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

TL;DR

This paper analyzes the convergence rate of a Markov chain used for generating uniform random vectors in _{p}^{k}, providing bounds on the number of steps needed based on the eigenvalues of the matrix A.

Contribution

It establishes precise conditions and step bounds for the Markov chain to approximate uniform distribution, depending on the eigenvalues of A.

Findings

If all eigenvalues have magnitude not equal to 1, then O((ln p)^2) steps suffice.

If A has eigenvalues that are roots of positive integers, then O(p^2) steps are necessary and sufficient.

The convergence rate depends critically on the spectral properties of matrix A.

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 $|\lambda_{i}|\not=1$ for all eigenvalues $\lambda_{i}$ 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 $\lambda_{i}$ that are roots of positive integer numbers, $|\lambda_{1}|=1$ and $|\lambda_{i}|>1$ for all $i\not=1$, then $O(p^{2}) $ steps are necessary and sufficient.