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We propose bit-parallel <inline-formula> <tex-math notation="LaTeX">$GF(2^{m})$ </tex-math></inline-formula> multipliers for irreducible trinomials using an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$GF(2^{m})$ </tex-math></inline-formula> depend on the choice of an irreducible trinomial <inline-formula> <tex-math notation="LaTeX">$x^{m}+x^{k}+1$ </tex-math></inline-formula> defining <inline-formula> <tex-math notation="LaTeX">$GF(2^{m})$ </tex-math></inline-formula> and values <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$m_{0}$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$m=nm_{0}$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$m=nm_{0}+1$ </tex-math></inline-formula>. It is possible to achieve a space-time tradeoff by choosing proper values for <inline-formula> <tex-math notation="LaTeX">$k,~n$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$m_{0}$ </tex-math></inline-formula>. For the purpose of a specific comparison, we compare the proposed multipliers with the best-known multipliers for an odd <inline-formula> <tex-math notation="LaTeX">$m\in [{399,450}]$ </tex-math></inline-formula> for which there exists an irreducible trinomial of degree <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$2T_{X}$ </tex-math></inline-formula> higher than the fastest multipliers, where <inline-formula> <tex-math notation="LaTeX">$T_{X}$ </tex-math></inline-formula> is the delay of one 2-input XOR gate.

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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

Low Space Complexity $GF(2^m)$ Multiplier for Trinomials Using $n$ -Term Karatsuba Algorithm