We study the computational complexity of a very basic problem, namely that of finding solutions to a very large set of random linear equations in a finite Galois Field modulo q (GF[q]). Using tools from statistical mechanics we are able to identify phase transitions in the structure of the solution space and to connect them to changes in performance of global algorithms. Crossing phase boundaries produces a dramatic increase in the memory requirements necessary to the algorithms. In turn, this causes the saturation of the upper bounds for the running time. We illustrate the results on the specific problem of integer factorization, which is of central interest for deciphering messages encrypted with the RSA cryptosystem.