Low Space Complexity <inline-formula> <tex-math notation="LaTeX">$GF(2^m)$ </tex-math></inline-formula> Multiplier for Trinomials Using <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-Term Karatsuba Algorithm

Abstract

We propose bit-parallel GF(2m) multipliers for irreducible trinomials using an n-term Karatsuba algorithm and Mastrovito approach, which are generalizations of the newly proposed multiplication scheme for a specific trinomial. The complexities of the proposed multipliers for GF(2m) depend on the choice of an irreducible trinomial xm + xk + 1 defining GF(2m) and values n, m0 such that m = nm0 or m = nm0 + 1. It is possible to achieve a space-time tradeoff by choosing proper values for k, n, and m0. For the purpose of a specific comparison, we compare the proposed multipliers with the best-known multipliers for an odd m ∈ [399, 450] for which there exists an irreducible trinomial of degree m. As a result, the proposed multipliers achieve the lowest space complexities among similar bit-parallel multipliers (they have roughly 40% reduced space complexities compared with the fastest multiplier). On the other hand, their time complexities match or are at most 2TX higher than the fastest multipliers, where TX is the delay of one 2-input XOR gate.

Keywords

• Bit-parallel multiplier
• trinomial
• Karatsuba algorithm
• Mastrovito approach
• shifted polynomial basis