Efficient Generation of Random Nonsingular Matrices

We present an eecient algorithm for generating an n n nonsingular matrix uniformly over a nite eld. This algorithm is useful for several cryptographic and checking applications. Over GFF2] our algorithm runs in expected time M(n) + O(n 2), where M(n) is the time needed to multiply two n n matrices, and the expected number of random bits it uses is n 2 + 3. (Over other nite elds we use n 2 + O(1) random eld elements on average.) This is more eecient than the standard method for solving this problem, both in terms of expected running time and the expected number of random bits used. The standard method is to generate random nn matrices until we produce one with nonzero determinant. In contrast, our technique directly produces a random matrix guaranteed to have non-zero determinant. We also introduce eecient algorithms for related problems such as uniformly generating singular matrices or matrices with xed determinant.