A versatile and scalable digit-serial/parallel multiplier architecture for finite fields GF(2/sup m/)

We present an architecture for digit-serial multiplication in finite fields GF(2/sup m/) with applications to cryptography. The proposed design uses polynomial basis representation and interleaves multiplication steps with degree reduction steps. An M-bit multiplier works with arbitrary irreducible polynomials and can be used for any binary field of order 2/sup m//spl les/2/sup M/. We introduce a new method for degree reduction which is significantly faster than previously reported iterative techniques. A representative example for a digit-size of d=4, illustrating the reduction circuit, is given. Experimental results show that the proposed method shortens the critical path of the reduction circuit by a factor of between 1.36 and 3.0 for digit-sizes ranging from d=4 to 16.

[1]  Huapeng Wu,et al.  Bit-Parallel Finite Field Multiplier and Squarer Using Polynomial Basis , 2002, IEEE Trans. Computers.

[2]  M. K. Ibrahim,et al.  Novel Radix Finite Field Multiplier for GF(2m) , 1997, J. VLSI Signal Process..

[3]  Keshab K. Parhi,et al.  Efficient finite field serial/parallel multiplication , 1996, Proceedings of International Conference on Application Specific Systems, Architectures and Processors: ASAP '96.

[4]  Keshab K. Parhi,et al.  Low-Energy Digit-Serial/Parallel Finite Field Multipliers , 1998 .

[5]  T. Itoh,et al.  A Fast Algorithm for Computing Multiplicative Inverses in GF(2^m) Using Normal Bases , 1988, Inf. Comput..

[6]  Gordon B. Agnew,et al.  An Implementation of Elliptic Curve Cryptosystems Over F2155 , 1993, IEEE J. Sel. Areas Commun..

[7]  M. Anwar Hasan,et al.  Look-Up Table Based Large Finite Field Multiplication in Memory Constrained Cryptosystems , 1999, IMACC.

[8]  Ricardo Dahab,et al.  High-Speed Software Multiplication in F2m , 2000, INDOCRYPT.

[9]  Mohammad K. Ibrahim,et al.  Bit-level pipelined digit serial GF(2/sup m/) multiplier , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[10]  Oscar Gustafsson.  A Digit-serial Polynomial Basis Gf(2 M ) Multiplier , 1998 .

[11]  Nigel P. Smart,et al.  A comparison of different finite fields for use in elliptic curve cryptosystems , 2000 .

[12]  Trieu-Kien Truong,et al.  A Comparison of VLSI Architecture of Finite Field Multipliers Using Dual, Normal, or Standard Bases , 1988, IEEE Trans. Computers.

[13]  Igor E. Shparlinski.  Finite Fields: Theory and Computation , 1999 .

[14]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: Preface , 1994 .

[15]  Chiou-Yng Lee,et al.  Bit-Parallel Systolic Multipliers for GF(2m) Fields Defined by All-One and Equally Spaced Polynomials , 2001, IEEE Trans. Computers.

[16]  Nigel P. Smart,et al.  Elliptic Curves in Cryptography: Preface , 1999 .

[17]  Tong Zhang,et al.  Systematic Design of Original and Modified Mastrovito Multipliers for General Irreducible Polynomials , 2001, IEEE Trans. Computers.

[18]  Jiang Zhang,et al.  Fast Algorithms for Elliptic Curve Cryptosystems over Binary Finite Field , 1999, ASIACRYPT.

[19]  Christof Paar,et al.  A super-serial Galois fields multiplier for FPGAs and its application to public-key algorithms , 1999, Seventh Annual IEEE Symposium on Field-Programmable Custom Computing Machines (Cat. No.PR00375).

[20]  Stafford E. Tavares,et al.  A Fast VLSI Multiplier for GF(2m) , 1986, IEEE J. Sel. Areas Commun..

[21]  Vijay K. Bhargava,et al.  Modular Construction of Low Complexity Parallel Multipliers for a Class of Finite Fields GF(2^m) , 1992, IEEE Trans. Computers.

[22]  C.-L. Wang,et al.  Digit-serial systolic multiplier for finite fields GF(2m) , 1998 .

[23]  ÇETIN K. KOÇ,et al.  Montgomery Multiplication in GF(2k) , 1998, Des. Codes Cryptogr..

[24]  Ronald C. Mullin,et al.  Optimal normal bases in GF(pn) , 1989, Discret. Appl. Math..

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

[26]  Dieter Gollmann,et al.  Architectures for Exponentiation in GF(2n) , 1986, CRYPTO.

[27]  Chun Pyo Hong,et al.  A Digit-Serial Systolic Multiplier for Finite Fields GF ( 2 m ) , .