New Low-Complexity Bit-Parallel Finite Field Multipliers Using Weakly Dual Bases

New structures of bit-parallel weakly dual basis (WDB) multipliers over the binary ground field are proposed. An upper bound on the size complexity of bit-parallel multiplier using an arbitrary generating polynomial is given. When the generating polynomial is an irreducible trinomial x/sup m/+x/sup k/+1, 1/spl les/k/spl les/[m/2], the structure of the proposed bit-parallel multiplier requires only m/sup 2/ two-input AND gates and at most m/sup 2/-1 XOR gates. The time delay is no greater than T/sub A/+([log/sub 2/ m]+2)T/sub x/, where T/sub A/ and T/sub X/ are the time delays of an AND gate and an XOR gate, respectively.

[1]  Rudolf Lide,et al.  Finite fields , 1983 .

[2]  Toshiya Itoh,et al.  Structure of Parallel Multipliers for a Class of Fields GF(2^m) , 1989, Inf. Comput..

[3]  V.K. Bhargava,et al.  A Modified Massey-Omura Parallel Multiplier for a Class of Finite Fields , 1993, IEEE Trans. Computers.

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

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

[6]  Thomas C. Bartee,et al.  Computation with Finite Fields , 1963, Inf. Control..

[7]  Trieu-Kien Truong,et al.  Systolic Multipliers for Finite Fields GF(2m) , 1984, IEEE Transactions on Computers.

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

[9]  Dieter Gollmann,et al.  Self-dual bases in $$\mathbb{F}_{q^n } $$ , 1993, Des. Codes Cryptogr..

[10]  Ian F. Blake,et al.  Bit Serial Multiplication in Finite Fields , 1990, SIAM J. Discret. Math..

[11]  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.

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

[13]  Stephen D. Cohen.  On irreducible polynomials of certain types in finite fields , 1969 .

[14]  Craig K. Rushforth,et al.  A Cellular-Array Multiplier for GF(2m) , 1971, IEEE Transactions on Computers.

[15]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[16]  Charles C. Wang,et al.  An Algorithm to Design Finite Field Multipliers Using a Self-Dual Normal Basis , 1987, IEEE Trans. Computers.

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

[18]  Ian F. Blake,et al.  Constructive problems for irreducible polynominals over finite fields , 1993, Information Theory and Applications.