mcl

A generic and fast pairing-based cryptography library.

Abstract

mcl is a library for pairing-based cryptography. The current version supports the optimal Ate pairing over BN curves.

Support architecture

• x86-64 Windows + Visual Studio
• x86, x86-64 Linux + gcc/clang
• ARM Linux
• ARM64 Linux
• (maybe any platform to be supported by LLVM)

Installation Requirements

• GMP and OpenSSL

apt install libgmp-dev libssl-dev

Create a working directory (e.g., work) and clone the following repositories.

mkdir work
cd work
git clone git://github.com/herumi/mcl
git clone git://github.com/herumi/cybozulib
git clone git://github.com/herumi/xbyak ; for only x86/x64
git clone git://github.com/herumi/cybozulib_ext ; for only Windows

• Cybozulib_ext is a prerequisite for running OpenSSL and GMP on VC (Visual C++).

Build and test on x86-64 Linux, macOS, ARM and ARM64 Linux

To make lib/libmcl.a and test it:

cod work/mcl
make test

To benchmark a pairing:

bin/bn_test.exe

To make sample programs:

make sample

if you want to change compiler options for optimization, then set CFLAGS_OPT_USER.

make CLFAGS_OPT_USER="-O2"

Build for 32-bit Linux

Build openssl and gmp for 32-bit mode and install <lib32>

make ARCH=x86 CFLAGS_USER="-I <lib32>/include" LDFLAGS_USER="-L <lib32>/lib -Wl,-rpath,<lib32>/lib"

Build for 64-bit Windows

1. make library

mklib.bat

2. make exe binary of sample\pairing.cpp

mk sample\pairing.cpp
bin/bn_test.exe

open mcl.sln and build or if you have msbuild.exe

msbuild /p:Configuration=Release

Build with cmake

For Linux,

mkdir build
cd build
cmake ..
make

For Visual Studio,

mkdir build
cd build
cmake .. -A x64
msbuild mcl.sln /p:Configuration=Release /m

Build for WASM(WebAssembly) (experimental)

mcl supports emcc (Emscripten) and test/bn_test.cpp runs on browers such as Firefox, Chrome and Edge(enable extended JavaScript at about:config).

• SHE on browser
• BLS signature on brower

Type

emcc -O3 -I ./include/ -I ../cybozulib/include/ src/fp.cpp test/bn_test.cpp -DNDEBUG -s WASM=1 -o t.html
emrun --no_browser --port 8080 --no_emrun_detect .

and open http://<address>:8080/t.html. The timing of a pairing on CurveFp254BNb is 2.8msec on 64-bit Firefox with Skylake 3.4GHz.

SELinux

mcl uses Xbyak JIT engine if it is available on x64 architecture, otherwise mcl uses a little slower functions generated by LLVM. The default mode enables SELinux security policy on CentOS, then JIT is disabled.

% sudo setenforce 1
% getenforce
Enforcing
% bin/bn_test.exe
JIT 0
pairing   1.496Mclk
finalExp 581.081Kclk

% sudo setenforce 0
% getenforce
Permissive
% bin/bn_test.exe
JIT 1
pairing   1.394Mclk
finalExp 546.259Kclk

Libraries

• libmcl.a ; static C++ library of mcl
• libmcl_dy.so ; shared C++ library of mcl
• libbn256.a ; static C library for mcl/bn256f.h
• libbn256_dy.so ; shared C library

If you want to remove '_dyof so files, thenmakeSHARE_BASENAME_SUF=`.

How to initialize pairing library

Call mcl::bn256::initPairing before calling any operations.

#include <mcl/bn256.hpp>
mcl::bn::CurveParam cp = mcl::bn::CurveFp254BNb; // or mcl::bn::CurveSNARK1
mcl::bn256::initPairing(cp);
mcl::bn256::G1 P(...);
mcl::bn256::G2 Q(...);
mcl::bn256::Fp12 e;
mcl::bn256::BN::pairing(e, P, Q);

1. (CurveFp254BNb) a BN curve over the 254-bit prime p = 36z^4 + 36z^3 + 24z^2 + 6z + 1 where z = -(2^62 + 2^55 + 1).
2. (CurveSNARK1) a BN curve over a 254-bit prime p such that n := p + 1 - t has high 2-adicity.

See test/bn_test.cpp.

Default constructor of Fp, Ec, etc.

A default constructor does not initialize the instance. Set a valid value before reffering it.

Definition of groups

The curve equation for a BN curve is:

E/Fp: y^2 = x^3 + b .

• the cyclic group G1 is instantiated as E(Fp)[n] where n := p + 1 - t;
• the cyclic group G2 is instantiated as the inverse image of E'(Fp^2)[n] under a twisting isomorphism phi from E' to E; and
• the pairing e: G1 x G2 -> Fp12 is the optimal ate pairing.

The field Fp12 is constructed via the following tower:

• Fp2 = Fp[u] / (u^2 + 1)
• Fp6 = Fp2[v] / (v^3 - Xi) where Xi = u + 1
• Fp12 = Fp6[w] / (w^2 - v)
• GT = { x in Fp12 | x^r = 1 }

Arithmetic operations

G1 and G2 is additive group and has the following operations:

• T::add(T& z, const T& x, const T& y); // z = x + y
• T::sub(T& z, const T& x, const T& y); // z = x - y
• T::neg(T& y, const T& x); // y = -x
• T::mul(T& z, const T& x, const INT& y); // z = y times scalar multiplication of x

Remark: &z == &x or &y are allowed. INT means integer type such as Fr, int and mpz_class.

