Fast Multiplication of the Algebraic Normal Forms of Two Boolean Functions

The contribution of this paper is twofold. Firstly, it proposes a simple algorithm which performs the multiplication of two n-variate boolean functions in their algebraic normal forms in O(n2) time and O(2) space. Secondly, it proposes a fast implementation (MultANFw) of the algorithm which works with w-bit words. Results for w = 8, 32 and 64 show that the 64-bit implementation is the fastest. To further analyze the performance, a sparse implementation has been done, which we call quadratic implementation. It has been observed that for a w-bit implementation, if the product of the number of monomials of the two input polynomials is < 2n−log2 , then the quadratic implementation performs better than MultANFw. It is also found that MultANFw performs much better than the algorithm internally used by SAGE for all the three variants, i.e., w = 8, 32 and 64. Our study also indicates that quadratic implementation performs better than SAGE.