Generating Uniform Random Vectors

In this paper, we study the rate of convergence of the Markov chain X n+1=AX n +b n mod p, where A is an integer matrix with nonzero integer eigenvalues and {b n } n is a sequence of independent and identically distributed integer vectors. If (lambda)i(not equal)±1 for all eigenvalues (lambda)i of A, then n=O((log p)2) steps are sufficient and n=O(log p) steps are necessary to have X n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalue (lambda)1=±1, and (lambda)i(not equal)±1 for all i(not equal)1, n=O(p2) steps are necessary and sufficient.