N-Term Karatsuba Algorithm and its Application to Multiplier Designs for Special Trinomials

Abstract

In this paper, we propose a new type of non-recursive Mastrovito multiplier for $\text {GF}(2^{m})$ using an $n$ -term Karatsuba algorithm (KA), where $\text {GF}(2^{m})$ is defined by an irreducible trinomial, $x^{m}+x^{k}+1, m=nk$ . We show that such a type of trinomial combined with the $n$ -term KA can fully exploit the spatial correlation of entries in related Mastrovito product matrices and lead to a low-complexity architecture. The optimal parameter $n$ is further studied. As the main contribution of this paper, the lower bound of the space complexity of our proposal is about $O({m^{2}}/{2})+m^{3/2})$ . Meanwhile, the time complexity matches the best Karatsuba multiplier known to date. To the best of our knowledge, it is the first time that Karatsuba-based multiplier has reached such a space complexity bound while maintaining a relatively low time delay.