61717; + background-color: #e3d2d2; } + .highlight .k { + color: #000000; + font-weight: bold; } + .highlight .o { + color: #000000; + font-weight: bold; } + .highlight .cm { + color: #999988; + font-style: italic; } + .highlight .cp { + color: #999999; + font-weight: bold; } + .highlight .c1 { + color: #999988; + font-style: italic; } + .highlight .cs { + color: #999999; + font-weight: bold; + font-style: italic; } + .highlight .gd { + color: #000000; + background-color: #ffdddd; } + .highlight .gd .x { + color: #000000; + background-color: #ffaaaa; } + .highlight .ge { + color: #000000; + font-style: italic; } + .highlight .gr { + color: #aa0000; } + .highlight .gh { + color: #999999; } + .highlight .gi { + color: #000000; + background-color: #ddffdd; } + .highlight .gi .x { + color: #000000; + background-color: #aaffaa; } + .highlight .go { + color: #888888; } + .highlight .gp { + color: #555555; } + .highlight .gs { + font-weight: bold; } + .highlight .gu { + color: #aaaaaa; } + .highlight .gt { + color: #aa0000; } + .highlight .kc { + color: #000000; + font-weight: bold; } + .highlight .kd { + color: #000000; + font-weight: bold; } + .highlight .kp { + color: #000000; + font-weight: bold; } + .highlight .kr { + color: #000000; + font-weight: bold; } + .highlight .kt { + color: #445588; } + .highlight .m { + color: #009999; } + .highlight .s { + color: #d14; } + .highlight .na { + color: #008080; } + .highlight .nb { + color: #0086B3; } + .highlight .nc { + color: #445588; + font-weight: bold; } + .highlight .no { + color: #008080; } + .highlight .ni { + color: #800080; } + .highlight .ne { + color: #990000; + font-weight: bold; } + .highlight .nf { + color: #990000; } + .highlight .nn { + color: #555555; } + .highlight .nt { + color: #000080; } + .highlight .nv { + color: #008080; } + .highlight .ow { + color: #000000; + font-weight: bold; } + .highlight .w { + color: #bbbbbb; } + .highlight .mf { + color: #009999; } + .highlight .mh { + color: #009999; } + .highlight .mi { + color: #009999; } + .highlight .mo { + color: #009999; } + .highlight .sb { + color: #d14; } + .highlight .sc { + color: #d14; } + .highlight .sd { + color: #d14; } + .highlight .s2 { + color: #d14; } + .highlight .se { + color: #d14; } + .highlight .sh { + color: #d14; } + .highlight .si { + color: #d14; } + .highlight .sx { + color: #d14; } + .highlight .sr { + color: #009926; } + .highlight .s1 { + color: #d14; } + .highlight .ss { + color: #990073; } + .highlight .bp { + color: #999999; } + .highlight .vc { + color: #008080; } + .highlight .vg { + color: #008080; } + .highlight .vi { + color: #008080; } + .highlight .il { + color: #009999; } diff --git a/docs/docsets/.docset/Contents/Resources/Documents/css/jazzy.css b/docs/docsets/.docset/Contents/Resources/Documents/css/jazzy.css new file mode 100644 index 0000000..0541d71 --- /dev/null +++ b/docs/docsets/.docset/Contents/Resources/Documents/css/jazzy.css @@ -0,0 +1,343 @@ +html, body, div, span, h1, h3, h4, p, a, code, em, img, ul, li, table, tbody, tr, td { + background: transparent; + border: 0; + margin: 0; + outline: 0; + padding: 0; + vertical-align: baseline; } + +body { + background-color: #f2f2f2; + font-family: Helvetica, freesans, Arial, sans-serif; + font-size: 14px; + -webkit-font-smoothing: subpixel-antialiased; + word-wrap: break-word; } + +h1, h2, h3 { + margin-top: 0.8em; + margin-bottom: 0.3em; + font-weight: 100; + color: black; } + +h1 { + font-size: 2.5em; } + +h2 { + font-size: 2em; + border-bottom: 1px solid #e2e2e2; } + +h4 { + font-size: 13px; + line-height: 1.5; + margin-top: 21px; } + +h5 { + font-size: 1.1em; } + +h6 { + font-size: 1.1em; + color: #777; } + +.section-name { + color: gray; + display: block; + font-family: Helvetica; + font-size: 22px; + font-weight: 100; + margin-bottom: 15px; } + +pre, code { + font: 0.95em Menlo, monospace; + color: #777; + word-wrap: normal; } + +p code, li code { + background-color: #eee; + padding: 2px 4px; + border-radius: 4px; } + +a { + color: #0088cc; + text-decoration: none; } + +ul { + padding-left: 15px; } + +li { + line-height: 1.8em; } + +img { + max-width: 100%; } + +blockquote { + margin-left: 0; + padding: 0 10px; + border-left: 4px solid #ccc; } + +.content-wrapper { + margin: 0 auto; + width: 980px; } + +header { + font-size: 0.85em; + line-height: 26px; + background-color: #414141; + position: fixed; + width: 100%; + z-index: 1; } + header img { + padding-right: 6px; + vertical-align: -4px; + height: 16px; } + header a { + color: #fff; } + header p { + float: left; + color: #999; } + header .header-right { + float: right; + margin-left: 16px; } + +#breadcrumbs { + background-color: #f2f2f2; + height: 27px; + padding-top: 17px; + position: fixed; + width: 100%; + z-index: 1; + margin-top: 26px; } + #breadcrumbs #carat { + height: 10px; + margin: 0 5px; } + +.sidebar { + background-color: #f9f9f9; + border: 1px solid #e2e2e2; + overflow-y: auto; + overflow-x: hidden; + position: fixed; + top: 70px; + bottom: 0; + width: 230px; + word-wrap: normal; } + +.nav-groups { + list-style-type: none; + background: #fff; + padding-left: 0; } + +.nav-group-name { + border-bottom: 1px solid #e2e2e2; + font-size: 1.1em; + font-weight: 100; + padding: 15px 0 15px 20px; } + .nav-group-name > a { + color: #333; } + +.nav-group-tasks { + margin-top: 5px; } + +.nav-group-task { + font-size: 0.9em; + list-style-type: none; + white-space: nowrap; } + .nav-group-task a { + color: #888; } + +.main-content { + background-color: #fff; + border: 1px solid #e2e2e2; + margin-left: 246px; + position: absolute; + overflow: hidden; + padding-bottom: 20px; + top: 70px; + width: 734px; } + .main-content p, .main-content a, .main-content code, .main-content em, .main-content ul, .main-content table, .main-content blockquote { + margin-bottom: 1em; } + .main-content p { + line-height: 1.8em; } + .main-content section .section:first-child { + margin-top: 0; + padding-top: 0; } + .main-content section .task-group-section .task-group:first-of-type { + padding-top: 10px; } + .main-content section .task-group-section .task-group:first-of-type .section-name { + padding-top: 15px; } + .main-content section .heading:before { + content: ""; + display: block; + padding-top: 70px; + margin: -70px 0 0; } + +.section { + padding: 0 25px; } + +.highlight { + background-color: #eee; + padding: 10px 12px; + border: 1px solid #e2e2e2; + border-radius: 4px; + overflow-x: auto; } + +.declaration .highlight { + overflow-x: initial; + padding: 0 40px 40px 0; + margin-bottom: -25px; + background-color: transparent; + border: none; } + +.section-name { + margin: 0; + margin-left: 18px; } + +.task-group-section { + padding-left: 6px; + border-top: 1px solid #e2e2e2; } + +.task-group { + padding-top: 0px; } + +.task-name-container a[name]:before { + content: ""; + display: block; + padding-top: 70px; + margin: -70px 0 0; } + +.item { + padding-top: 8px; + width: 100%; + list-style-type: none; } + .item a[name]:before { + content: ""; + display: block; + padding-top: 70px; + margin: -70px 0 0; } + .item code { + background-color: transparent; + padding: 0; } + .item .token, .item .direct-link { + padding-left: 3px; + margin-left: 15px; + font-size: 11.9px; + transition: all 300ms; } + .item .token-open { + margin-left: 0px; } + .item .discouraged { + text-decoration: line-through; } + .item .declaration-note { + font-size: .85em; + color: gray; + font-style: italic; } + +.pointer-container { + border-bottom: 1px solid #e2e2e2; + left: -23px; + padding-bottom: 13px; + position: relative; + width: 110%; } + +.pointer { + background: #f9f9f9; + border-left: 1px solid #e2e2e2; + border-top: 1px solid #e2e2e2; + height: 12px; + left: 21px; + top: -7px; + -webkit-transform: rotate(45deg); + -moz-transform: rotate(45deg); + -o-transform: rotate(45deg); + transform: rotate(45deg); + position: absolute; + width: 12px; } + +.height-container { + display: none; + left: -25px; + padding: 0 25px; + position: relative; + width: 100%; + overflow: hidden; } + .height-container .section { + background: #f9f9f9; + border-bottom: 1px solid #e2e2e2; + left: -25px; + position: relative; + width: 100%; + padding-top: 10px; + padding-bottom: 5px; } + +.aside, .language { + padding: 6px 12px; + margin: 12px 0; + border-left: 5px solid #dddddd; + overflow-y: hidden; } + .aside .aside-title, .language .aside-title { + font-size: 9px; + letter-spacing: 2px; + text-transform: uppercase; + padding-bottom: 0; + margin: 0; + color: #aaa; + -webkit-user-select: none; } + .aside p:last-child, .language p:last-child { + margin-bottom: 0; } + +.language { + border-left: 5px solid #cde9f4; } + .language .aside-title { + color: #4b8afb; } + +.aside-warning, .aside-deprecated, .aside-unavailable { + border-left: 5px solid #ff6666; } + .aside-warning .aside-title, .aside-deprecated .aside-title, .aside-unavailable .aside-title { + color: #ff0000; } + +.graybox { + border-collapse: collapse; + width: 100%; } + .graybox p { + margin: 0; + word-break: break-word; + min-width: 50px; } + .graybox td { + border: 1px solid #e2e2e2; + padding: 5px 25px 5px 10px; + vertical-align: middle; } + .graybox tr td:first-of-type { + text-align: right; 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+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+ +

