Signed-digit multi-comb ecmult_gen algorithm

This introduces the signed-digit multi-comb multiplication algorithm
for constant-time G multiplications (ecmult_gen). It is based on
section 3.3 of "Fast and compact elliptic-curve cryptography" by
Mike Hamburg (see https://eprint.iacr.org/2012/309).

Original implementation by Peter Dettman, with changes by Pieter Wuille
to use scalars for recoding, and additional comments.

src/ecmult_gen.h

@@ -1,5 +1,5 @@
 /***********************************************************************
- * Copyright (c) 2013, 2014 Pieter Wuille                              *
+ * Copyright (c) Pieter Wuille, Peter Dettman                          *
  * Distributed under the MIT software license, see the accompanying    *
  * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
  ***********************************************************************/
@@ -10,31 +10,105 @@
 #include "scalar.h"
 #include "group.h"
 
-#ifndef ECMULT_GEN_PREC_BITS
-#  define ECMULT_GEN_PREC_BITS 4
-#  ifdef DEBUG_CONFIG
-#     pragma message DEBUG_CONFIG_MSG("ECMULT_GEN_PREC_BITS undefined, assuming default value")
+
+/* Configuration parameters for the signed-digit multi-comb algorithm:
+ *
+ * - COMB_BLOCKS is the number of blocks the input is split into. Each
+ *   has a corresponding table.
+ * - COMB_TEETH is the number of bits simultaneously covered by one table.
+ *
+ * The comb's spacing (COMB_SPACING), or the distance between the teeth,
+ * is defined as ceil(256 / (COMB_BLOCKS * COMB_TEETH)). Each block covers
+ * COMB_SPACING * COMB_TEETH consecutive bits in the input.
+ *
+ * The size of the precomputed table is COMB_BLOCKS * (1 << (COMB_TEETH - 1))
+ * secp256k1_ge_storages.
+ *
+ * The number of point additions equals COMB_BLOCKS * COMB_SPACING. Each point
+ * addition involves a cmov from (1 << (COMB_TEETH - 1)) table entries and a
+ * conditional negation.
+ *
+ * The number of point doublings is COMB_SPACING - 1. */
+
+#if defined(EXHAUSTIVE_TEST_ORDER)
+/* We need to control these values for exhaustive tests because
+ * the table cannot have infinities in them (secp256k1_ge_storage
+ * doesn't support infinities) */
+#  undef COMB_BLOCKS
+#  undef COMB_TEETH
+#  if EXHAUSTIVE_TEST_ORDER > 32
+#    define COMB_BLOCKS 52
+#    define COMB_TEETH 5
+#  elif EXHAUSTIVE_TEST_ORDER > 16
+#    define COMB_BLOCKS 64
+#    define COMB_TEETH 4
+#  elif EXHAUSTIVE_TEST_ORDER > 8
+#    define COMB_BLOCKS 86
+#    define COMB_TEETH 3
+#  elif EXHAUSTIVE_TEST_ORDER > 4
+#    define COMB_BLOCKS 128
+#    define COMB_TEETH 2
+#  else
+#    define COMB_BLOCKS 256
+#    define COMB_TEETH 1
 #  endif
-#endif
+#else /* !defined(EXHAUSTIVE_TEST_ORDER) */
+/* Use (11, 6) as default configuration, which results in a 22 kB table. */
+#  ifndef COMB_BLOCKS
+#    define COMB_BLOCKS 11
+#    ifdef DEBUG_CONFIG
+#      pragma message DEBUG_CONFIG_MSG("COMB_BLOCKS undefined, assuming default value")
+#    endif
+#  endif
+#  ifndef COMB_TEETH
+#    define COMB_TEETH 6
+#    ifdef DEBUG_CONFIG
+#      pragma message DEBUG_CONFIG_MSG("COMB_TEETH undefined, assuming default value")
+#    endif
+#  endif
+#endif /* defined(EXHAUSTIVE_TEST_ORDER) */
 
