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 <math>P = NP</math>.This paper reiterates a non-approximative approach for finding the shortest linear straight-line program. After showing how to search for a circuit of XOR gates with the minimal number of such gates by a reduction of the associated decision problem ("Is there a program of length <math>k</math>?") to satisfiability of propositional logic, we show that using modern SAT solvers, provably optimal solutions to interesting problem instances from cryptography can be obtained. We substantiate this claim by a case study on optimizing the AES S-Box.