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

In this paper, we propose a new type of non-recursive Mastrovito multiplier for GF(2^m) using an n-term Karatsuba algorithm (KA), where 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(m²/2) + 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.