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In this paper a new bit-parallel structure for a multiplier with low complexity in Galois fields is introduced. The multiplier operates over composite fields GF((2n)m), with k = nm. The Karatsuba-Ofman algorithm is investigated and applied to the multiplication of polynomials over GF(2n). It is shown that this operation has a complexity of order $O(k^{{\rm log}_23})$ under certain constraints regarding k. A complete set of primitive field polynomials for composite fields is provided which perform modulo reduction with low complexity. As a result, multipliers for fields GF(2k) 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.