High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves

This library provides functionality to compute the optimal ate pairing over Barreto-Naehrig (BN) curves. It is released under the BSD 3-Clause License.

Now I'm developing a new pairing library mcl, which is more portable and supports larger primes than this library though it is a little slower.

History

• 2015/May/15: add java api
• 2014/Jun/15: support a BN curve for SNARKs, incorporating code from libsnark
• 2013/Jun/02: support mulx on Haswell
• 2013/Mar/08: add elliptic curve class
• 2012/Jan/30: rewrite ate pairing according to Faster explicit formulas for computing pairings over ordinary curves
• 2010/Sep/8: change twist xi from u + 12 to u
• 2010/Jul/15: use cyclotomic squaring for final exponentiation
• 2010/Jun/18: first release

Overview

The following two BN curves are supported:

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

By default, the first curve (we call it as CurveFp254BNb) is used; when setting the flag SUPPORT_SNARK, the second curve (we call it as CurveSNARK) is used instead.

• CurveFp254BNb The value of z is found by [NASKM] first. The curve instantiated by z is investigated by [PSNB] for an efficient implementation. Our library implements a fast algorithm, which is proposed by [AKLGL] for this curve. The performance of this library is competitive to the state-of-the-art implementation report in [ABLR].

• CurveSNARK Support for the second curve builds on code provided by SCIPR Lab in libsnark. The curve was specifically selected for speeding up Succinct Non-interactive ARguments of Knowledge (SNARKs), which benefit from its high 2-adicity (see [BCGTV13] and [BCTV14]).

Pairing computations on the first curve are more efficient, and the performance numbers reported below (and in our papers) are achieved using this curve (which is prefered for applications that do not benefit from high 2-adicity). Note that the old parameters in [BDMOHT] are not used now.

Parameters

The curve equation for a BN curve is:

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

The two supported BN curves have the following parameters:

1. b = 2 and p = 16798108731015832284940804142231733909889187121439069848933715426072753864723; and
2. b = 3 and p = 21888242871839275222246405745257275088696311157297823662689037894645226208583.

As usual,

• the cyclic group G1 (aka Ec1) is instantiated as E(Fp)[n] where n := p + 1 - t;
• the cyclic group G2 (aka Ec2) is instantiated as the inverse image of E'(Fp^2)[n] under a twisting isomorphism 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)

Requirements

• OS: 64-bit Windows; 64-bit Linux; Mac OS X
• CPU: x64 Intel; AMD processor
• C++ compiler: Visual Studio 2012; gcc 4.4.1 or later

Build instructions

Windows

> git clone git://github.com/herumi/xbyak.git
> git clone git://github.com/herumi/ate-pairing.git
> git clone git://github.com/herumi/cybozulib-ext.git ; compiled binary of mpir

Open ate/ate.sln and compile test_bn with Release mode. The produced binary is ate/x64/Release/test_bn.exe.

Cygwin

Install mingw64-x86_64-gcc-g++ (run Cygwin setup and search mingw64). Then use the following commands:

PATH=/usr/x86_64-w64-mingw32/sys-root/mingw/bin/:$PATH
make -j
test/bn.exe

Note that test/bn.exe uses mulx if possible; if you do not want to use it, run the executable as test/bn.exe -mulx 0. (This allows you to verify the difference with/without mulx on Haswell.)

Linux

Use the following commands:

$ git clone git://github.com/herumi/xbyak.git
$ git clone git://github.com/herumi/ate-pairing.git
$ cd ate-pairing
$ make -j
$ test/bn

The library xbyak is a x86/x86-64 JIT assembler for C++, developed for efficient pairing implementations. (See also this webpage.) Note that binaries other than test/bn are used for testing purposes only.

• This implementation uses dynamically-generated code, so you will get the error zmInit ERR:can't protect if execution of code on the heap is disallowed by some modern systems. For example, on Fedora 20, run sudo setsebool -P allow_execheap 1 to allow execution to solve this.

By the default, the first BN curve is used. If instead you want to use the second BN curve (specialized to SNARKs), modify the fourth line above to:

$ make -j SUPPORT_SNARK=1

• REMARK. You defined BN_SUPPORT_SNARK macro for a compile when if you use a library(libzm.a) made by SUPPORT_SNARK=1.