Description

+ +

The BigInt package provides arbitrary-precision integer arithmetic in Swift. +It contains functionality comparable to that of the Java BigInteger class.

+ +

BigInt requires Swift 5.0.

+ +

Usage

+ +

Using Swift Package Manager, in your projects Package.swift file add a dependency like

+
dependencies: [
+// Dependencies declare other packages that this package depends on.
+.package(url: "https://github.com/leif-ibsen/BigInt", from: "1.0.0"),
+],
+
+ +

References

+ +

Algorithms from the following books have been used in the implementation. +There are references in the source code where appropriate.

+ +
    +
  • Donald E. Knuth: Seminumerical Algorithms. Addison-Wesley 1971
    • +
  • Crandall and Pomerance: Prime Numbers - A Computational Perspective. Second Edition, Springer 2005
    • +
  • Henry S. Warren, Jr: Hacker’s Delight. Second Edition, Addison-Wesley 2013
    • +
  • Menezes, Oorschot, Vanstone: Handbook of Applied Cryptography. CRC Press 1996
    • +
+ +

Acknowledgement

+ +

The BitSieve class used in the implementation is a translation to Swift of the corresponding class from Java BigInteger. +The GCD algorithm and the Karatsuba and ToomCook multiplication algorithms are modelled after the corresponding algorithms in Java BigInteger.

+ +
+
+ +
+
+ + + diff --git a/docs/docsets/.docset/Contents/Resources/Documents/js/jazzy.js b/docs/docsets/.docset/Contents/Resources/Documents/js/jazzy.js new file mode 100755 index 0000000..c31dc05 --- /dev/null +++ b/docs/docsets/.docset/Contents/Resources/Documents/js/jazzy.js @@ -0,0 +1,59 @@ +window.jazzy = {'docset': false} +if (typeof window.dash != 'undefined') { + document.documentElement.className += ' dash' + window.jazzy.docset = true +} +if (navigator.userAgent.match(/xcode/i)) { + document.documentElement.className += ' xcode' + window.jazzy.docset = true +} + +function toggleItem($link, $content) { + var animationDuration = 300; + $link.toggleClass('token-open'); + $content.slideToggle(animationDuration); +} + +function itemLinkToContent($link) { + return $link.parent().parent().next(); +} + +// On doc load + hash-change, open any targetted item +function openCurrentItemIfClosed() { + if (window.jazzy.docset) { + return; + } + var $link = $(`.token[href="${location.hash}"]`); + 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+
+

Docs (100% documented)

+
+
+
+ +
+
+ +
+
+
+ +

Description

+ +

The BigInt package provides arbitrary-precision integer arithmetic in Swift. +It contains functionality comparable to that of the Java BigInteger class.

+ +

BigInt requires Swift 5.0.

+ +

Usage

+ +

Using Swift Package Manager, in your projects Package.swift file add a dependency like

+
dependencies: [
+// Dependencies declare other packages that this package depends on.
+.package(url: "https://github.com/leif-ibsen/BigInt", from: "1.0.0"),
+],
+
+ +

References

+ +

Algorithms from the following books have been used in the implementation. +There are references in the source code where appropriate.

+ +
    +
  • Donald E. Knuth: Seminumerical Algorithms. Addison-Wesley 1971
    • +
  • Crandall and Pomerance: Prime Numbers - A Computational Perspective. Second Edition, Springer 2005
    • +
  • Henry S. Warren, Jr: Hacker’s Delight. Second Edition, Addison-Wesley 2013
    • +
  • Menezes, Oorschot, Vanstone: Handbook of Applied Cryptography. CRC Press 1996
    • +
+ +

Acknowledgement

+ +

The BitSieve class used in the implementation is a translation to Swift of the corresponding class from Java BigInteger. +The GCD algorithm and the Karatsuba and ToomCook multiplication algorithms are modelled after the corresponding algorithms in Java BigInteger.

+ +
+
+ +
+
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