Synthesizing shortest linear straight-line programs over GF(2) using SAT

Non-trivial linear straight-line programs over the Galois field of two elements occur frequently in applications such as encryption or high-performance computing. Finding the shortest linear straight-line program for a given set of linear forms is known to be MaxSNP-complete, i.e., there is no ε-approximation for the problem unless P = NP. This paper presents a non-approximative approach for finding the shortest linear straight-line program. In other words, we show how to search for a circuit of XOR gates with the minimal number of such gates. The approach is based on a reduction of the associated decision problem ("Is there a program of length k?") to satisfiability of propositional logic. Using modern SAT solvers, optimal solutions to interesting problem instances can be obtained.