Generating uniform random vectors

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