GF(2^m) Multiplication and Division Over the Dual Basis

In this paper an algorithm for GF(2/sup m/) multiplication/division is presented and a new, more generalized definition of duality is proposed. From these the bit-serial Berlekamp multiplier is derived and shown to be a specific case of a more general class of multipliers. Furthermore, it is shown that hardware efficient, bit-parallel dual basis multipliers can also be designed. These multipliers have a regular structure, are easily extended to different GF(2/sup m/) and hence suitable for VLSI implementations. As in the bit-serial case these bit-parallel multipliers can also be hardwired to carry out constant multiplication. These constant multipliers have reduced hardware requirements and are also simple to design. In addition, the multiplication/division algorithm also allows a bit-serial systolic finite field divider to be designed. This divider is modular, independent of the defining irreducible polynomial for the field, easily expanded to different GF(2/sup m/) and its longest delay path is independent of m.

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