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<p><b>Abstract</b>—In this paper a new bit-parallel structure for a multiplier with low complexity in Galois fields is introduced. The multiplier operates over <it>composite fields GF</it>((2<super><it>n</it></super>)<super><it>m</it></super>), with <it>k</it> = <it>nm</it>. The Karatsuba-Ofman algorithm is investigated and applied to the multiplication of polynomials over <it>GF</it>(2<super><it>n</it></super>). It is shown that this operation has a complexity of order <tmath>$O(k^{{\rm log}_23})$</tmath> under certain constraints regarding <it>k</it>. 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 <it>GF</it>(2<super><it>k</it></super>) up to <it>k</it> = 32 with low gate counts and low delays are listed. The architectures are highly modular and thus well suited for VLSI implementation.</p>
Finite field multiplication, bit parallel multiplication, composite fields, polynomial multiplication, Karatsuba Ofman algorithm, primitive polynomials, VLSI architecture.
Christof Paar, "A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields", IEEE Transactions on Computers, vol. 45, no. , pp. 856-861, July 1996, doi:10.1109/12.508323
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