Proceedings on Advances in cryptology---CRYPTO '86
Efficient Algorithms for Pairing-Based Cryptosystems
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
The Weil Pairing, and Its Efficient Calculation
Journal of Cryptology
Low-Weight Polynomial Form Integers for Efficient Modular Multiplication
IEEE Transactions on Computers
Montgomery Reduction Algorithm for Modular Multiplication Using Low-Weight Polynomial Form Integers
ARITH '07 Proceedings of the 18th IEEE Symposium on Computer Arithmetic
On Software Parallel Implementation of Cryptographic Pairings
Selected Areas in Cryptography
Pairing-Friendly elliptic curves of prime order
SAC'05 Proceedings of the 12th international conference on Selected Areas in Cryptography
IEEE Transactions on Information Theory
Implementing cryptographic pairings over barreto-naehrig curves
Pairing'07 Proceedings of the First international conference on Pairing-Based Cryptography
INDOCRYPT '09 Proceedings of the 10th International Conference on Cryptology in India: Progress in Cryptology
New software speed records for cryptographic pairings
LATINCRYPT'10 Proceedings of the First international conference on Progress in cryptology: cryptology and information security in Latin America
High-speed software implementation of the optimal ate pairing over Barreto-Naehrig curves
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
Compact hardware for computing the tate pairing over 128-bit-security supersingular curves
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
High speed flexible pairing cryptoprocessor on FPGA platform
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
A family of implementation-friendly BN elliptic curves
Journal of Systems and Software
Faster explicit formulas for computing pairings over ordinary curves
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
Parallelizing the weil and tate pairings
IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
Core based architecture to speed up optimal ate pairing on FPGA platform
Pairing'12 Proceedings of the 5th international conference on Pairing-Based Cryptography
Secure dual-core cryptoprocessor for pairings over Barreto-Naehrig curves on FPGA platform
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
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This paper describes a new method to speed up $\mathbb{F}_p$-arithmetic for Barreto-Naehrig (BN) curves. We explore the characteristics of the modulus defined by BN curves and choose curve parameters such that $\mathbb{F}_p$ multiplication becomes more efficient. The proposed algorithm uses Montgomery reduction in a polynomial ring combined with a coefficient reduction phase using a pseudo-Mersenne number. With this algorithm, the performance of pairings on BN curves can be significantly improved, resulting in a factor 5.4 speed-up compared with the state-of-the-art hardware implementations. Using this algorithm, we implemented a pairing processor in hardware, which runs at 204 MHz and finishes one ate and R-ate pairing computation over a 256-bit BN curve in 4.22 ms and 2.91 ms, respectively.