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Recently, hybrid multiplication schemes over the binary extension field <inline-formula> <tex-math notation="LaTeX">$GF(2^{m})$ </tex-math></inline-formula> based on <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-term Karatsuba algorithm (KA) have been proposed for irreducible trinomials. Their complexities depend on a decomposition of <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> and the choice of a generation polynomial. However, these multipliers have some limitations on a decomposition of <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> or generation polynomial <inline-formula> <tex-math notation="LaTeX">$x^{m}+x^{k}+1$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$m\geq 2k$ </tex-math></inline-formula>. In this paper, we loosen such limited conditions. We present a new hybrid bit-parallel multiplier based on <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-term KA for any irreducible trinomial <inline-formula> <tex-math notation="LaTeX">$x^{m}+x^{k}+1\,\,(0< k< m)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is decomposed as <inline-formula> <tex-math notation="LaTeX">$m=nm_{0}+r$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$0< r< m_{0}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$1< n$ </tex-math></inline-formula>. (Here, various values for <inline-formula> <tex-math notation="LaTeX">$n,m_{0}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> may be chosen.) To this end, we generalize the previously proposed multiplication scheme for <inline-formula> <tex-math notation="LaTeX">$x^{nm_{0}+1}+x^{k}+1$ </tex-math></inline-formula> into <inline-formula> <tex-math notation="LaTeX">$x^{nm_{0}+r}+x^{k}+1$ </tex-math></inline-formula>. We evaluate the explicit complexity of the proposed multiplier. Specific comparisons show that the proposed multiplier achieves the lowest space complexity with the same or lower time complexity among hybrid multipliers. Compared to the fastest multipliers, the time complexity of the proposed multiplier costs only <inline-formula> <tex-math notation="LaTeX">$T_{X}$ </tex-math></inline-formula> higher while its space complexity is much lower (it has roughly 40% reduced space complexity), where <inline-formula> <tex-math notation="LaTeX">$T_{X}$ </tex-math></inline-formula> is the delay of one 2-input XOR gate.

Efficient Bit-Parallel Multiplier for All Trinomials Based on n-Term Karatsuba Algorithm