Parallel Factorization of Boolean Polynomials

Abstract

Polynomial factorization is a classical algorithmic problem in algebra, which has a wide range of applications. Of special interest is factorization over finite fields, among which the field of order two is probably the most important one due to the relationship to Boolean functions. In particular, factorization of Boolean polynomials corresponds to decomposition of Boolean functions given in the Algebraic Normal Form. It has been also shown that factorization provides a solution to decomposition of functions given in the full DNF (i.e., by a truth table), for positive DNFs, and for cartesian decomposition of relational datatables. These applications show the importance of developing fast and practical factorization algorithms. In the paper, we consider some recently proposed polynomial time factorization algorithms for Boolean polynomials and describe a parallel MIMD implementation thereof, which exploits both the task and data level parallelism. We report on an experimental evaluation, which has been conducted on logic circuit synthesis benchmarks and synthetic polynomials, and show that our implementation significantly improves the efficiency of factorization. Finally, we report on the performance benefits obtained from a parallel algorithm when executed on a massively parallel many core architecture (Redefine).