This is an implementation of Brakerski's somewhat homomorphic encryption
system (http://eprint.iacr.org/2012/078.pdf). Much of the algebraic and number
theoretic facilities are taken from HElib (https://github.com/shaih/HElib).
This library leverages the NTL library over GMP, and is distributed under the
terms of the GNU General Public License (GPL).

A more complete description of the implementation details is available here:
http://cs.stanford.edu/~dwu4/FHE-SI_Report.pdf.

===========
Basic Setup
=========== 

To compile the system, simply run make in the main directory. Note that you
will need to install the NTL library prior to compiling and running the
system. NTL is available for download at http://www.shoup.net/ntl/. If
compilation fails due to NTL (missing header files or linker errors), you will
have to modify the Makefile accordingly. If the header files are not present
in a default directory, adding -I/path/to/NTL/header/files to the CFLAGS
should fix the problem. If there is a linker problem, add the path to the NTL
library to LDPATH: -L/path/to/NTL/library.

===================
Running Basic Tests
===================

Several test programs are included to illustrate usage of the system. We describe them below.

Test_AddMul: ./tests/TestAddMul_x logQ p generator [seed]
---------------------------------------------------------

This test exercises the basic homomorphic operations of the Brakerski system.
The first three parameters pertain to the SWHE system:
  logQ: the ciphertext modulus to use (q = 2^logQ)
  p: the plaintext modulus to use (should be a safe prime)
  generator: the generator for the plaintext space (a generator 
             of the multiplicative group Z/(p-1)Z)

If a seed is not provided, Test_AddMul will test the system against a series
of randomly generated test cases. The test will report whether any of the test
cases failed. This can be used to verify that the homomorphic operations of
the system are properly behaving. If a seed is provided, the will just run the
test using the specified seed for the random number generator. This can be
used to debug specific test cases.

Small parameters that demonstrate the basic functionality are:
  ./tests/TestAddMul_x 80 23 7
Note that these parameters do not constitute a secure choice of parameters. More
information on choosing parameters to guarantee a certain level of security may 
be found here: http://cs.stanford.edu/~dwu4/FHE-SI_Report.pdf.

Test_General: ./tests/Test_General_x
-----------------------------------

This is a test that is similar to the Test_General from HELib. Not all of the
functionality supported by HELib is present in our implementation. It should,
however, demonstrate some of the facilities that is supported by our
implementation.

Test_Regression: ./tests/Test_Regression_x datafile p generator
---------------------------------------------------------------

This test showcases running linear regression on encrypted data. The datafile
is the path to an input data file. The remaining parameters pertain to the
SWHE. As in Test_AddMul, the parameter p is the plaintext modulus and the
parameter generator is a generator for the plaintext space. Again, p should be
a safe prime.

The datafile consists of the data points that we regress over. The datafile
format is specified as follows. The first line consists of two numbers,
separated by a space. The first number is the dimension d and the second
number specifies the number of points. What follows is a series of lines. Each
of the subsequent lines contains d + 1 numbers. The first d values are the
independent variables and the last value is the dependent variable.

A Python script is provided to generate random data to run this test:

  ./scripts/generateRandomData.py filename d N [nFiles]

The filename specifies the output datafile. The next parameter d specifies the
data dimension and lastly, N specifies the number of data points to generate.
The optional nFiles parameter allows you to split the data points into
multiple files, for instance, to perform parallel execution of linear
regression.

More concrete details on regression analysis is provided in the report:
http://cs.stanford.edu/~dwu4/FHE-SI_Report.pdf.

Test_Statistics: ./tests/Test_Statistics_x datafile p generator
---------------------------------------------------------------

This test demonstrates computing basic statistics on encrypted data. The
parameters are the same as that of Test_Regression. More detailed information
is available in the report.

=============
Serialization
=============

Public keys, secret keys, and ciphertexts are all serializable and can be
saved to and loaded from files. Refer to "Serialization.h" to see how this can
be done.