-#ifdef DEBUG_CONFIG
-#  pragma message DEBUG_CONFIG_DEF(ECMULT_GEN_PREC_BITS)
+/* Range checks on the parameters. */
+#if !(1 <= COMB_BLOCKS && COMB_BLOCKS <= 256)
+#  error "COMB_BLOCKS must be in the range [1, 256]"
 #endif
+#if !(1 <= COMB_TEETH && COMB_TEETH <= 8)
+#  error "COMB_TEETH must be in the range [1, 8]"
+#endif
+
+/* The remaining COMB_* parameters are derived values, don't modify these. */
+/* - The distance between the teeth of each comb. */
+#define COMB_SPACING CEIL_DIV(256, COMB_BLOCKS * COMB_TEETH)
+/* - The number of bits covered by all the blocks; must be at least 256. */
+#define COMB_BITS (COMB_BLOCKS * COMB_TEETH * COMB_SPACING)
+/* - The number of entries per table. */
+#define COMB_POINTS (1 << (COMB_TEETH - 1))
 
-#if ECMULT_GEN_PREC_BITS != 2 && ECMULT_GEN_PREC_BITS != 4 && ECMULT_GEN_PREC_BITS != 8
-#  error "Set ECMULT_GEN_PREC_BITS to 2, 4 or 8."
+/* Additional sanity checks. */
+#if (COMB_BLOCKS - 1) * COMB_TEETH * COMB_SPACING >= 256
+#  error "COMB_BLOCKS can be reduced"
+#endif
+#if COMB_BLOCKS * (COMB_TEETH - 1) * COMB_SPACING >= 256
+#  error "COMB_TEETH can be reduced"
 #endif
 
-#define ECMULT_GEN_PREC_G(bits) (1 << bits)
-#define ECMULT_GEN_PREC_N(bits) (256 / bits)
+#ifdef DEBUG_CONFIG
+#  pragma message DEBUG_CONFIG_DEF(COMB_BLOCKS)
+#  pragma message DEBUG_CONFIG_DEF(COMB_TEETH)
+#endif
 
 typedef struct {
     /* Whether the context has been built. */
     int built;
 
-    /* Blinding values used when computing nG as (n-b)G + bG. */
-    secp256k1_scalar scalar_offset; /* -b */
-    secp256k1_ge ge_offset;  /* bG */
+    /* Values chosen such that
+     *
+     *   n*G == comb(n + (2^COMB_BITS-1)/2 + scalar_offset, G/2) + ge_offset.
+     *
+     * This expression lets us use scalar blinding and optimize the comb precomputation. See
+     * ecmult_gen_impl.h for more details. */
+    secp256k1_scalar scalar_offset;
+    secp256k1_ge ge_offset;
 } secp256k1_ecmult_gen_context;
 
 static void secp256k1_ecmult_gen_context_build(secp256k1_ecmult_gen_context* ctx);

src/ecmult_gen_compute_table.h

@@ -1,5 +1,5 @@
 /***********************************************************************
- * Copyright (c) 2013, 2014, 2015 Pieter Wuille, Gregory Maxwell       *
+ * Copyright (c) Pieter Wuille, Gregory Maxwell                        *
  * Distributed under the MIT software license, see the accompanying    *
  * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
  ***********************************************************************/
@@ -9,6 +9,6 @@
 
 #include "ecmult_gen.h"
 
-static void secp256k1_ecmult_gen_compute_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int bits);
+static void secp256k1_ecmult_gen_compute_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int blocks, int teeth);
 
 #endif /* SECP256K1_ECMULT_GEN_COMPUTE_TABLE_H */

src/ecmult_gen_compute_table_impl.h

@@ -1,5 +1,5 @@
 /***********************************************************************
- * Copyright (c) 2013, 2014, 2015 Pieter Wuille, Gregory Maxwell       *
+ * Copyright (c) Pieter Wuille, Gregory Maxwell, Peter Dettman         *
  * Distributed under the MIT software license, see the accompanying    *
  * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
  ***********************************************************************/
@@ -10,74 +10,96 @@
 #include "ecmult_gen_compute_table.h"
 #include "group_impl.h"
 #include "field_impl.h"
+#include "scalar_impl.h"
 #include "ecmult_gen.h"
 #include "util.h"
 
