Generating uniform random vectors in $\QTR{bf}{Z}_{p}^{k}$: the general case

Abstract

This paper is about 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 eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of $Q^{k}$ invariant under $A$. If $|λ_{i}|\not=1$ for all eigenvalues $λ_{i}$ of $A$, then $n=O((\ln p)^{2}) $ steps are sufficient and $n=O(\ln p)$ steps are necessary to have $X_{n}$ sampling from a nearly uniform distribution. Conversely, if $A$ has the eigenvalues $λ_{i}$ that are roots of positive integer numbers, $|λ_{1}|=1$ and $|λ_{i}|>1$ for all $i\not=1$, then $O(p^{2}) $ steps are necessary and sufficient.