Recently, hybrid multiplication schemes over the binary extension field GF (2 m) based on n-term Karatsuba algorithm (KA) have been proposed for irreducible trinomials. Their complexities depend on a decomposition of m and the choice of a generation polynomial. However, these multipliers have some limitations on a decomposition of m or generation polynomial x m + x k + 1 such that m ≥ 2k. In this paper, we loosen such limited conditions. We present a new hybrid bit-parallel multiplier based on n-term KA for any irreducible trinomial x m + x k + 1 (0 < k < m), where m is decomposed as m = nm 0 + r with 0 < r < m 0 and 1 < n. (Here, various values for n, m 0 and r may be chosen.) To this end, we generalize the previously proposed multiplication scheme for x nm0+1 + x k + 1 into x nm0+r + x k + 1. 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 T X higher while its space complexity is much lower (it has roughly 40% reduced space complexity), where T X is the delay of one 2-input XOR gate. INDEX TERMS Hybrid bit-parallel multiplier, trinomial, Karatsuba algorithm, Mastrovito approach, shifted polynomial basis

Recently, new multiplication schemes over the binary extension field GF(2 m) based on an n-term Karatsuba algorithm have been proposed for irreducible trinomials. In this paper, we extend these schemes for trinomials to any irreducible polynomials. We introduce some new types of pentanomials and propose multipliers for those pentanomials utilizing the extended schemes. We evaluate the rigorous space and time complexities of the proposed multipliers, and compare those with similar bit-parallel multipliers for pentanomials. As a main contribution, the best space complexities of our multipliers are 1 2 m 2 + O(m 3 2) AND gates and 1 2 m 2 + O(m 3 2) XOR gates, which nearly correspond to the best results for trinomials. Also, specific comparisons for three fields GF(2 163), GF(2 283), and GF(2 571) recommended by NIST show that the proposed multiplier has roughly 40% reduced space complexity compared to the fastest multipliers, while it costs a few more XOR gate delay. It is noticed that our space complexity gain is much greater than the time complexity loss. Moreover, the proposed multiplier has about 21% reduced space complexity than the bestknown space efficient multipliers, while having the same time complexity. The results show that the proposed multipliers are the best space optimized multipliers. INDEX TERMS Bit-parallel multiplier, Karatsuba algorithm, Mastrovito approach, pentanomial, shifted polynomial basis.

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