-static void secp256k1_ecmult_gen_compute_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int bits) {
-    int g = ECMULT_GEN_PREC_G(bits);
-    int n = ECMULT_GEN_PREC_N(bits);
+static void secp256k1_ecmult_gen_compute_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int blocks, int teeth) {
+    size_t points = ((size_t)1) << (teeth - 1);
+    size_t points_total = points * blocks;
+    int spacing = (256 + blocks * teeth - 1) / (blocks * teeth);
+    secp256k1_ge* prec = checked_malloc(&default_error_callback, points_total * sizeof(*prec));
+    secp256k1_gej* ds = checked_malloc(&default_error_callback, teeth * sizeof(*ds));
+    secp256k1_gej* vs = checked_malloc(&default_error_callback, points_total * sizeof(*vs));
+    secp256k1_gej u;
+    size_t vs_pos = 0;
+    secp256k1_scalar half;
+    int block, i;
 
-    secp256k1_ge* prec = checked_malloc(&default_error_callback, n * g * sizeof(*prec));
-    secp256k1_gej gj;
-    secp256k1_gej nums_gej;
-    int i, j;
+    VERIFY_CHECK(points_total > 0);
 
-    VERIFY_CHECK(g > 0);
-    VERIFY_CHECK(n > 0);
-
-    /* get the generator */
-    secp256k1_gej_set_ge(&gj, gen);
-
-    /* Construct a group element with no known corresponding scalar (nothing up my sleeve). */
+    /* u is the running power of two times gen we're working with, initially gen/2. */
+    secp256k1_scalar_half(&half, &secp256k1_scalar_one);
+    secp256k1_gej_set_infinity(&u);
+    for (i = 255; i >= 0; --i) {
+        /* Use a very simple multiplication ladder to avoid dependency on ecmult. */
+        secp256k1_gej_double_var(&u, &u, NULL);
+        if (secp256k1_scalar_get_bits(&half, i, 1)) {
+            secp256k1_gej_add_ge_var(&u, &u, gen, NULL);
+        }
+    }
+#ifdef VERIFY
     {
-        static const unsigned char nums_b32[33] = "The scalar for this x is unknown";
-        secp256k1_fe nums_x;
-        secp256k1_ge nums_ge;
-        int r;
-        r = secp256k1_fe_set_b32_limit(&nums_x, nums_b32);
-        (void)r;
-        VERIFY_CHECK(r);
-        r = secp256k1_ge_set_xo_var(&nums_ge, &nums_x, 0);
-        (void)r;
-        VERIFY_CHECK(r);
-        secp256k1_gej_set_ge(&nums_gej, &nums_ge);
-        /* Add G to make the bits in x uniformly distributed. */
-        secp256k1_gej_add_ge_var(&nums_gej, &nums_gej, gen, NULL);
+        /* Verify that u*2 = gen. */
+        secp256k1_gej double_u;
+        secp256k1_gej_double_var(&double_u, &u, NULL);
+        VERIFY_CHECK(secp256k1_gej_eq_ge_var(&double_u, gen));
     }
+#endif
 