Usage

See the function sample2() in sample.cpp. Also, use can use mpz_class for scalar multiplication of points on the elliptic curves, if MIE_ATE_USE_GMP is defined. For instance:

using namespace bn;
Param::init();
const Ec2 g2(...);
const Ec1 g1(...);
mpz_class a("123456789");
mpz_class b("98765432");
Ec1 g1a = g1 * a;
Ec2 g2b = g2 * b;
Fp12 e;
opt_atePairing(e, g2b, g1a);

Usage for Java

See java.md. A sample code is BN254Test.java.

Operation costs

Let mu be the cost of unreduced multiplication producing double-precision result (i.e., 256-bit int x 256-bit int to 512-bit int); and let r be the cost of modular reduction of double-precision integers (i.e., 512-bit int to 256-bit int in Fp). Then, for us,

• Fp::mul = mu + r
• Fp2::mul = 3mu + 2r
• Fp2::square = 2mu + 2r

Next, we compare the costs of our library with the one of [AKLGL10]:

Phase	[AKLGL10]	This work
Miller loop	6792mu + 3022r	6785mu + 3022r
Final exponentiation	3753mu + 2006r	3526mu + 1932r
Optimal ate pairing	10545mu + 5028r	10311mu + 4954r

Note: [Table 2 in p. 17, AKLGL10] does not contain the cost of (m, r) so we have added the costs of (282m + 6mu + 4r) and (30m + 75mu + 50r) to ML and FE respectively.

Finally, at the moment, our implementation does not support the algorithm in PSNB10.

Benchmark

The cost of a pairing is 1.17M clock cycles on Core i7 4700MQ (Haswell) 2.4GHz processor with TurboBoost disabled. Below, we also include clock cycle counts on Core i7 2600 3.4GHz, Xeon X5650 2.6GHz, and Core i7 4700MQ 2.4GHz. The formal benchmark is written in [ZPMRTH].

% sudo sh -c "echo 0 > /sys/devices/system/cpu/cpufreq/boost"
% cat /sys/devices/system/cpu/cpufreq/boost
0

operation	i7 2600	Xeon X5650	Haswell	Haswell with mulx
TurboBoost	on	on	off	off
mu	50	60	42	38
r	80	98	69	65
Fp:mul	124	146	98	90
Fp2:mul	360	412
Fp2:square	288	335
G1::double	1150	1300
G1::add	2200	2600
G2::double	2500	2900
G2::add	5650	6500
Fp12::square	4500	5150
Fp12::mul	6150	7000
Miller loop	0.83M	0.97M	0.82M	0.71M
final_exp	0.53M	0.63M	0.51M	0.46M
pairing	1.36M	1.60M	1.33M	1.17M

References

• [ABLR] The Realm of the Pairings (Invited Talk), Diego F. Aranha, Paulo S. L. M. Barreto, Patrick Longa, and Jefferson E. Ricardini, SAC 2013, (preprint, slide)

• [NASKM] Integer Variable chi-Based Ate Pairing, Y. Nogami, M. Akane, Y. Sakemi, H. Kato, and Y. Morikawa, Pairing 2008

• [PSNB] A Family of Implementation-Friendly BN Elliptic Curves, G.C.C.F. Pereira, M.A. Simplicio Jr, M. Naehrig, P.S.L.M. Barreto, J. Systems and Software 2011, (preprint)

• [AKLGL] Faster Explicit Formulas for Computing Pairings over Ordinary Curves, D.F. Aranha, K. Karabina, P. Longa, C.H. Gebotys, J. Lopez, EUROCRYPTO 2011, (preprint)

• [BDMOHT] 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)

• A Fast Implementation of the Optimal Ate Pairing over BN curve on Intel Haswell Processor, Shigeo Mitsunari, IACR ePrint 2013/362

• Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture, Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza, USENIX Security 2014

• [ZPMRTH] Software implementation of an Attribute-Based Encryption scheme, Eric Zavattoni and Luis J. Dominguez Perez and Shigeo Mitsunari and Ana H. Sanchez-Ramirez and Tadanori Teruya and Francisco Rodriguez-Henriquez, IEEE Transactions on Computers, To appear, (preprint, project Web page and source code)

• This library's old webpage

Authors

• MITSUNARI Shigeo (herumi@nifty.com)
• TERUYA Tadanori (tadanori.teruya@gmail.com)

Contributors

• Alessandro Chiesa (alexch@mit.edu)
• Madars Virza (madars@mit.edu)