T::mul uses GLV method then G2::mul returns wrong value if x is not in G2. Use T::mulGeneric(T& z, const T& x, const INT& y) for x in phi^-1(E'(Fp^2)) - G2.

Fp, Fp2, Fp6 and Fp12 have the following operations:

• T::add(T& z, const T& x, const T& y); // z = x + y
• T::sub(T& z, const T& x, const T& y); // z = x - y
• T::mul(T& z, const T& x, const T& y); // z = x * y
• T::div(T& z, const T& x, const T& y); // z = x / y
• T::neg(T& y, const T& x); // y = -x
• T::inv(T& y, const T& x); // y = 1/x
• T::pow(T& z, const T& x, const INT& y); // z = x^y
• Fp12::unitaryInv(T& y, const T& x); // y = conjugate of x

Remark: Fp12::mul uses GLV method then returns wrong value if x is not in GT. Use Fp12::mulGeneric for x in Fp12 - GT.

Map To points

• BN::mapToG1(G1& P, const Fp& x);
• BN::mapToG2(G2& P, const Fp2& x);

These functions maps x into Gi according to [Faster hashing to G2].

String format of G1 and G2

G1 and G2 have three elements of Fp (x, y, z) for Jacobi coordinate. normalize() method normalizes it to affine coordinate (x, y, 1) or (0, 0, 0).

getStr() method gets

• 0 ; infinity
• 1 <x> <y> ; not compressed format
• 2 <x> ; compressed format for even y
• 3 <x> ; compressed format for odd y

Verify an element in G2

G2::isValid() checks that the element is in the curve of G2 and the order of it is r. G2::set(), G2::setStr and operator<< also check the order. If you check it out of the library, then you can stop the verification by calling G2::setOrder(0).

Benchmark

A benchmark of a BN curve CurveFp254BNb(2016/12/25).

• x64, x86 ; Inte Core i7-6700 3.4GHz(Skylake) upto 4GHz on Ubuntu 16.04.
  • sudo cpufreq-set -g performance
• arm ; 900MHz quad-core ARM Cortex-A7 on Raspberry Pi2, Linux 4.4.11-v7+
• arm64 ; 1.2GHz ARM Cortex-A53 HiKey

software	x64	x86	arm	arm64(msec)
ate-pairing	0.21	-	-	-
mcl	0.31	1.6	22.6	3.9
TEPLA	1.76	3.7	37	17.9
RELIC PRIME=254	0.30	3.5	36	-
MIRACL ake12bnx	4.2	-	78	-
NEONabe	-	-	16	-

• compile option for RELIC

cmake -DARITH=x64-asm-254 -DFP_PRIME=254 -DFPX_METHD="INTEG;INTEG;LAZYR" -DPP_METHD="LAZYR;OATEP"

384-bit curve

see test/bn384_test.cpp Benchmark on Skylake(3.4GHz)

# mcl::bn::CurveFp382_1 ; -(2^94 + 2^76 + 2^72 + 1)
pairing   3.163Mclk  ; 0.93msec

# mcl::bn::CurveFp382_2 ; -(2^94 + 2^78 + 2^67 + 2^64 + 2^48 + 1)
pairing   3.261Mclk  ; 0.96msec

How to make asm files (optional)

The asm files generated by this way are already put in src/asm, then it is not necessary to do this.

Install LLVM.

make MCL_USE_LLVM=1 LLVM_VER=<llvm-version> UPDATE_ASM=1

For example, specify -3.8 for <llvm-version> if opt-3.8 and llc-3.8 are installed.

If you want to use Fp with 1024-bit prime on x86-64, then

make MCL_USE_LLVM=1 LLVM_VER=<llvm-version> UPDATE_ASM=1 MCL_MAX_BIT_SIZE=1024

Java API

See java.md

License

modified new BSD License http://opensource.org/licenses/BSD-3-Clause

The original source of the followings are https://github.com/aistcrypt/Lifted-ElGamal . These files are licensed by BSD-3-Clause and are used for only tests.

include/mcl/elgamal.hpp
include/mcl/window_method.hpp
test/elgamal_test.cpp
test/window_method_test.cpp
sample/vote.cpp

This library contains mie and Lifted-ElGamal.

References

• ate-pairing
• Faster Explicit Formulas for Computing Pairings over Ordinary Curves, D.F. Aranha, K. Karabina, P. Longa, C.H. Gebotys, J. Lopez, EUROCRYPTO 2011, (preprint)
• High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves, Jean-Luc Beuchat, Jorge Enrique González Díaz, Shigeo Mitsunari, Eiji Okamoto, Francisco Rodríguez-Henríquez, Tadanori Teruya, Pairing 2010, (preprint)
• Faster hashing to G2,Laura Fuentes-Castañeda, Edward Knapp, Francisco Rodríguez-Henríquez, SAC 2011, (preprint)
• Skew Frobenius Map and Efficient Scalar Multiplication for Pairing–Based Cryptography, Y. Sakemi, Y. Nogami, K. Okeya, Y. Morikawa, CANS 2008.

Author

光成滋生 MITSUNARI Shigeo(herumi@nifty.com)