Low-complexity design of bit-parallel dual-basis multiplier over GF(2m)

Recently, information security is heavily dependent on cryptosystems such as Rivest-Shamir-Adleman algorithm (RSA algorithm) and elliptic curve cryptosystem (ECC). RSA can provide higher security level than ECC, but it is not suitable for the resource-constrained devices such as smart phones or embedded system. Thus, ECC is attracted on application in resource-constrained devices because it can achieve the same security level, but uses less key length than RSA. Galois or finite field multiplication is the core arithmetic operation of ECC. There are three popular bases in the finite field over GF(2m), polynomial basis, normal basis and dual basis (DB). Each basis representation has its own advantages. In this study, the authors will introduce a low-complexity bit-parallel DB multiplier using the multiplexer approach. Compared with the related work, our design saves up to 60% of space complexity.

[1]  Masao Kasahara,et al.  Efficient bit-serial multiplication and the discrete-time Wiener-Hopf equation over finite fields , 1989, IEEE Trans. Inf. Theory.

[2]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[3]  Elwyn R. Berlekamp,et al.  Bit-serial Reed - Solomon encoders , 1982, IEEE Transactions on Information Theory.

[4]  Trieu-Kien Truong,et al.  The use of finite fields to compute convolutions , 1975, IEEE Trans. Inf. Theory.

[5]  M. Anwar Hasan,et al.  A New Construction of Massey-Omura Parallel Multiplier over GF(2m) , 2002, IEEE Trans. Computers.

[6]  Johann Großschädl,et al.  A Bit-Serial Unified Multiplier Architecture for Finite Fields GF(p) and GF(2m) , 2001, CHES.

[7]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[8]  Christof Paar,et al.  A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields , 1996, IEEE Trans. Computers.

[9]  M.A. Hasan,et al.  New Low-Complexity Bit-Parallel Finite Field Multipliers Using Weakly Dual Bases , 1998, IEEE Trans. Computers.

[10]  Low-complexity finite field multiplier using irreducible trinomials , 2003 .

[11]  C. Chiou,et al.  Scalable and Systolic Architecture for Computing Double Exponentiation Over GF(2m) , 2006 .

[12]  Chiou-Yng Lee,et al.  Multiplexer-based double-exponentiation for normal basis of GF(2m) , 2005, Comput. Secur..

[13]  Shun-Shii Lin,et al.  Self-checking alternating logic bit-parallel gaussian normal basis multiplier with type-t , 2011, IET Inf. Secur..

[14]  Jim-Min Lin,et al.  Concurrent Error Detection in a Bit-Parallel Systolic Multiplier for Dual Basis of GF(2m) , 2005, J. Electron. Test..

[15]  A. P. Chandrakasan,et al.  An energy-efficient reconfigurable public-key cryptography processor , 2001, IEEE J. Solid State Circuits.

[16]  Jim-Min Lin,et al.  Unified dual-field multiplier in GF(P) and GF(2k) , 2009, IET Inf. Secur..

[17]  Erkay Savas,et al.  Multiplier architectures for GF(p) and GF(2n) , 2004 .

[18]  Jim-Min Lin,et al.  Concurrent Error Detection in a Polynomial Basis Multiplier over GF(2m) , 2006, J. Electron. Test..

[19]  Servaas Vandenberghe,et al.  A Fast Software Implementation for Arithmetic Operations in GF(2n) , 1996, ASIACRYPT.

[20]  William Stallings,et al.  Cryptography and Network Security: Principles and Practice , 1998 .

[21]  Wen-Yew Liang,et al.  Low Cost Dual-Basis Multiplier over GF (2 m ) Using Multiplexer Approach , 2012 .

[22]  Erkay Savas,et al.  A Scalable and Unified Multiplier Architecture for Finite Fields GF(p) and GF(2m) , 2000, CHES.

[23]  Hilarie K. Orman,et al.  Fast Key Exchange with Elliptic Curve Systems , 1995, CRYPTO.

[24]  Jim-Min Lin,et al.  Concurrent error detection in semi-systolic dual basis multiplier over GF(2m) using self-checking alternating logic , 2010, IET Circuits Devices Syst..

[25]  Chiou-Yng Lee,et al.  Efficient Design of Low-Complexity Bit-Parallel Systolic Hankel Multipliers to Implement Multiplication in Normal and Dual Bases of GF (2m) , 2005, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[26]  Dingyi Pei,et al.  A VLSI DEsign for Computing Exponentiations in GF(2^m) and Its Application to Generate Pseudorandom Number Sequences , 1990, IEEE Trans. Computers.

[27]  Johannes Wolkerstorfer,et al.  Dual-Field Arithmetic Unit for GF(p) and GF(2m) , 2002, CHES.

[28]  Jim-Min Lin,et al.  Low-complexity bit-parallel dual basis multipliers using the modified Booth's algorithm , 2005, Comput. Electr. Eng..

[29]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

[30]  Richard E. Blahut,et al.  Fast Algorithms for Digital Signal Processing , 1985 .

[31]  A. Menezes,et al.  Applications of Finite Fields , 1992 .

[32]  Chin-Chen Chang,et al.  Concurrent Error Detection and Correction in Gaussian Normal Basis Multiplier over GF(2^m) , 2009, IEEE Transactions on Computers.

[33]  Chiou-Yng Lee.  Low complexity bit-parallel systolic multiplier over GF(2m) using irreducible trinomials , 2003 .

[34]  Akashi Satoh,et al.  A Scalable Dual-Field Elliptic Curve Cryptographic Processor , 2003, IEEE Trans. Computers.

[35]  Mohammed Benaissa,et al.  GF(2^m) Multiplication and Division Over the Dual Basis , 1996, IEEE Trans. Computers.

[36]  Chin-Chen Chang,et al.  Concurrent error detection and correction in dual basis multiplier over GF(2m) , 2009, IET Circuits Devices Syst..

[37]  Irving S. Reed,et al.  Galois Switching Functions and Their Applications , 1975, IEEE Transactions on Computers.