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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.



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.


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)


  • 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


> git clone git://
> git clone git://
> git clone 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.


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

make -j

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.)


Use the following commands:

$ git clone git://
$ git clone 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.


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;
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 A sample code is

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.


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
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



  • MITSUNARI Shigeo (
  • TERUYA Tadanori (


  • Alessandro Chiesa (
  • Madars Virza (