Better Circuits for Binary Polynomial Multiplication

We develop a new and simple way to describe Karatsuba-like algorithms for multiplication of polynomials over $F_2$ . We restrict the search of small circuits to a class of circuits we call symmetric bilinear. These are circuits in which AND gates only compute functions of the form ${\sum }_{i\in S}{a}_i\cdot {\sum }_{i\in S}{b}_i$ (S ⊆ {0,…, n - 1}). These techniques yield improved recurrences for M(kn), the number of gates used in a circuit that multiplies two kn-term polynomials, for k = 4, 5, 6, and 7. We built and verified the circuits for n-term binary polynomial multiplication for values of n of practical interest. Circuits for n up to 100 are posted at http://cs-www.cs.yale.edu/homes/peralta/CircuitStuff/BinPolMult.tar.gz.