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Overview

bls48 is an Optimal Ate Pairing(OAP) on BLS Curve implementation in C++. This implementation is based on the drafts below;

• Y. Kiyomura, et al. "Secure and Eifficient Pairing at 256-Bit Secure Level" (2017)
• C. Costello, K. Lauter, M. Naehrig, "Attractive Subfamilies of BLS Curves for Implementing High-Security Pairings." (2011)
• draft-by-Kato(not submitted)

Furthermore, this implementation uses libsnark library.

• https://github.com/scipr-lab/libsnark/tree/deprecated-master (branch : deprecated-master)

We are motivated by the security consideration for pairing on any curve give by T. Kim.

• T. Kim et al. "The extended tower number field sieve: A new complexity for the medium prime case. In Advances in Cryptology", (2016).

Kim's attack made us to update the security parameters or to use alternative curves in OAP. BLS-48 cuerve is one of the pairing friendly curves, and recommended parameters for realizing 256-bit security is written in [Kiyomura].

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Build instructions

This implementation relies on the following:

• C++ build environment
• GMP for certain bit-integer arithmetic
• libprocps for reporting memory usage
• GTest for some of the unit tests

We have tested these only Linux so far(Ubuntu14.04). For example, on a fresh install of Ubuntu 14.04, install the following packages:

$ sudo apt-get install build-essential git libgmp3-dev libprocps3-dev libgtest-dev python-markdown libboost-all-dev libssl-dev

In the case of Ubuntu 16.04, install the following packages:

$ sudo apt-get install build-essential git libgmp3-dev libprocps4-dev libgtest-dev python-markdown libboost-all-dev libssl-dev

Then, to compile, run:

$ make

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Timing Test

• Ubuntu14.04　　 Intel(R) Core(TM) i5-4210M CPU @ 2.60GHz, 4 cores RAM 8GB

Security Level	100	256
Curve Parameter	BN256	BLS48
Scalar Multiplication on G1	0.08[ms]	2.32[ms]
Scalar Multiplication on G2	1.11[ms]	160.09[ms]
Pairing	2.81[ms]	509[ms]

• Ubuntu16.04　　 Intel(R) Core(TM) i7 CPU 950 @ 3.07GHz, 8 cores RAM 24GB

Security Level	100	256
Curve Parameter	BN256	BLS48
Scalar Multiplication on G1	0.06[ms]	3.05[ms]
Scalar Multiplication on G2	1.24[ms]	230.61[ms]
Pairing	4.05[ms]	740.69[ms]

• Raspberry Pi 3 model B Linux devuan 4.6.3-gb48d47a(64bit OS) CPU Cortex-A53 ARMv8 64bit @ 1.2GHz, 4 cores RAM 1GB

Security Level	100	256
Curve Parameter	BN256	BLS48
Scalar Multiplication on G1	6.68[ms]	41.15[ms]
Scalar Multiplication on G2	24.23[ms]	2643.23[ms]
Pairing	58[ms]	8549.02[ms]

• Raspberry Pi model B Ubuntu16.04.2 LTS xenial(32bit OS) CPU Cortex-A53 ARMv8 64bit @ 1.2GHz, 4 cores RAM 1GB

Security Level	100	256
Curve Parameter	BN256	BLS48
Scalar Multiplication on G1	5.39[ms]	38.42[ms]
Scalar Multiplication on G2	21.81[ms]	2660.55[ms]
Pairing	53.03[ms]	8585.53[ms]

• Mac Mac OS Sierra Intel Core i7 @ 3.5 GHz, 2 cores RAM 16 GB

Security Level	100	256
Curve Parameter	BN256	BLS48
Scalar Multiplication on G1	0[ms]	1.07[ms]
Scalar Multiplication on G2	0.14[ms]	112.86[ms]
Pairing	2.11[ms]	373.02[ms]

• Windows (Using MSYS2) Windows10 Intel(R) Core(TM) i7-7500U CPU @ 2.70GHz, 4 cores RAM 16 GB

Security Level	100	256
Curve Parameter	BN256	BLS48
Scalar Multiplication on G1	0[ms]	1.5[ms]
Scalar Multiplication on G2	0.9[ms]	127.74[ms]
Pairing	2.25[ms]	409.8[ms]