Space Efficient $GF(2^m)$ Multiplier for Special Pentanomials Based on $n$ -Term Karatsuba Algorithm

Abstract

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 $\frac {1}2m^{2}+O\left({m^{\frac {3}2}}\right)$ AND gates and $\frac {1}2m^{2}+O\left({m^{\frac {3}2}}\right)$ 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 best-known space efficient multipliers, while having the same time complexity. The results show that the proposed multipliers are the best space optimized multipliers.