-    /* compute prec. */
-    {
-        secp256k1_gej gbase;
-        secp256k1_gej numsbase;
-        secp256k1_gej* precj = checked_malloc(&default_error_callback, n * g * sizeof(*precj));  /* Jacobian versions of prec. */
-        gbase = gj; /* PREC_G^j * G */
-        numsbase = nums_gej; /* 2^j * nums. */
-        for (j = 0; j < n; j++) {
-            /* Set precj[j*PREC_G .. j*PREC_G+(PREC_G-1)] to (numsbase, numsbase + gbase, ..., numsbase + (PREC_G-1)*gbase). */
-            precj[j*g] = numsbase;
-            for (i = 1; i < g; i++) {
-                secp256k1_gej_add_var(&precj[j*g + i], &precj[j*g + i - 1], &gbase, NULL);
+    for (block = 0; block < blocks; ++block) {
+        int tooth;
+        /* Here u = 2^(block*teeth*spacing) * gen/2. */
+        secp256k1_gej sum;
+        secp256k1_gej_set_infinity(&sum);
+        for (tooth = 0; tooth < teeth; ++tooth) {
+            /* Here u = 2^((block*teeth + tooth)*spacing) * gen/2. */
+            int bit_off;
+            /* Make sum = sum(2^((block*teeth + t)*spacing), t=0..tooth) * gen/2. */
+            secp256k1_gej_add_var(&sum, &sum, &u, NULL);
+            /* Make u = 2^((block*teeth + tooth)*spacing + 1) * gen/2. */
+            secp256k1_gej_double_var(&u, &u, NULL);
+            /* Make ds[tooth] = u = 2^((block*teeth + tooth)*spacing + 1) * gen/2. */
+            ds[tooth] = u;
+            /* Make u = 2^((block*teeth + tooth + 1)*spacing) * gen/2. */
+            for (bit_off = 1; bit_off < spacing; ++bit_off) {
+                secp256k1_gej_double_var(&u, &u, NULL);
             }
-            /* Multiply gbase by PREC_G. */
-            for (i = 0; i < bits; i++) {
-                secp256k1_gej_double_var(&gbase, &gbase, NULL);
-            }
-            /* Multiply numbase by 2. */
-            secp256k1_gej_double_var(&numsbase, &numsbase, NULL);
-            if (j == n - 2) {
-                /* In the last iteration, numsbase is (1 - 2^j) * nums instead. */
-                secp256k1_gej_neg(&numsbase, &numsbase);
-                secp256k1_gej_add_var(&numsbase, &numsbase, &nums_gej, NULL);
+        }
+        /* Now u = 2^((block*teeth + teeth)*spacing) * gen/2
+         *       = 2^((block+1)*teeth*spacing) * gen/2       */
+
+        /* Next, compute the table entries for block number block in Jacobian coordinates.
+         * The entries will occupy vs[block*points + i] for i=0..points-1.
+         * We start by computing the first (i=0) value corresponding to all summed
+         * powers of two times G being negative. */
+        secp256k1_gej_neg(&vs[vs_pos++], &sum);
+        /* And then teeth-1 times "double" the range of i values for which the table
+         * is computed: in each iteration, double the table by taking an existing
+         * table entry and adding ds[tooth]. */
+        for (tooth = 0; tooth < teeth - 1; ++tooth) {
+            size_t stride = ((size_t)1) << tooth;
+            size_t index;
+            for (index = 0; index < stride; ++index, ++vs_pos) {
+                secp256k1_gej_add_var(&vs[vs_pos], &vs[vs_pos - stride], &ds[tooth], NULL);
             }
         }
-        secp256k1_ge_set_all_gej_var(prec, precj, n * g);
-        free(precj);
     }
-    for (j = 0; j < n; j++) {
-        for (i = 0; i < g; i++) {
-            secp256k1_ge_to_storage(&table[j*g + i], &prec[j*g + i]);
+    VERIFY_CHECK(vs_pos == points_total);
+
+    /* Convert all points simultaneously from secp256k1_gej to secp256k1_ge. */
+    secp256k1_ge_set_all_gej_var(prec, vs, points_total);
+    /* Convert all points from secp256k1_ge to secp256k1_ge_storage output. */
+    for (block = 0; block < blocks; ++block) {
+        size_t index;
+        for (index = 0; index < points; ++index) {
+            secp256k1_ge_to_storage(&table[block * points + index], &prec[block * points + index]);
         }
     }
+
+    /* Free memory. */
+    free(vs);
+    free(ds);
     free(prec);
 }