On Software Parallel Implementation of Cryptographic Pairings

A significant amount of research has focused on methods to improve the efficiency of cryptographic pairings; in part this work is motivated by the wide range of applications for such primitives. Although numerous hardware accelerators for pairing evaluation have used parallelism within extension field arithmetic to improve efficiency, thus far less emphasis has been placed on software exploitation of similar. In this paper we focus on parallelism within one pairing evaluation (intra-pairing), and parallelism between different pairing evaluations (inter-pairing). We identify several methods for exploiting such parallelism (extending previous results in the context of ECC) and show that it is possible to accelerate pairing evaluation by a significant factor in comparison to a naive approach.

[1]  Joos Vandewalle,et al.  SHA: A Design for Parallel Architectures? , 1997, EUROCRYPT.

[2]  Alfred Menezes,et al.  Pairing-Based Cryptography at High Security Levels , 2005, IMACC.

[3]  M. Scott.  Implementing cryptographic pairings , 2007 .

[4]  Frederik Vercauteren,et al.  The Eta Pairing Revisited , 2006, IEEE Transactions on Information Theory.

[5]  Ingrid Verbauwhede,et al.  Montgomery Modular Multiplication Algorithm on Multi-Core Systems , 2007, 2007 IEEE Workshop on Signal Processing Systems.

[6]  Mitsuru Matsui,et al.  On the Power of Bitslice Implementation on Intel Core2 Processor , 2007, CHES.

[7]  Helger Lipmaa,et al.  IDEA: A Cipher For Multimedia Architectures? , 1998, Selected Areas in Cryptography.

[8]  Ruby B. Lee,et al.  PLX: a fully subword-parallel instruction set architecture for fast scalable multimedia processing , 2002, Proceedings. IEEE International Conference on Multimedia and Expo.

[9]  Alfred Menezes,et al.  Guide to Elliptic Curve Cryptography , 2004, Springer Professional Computing.

[10]  Vijay Kumar,et al.  Efficient galois field arithmetic on SIMD architectures , 2003, SPAA '03.

[11]  Mitsuru Matsui,et al.  Performance Analysis and Parallel Implementation of Dedicated Hash Functions , 2002, EUROCRYPT.

[12]  Nigel P. Smart,et al.  On Computing Products of Pairings , 2006, IACR Cryptol. ePrint Arch..

[13]  Paulo S. L. M. Barreto,et al.  Efficient Algorithms for Pairing-Based Cryptosystems , 2002, CRYPTO.

[14]  Tolga Acar,et al.  Analyzing and comparing Montgomery multiplication algorithms , 1996, IEEE Micro.

[15]  Ingrid Verbauwhede,et al.  Elliptic curve cryptography on embedded multicore systems , 2008, Des. Autom. Embed. Syst..

[16]  Paulo S. L. M. Barreto,et al.  Efficient Hardware for the Tate Pairing Calculation in Characteristic Three , 2005, CHES.

[17]  Jacques Jean-Alain Michael Fournier,et al.  Vector microprocessors for cryptography , 2007 .

[18]  Paulo S. L. M. Barreto,et al.  Efficient pairing computation on supersingular Abelian varieties , 2007, IACR Cryptol. ePrint Arch..

[19]  P. L. Montgomery.  Modular multiplication without trial division , 1985 .

[20]  Tsuyoshi Takagi,et al.  Fast Elliptic Curve Multiplications with SIMD Operations , 2004, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[21]  Kazumaro Aoki,et al.  Elliptic Curve Arithmetic Using SIMD , 2001, ISC.

[22]  Nigel P. Smart,et al.  Parallel cryptographic arithmetic using a redundant Montgomery representation , 2004, IEEE Transactions on Computers.

[23]  Craig S. K. Clapp,et al.  Optimizing a Fast Stream Cipher for VLIW, SIMD, and Superscalar Processors , 1997, FSE.

[24]  Tolga Acar,et al.  High-speed algorithms and architectures for number-theoretic cryptosystems , 1998 .

[25]  Ricardo Dahab,et al.  Implementing Cryptographic Pairings over Barreto-Naehrig Curves , 2007, Pairing.

[26]  Eli Biham,et al.  A Fast New DES Implementation in Software , 1997, FSE.

[27]  Soonhak Kwon,et al.  Efficient Tate Pairing Computation for Elliptic Curves over Binary Fields , 2005, ACISP.

[28]  Ricardo Dahab,et al.  Multiplication and Squaring on Pairing-Friendly Fields , 2006, IACR Cryptol. ePrint Arch..

[29]  Hovav Shacham,et al.  Short Signatures from the Weil Pairing , 2001, J. Cryptol..

[30]  Nigel P. Smart,et al.  High Security Pairing-Based Cryptography Revisited , 2006, ANTS.

[31]  Paulo S. L. M. Barreto,et al.  Pairing-Friendly Elliptic Curves of Prime Order , 2005, Selected Areas in Cryptography.

[32]  Steven D. Galbraith,et al.  Implementing the Tate Pairing , 2002, ANTS.

[33]  Iwan M. Duursma,et al.  Tate Pairing Implementation for Hyperelliptic Curves y2 = xp-x + d , 2003, ASIACRYPT.

[34]  Darrel HANKERSON,et al.  Software Implementation of Pairings , 2009, Identity-Based Cryptography.