A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields

In this paper a new bit-parallel structure for a multiplier with low complexity in Galois elds is introduced. The multiplier operates over composite elds GF((2 n) m), with k = nm. The Karatsuba-Ofman algorithm is investigated and applied to the multiplication of polynomials over GF(2 n). It is shown that this operation has a complexity of order O(k log 2 3) under certain constraints regarding k. A complete set of primitive eld polynomials for composite elds is provided which perform modulo reduction with low complexity. As a result, multipliers for elds GF(2 k) up to k = 32 with low gate counts and low delays are listed. The architectures are highly modular and thus well suited for VLSI implementation.