Abstract
At Crypto 2009 Bernstein proposed two optimized Karatsuba formulas for binary polynomial multiplication. Bernstein obtained these optimizations by re-expressing the reconstruction of one or two recursions of the Karatsuba formula. In this paper we present a generalization of these optimizations. Specifically, we optimize the reconstruction of s recursions of the Karatsuba formula for s >= 1. To reach this goal, we express the recursive reconstruction through a tree and re-organize this tree to derive an optimized recursive reconstruction of depth $s$. When we apply this approach to a recursion of depth s=\log_2(n) we obtain a parallel multiplier with a space complexity of 5.25 n^(\log_2(3))+O(n) XOR gates and n^(\log_2(3)) AND gates and with a delay of 2log_2(n) D_X+D_A.