RNS modular multiplication through reduced base extensions

The paper describes a new RNS (residue number system) modular multiplication algorithm, for finite field arithmetic over FP, based on a reduced number of moduli in base extensions with only 3n=2 moduli instead of 2n for standard ones. Our algorithm reduces both the number of elementary modular multiplications (EMMs) and the number of stored precomputations for large asymmetric cryptographic applications such as elliptic curve cryptography or Diffie-Hellman (DH) cryptosystem. It leads to faster operations and smaller circuits.

[1]  Thanos Stouraitis,et al.  Efficient RNS Implementation of Elliptic Curve Point Multiplication Over ${\rm GF}(p)$ , 2013, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[2]  Thanos Stouraitis,et al.  An RNS Implementation of an $F_{p}$ Elliptic Curve Point Multiplier , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[3]  Atsushi Shimbo,et al.  Cox-Rower Architecture for Fast Parallel Montgomery Multiplication , 2000, EUROCRYPT.

[4]  Tim Güneysu,et al.  Exploiting the Power of GPUs for Asymmetric Cryptography , 2008, CHES.

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

[6]  Anatolij A. Karatsuba,et al.  Multiplication of Multidigit Numbers on Automata , 1963 .

[7]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[8]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[9]  Tanja Lange,et al.  Handbook of Elliptic and Hyperelliptic Curve Cryptography , 2005 .

[10]  Andrew M. Odlyzko,et al.  Discrete Logarithms: The Past and the Future , 2000, Des. Codes Cryptogr..

[11]  Jean-Claude Bajard,et al.  An RNS Montgomery Modular Multiplication Algorithm , 1998, IEEE Trans. Computers.

[12]  晋輝 趙,et al.  H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen and F. Vercauteren (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Math. Appl. (Boca Raton)., Chapman & Hall/CRC, 2006年，xxxiv + 808ページ． , 2009 .

[13]  Ramdas Kumaresan,et al.  Fast Base Extension Using a Redundant Modulus in RNS , 1989, IEEE Trans. Computers.

[14]  Filippo Gandino,et al.  An Algorithmic and Architectural Study on Montgomery Exponentiation in RNS , 2012, IEEE Transactions on Computers.

[15]  Jean-Claude Bajard,et al.  Modular multiplication and base extensions in residue number systems , 2001, Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001.

[16]  Reinhard Posch,et al.  Modulo Reduction in Residue Number Systems , 1995, IEEE Trans. Parallel Distributed Syst..

[17]  Julien Eynard,et al.  Fault Detection in RNS Montgomery Modular Multiplication , 2013, 2013 IEEE 21st Symposium on Computer Arithmetic.

[18]  Atsushi Shimbo,et al.  Implementation of RSA Algorithm Based on RNS Montgomery Multiplication , 2001, CHES.

[19]  Laurent Imbert,et al.  Leak Resistant Arithmetic , 2004, CHES.

[20]  Leonel Sousa,et al.  RNS-Based Elliptic Curve Point Multiplication for Massive Parallel Architectures , 2012, Comput. J..

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

[22]  Harvey L. Garner,et al.  RESIDUE NUMBER SYSTEM ENHANCEMENTS FOR PROGRAMMABLE PROCESSORS , 2008 .

[23]  Marc Joye,et al.  The Montgomery Powering Ladder , 2002, CHES.

[24]  Nicolas Guillermin.  A High Speed Coprocessor for Elliptic Curve Scalar Multiplications over \mathbbFp\mathbb{F}_p , 2010, CHES.

[25]  Richard I. Tanaka,et al.  Residue arithmetic and its applications to computer technology , 1967 .

[26]  Taher ElGamal,et al.  A public key cyryptosystem and signature scheme based on discrete logarithms , 1985 .