Generating Random Vectors in (Z/pZ)^d Via an Affine Random Process

Details Generating Random Vectors in (Z/pZ)^d Via an Affine Random Process Generating Random Vectors in (Z/pZ)^d Via an Affine Random Process

2007-01-20

arxiv.org/abs/math/0701570v2

Mathematics

Probability

This version incorporates some changes suggested by a referee and is the final pre-publication version. The published version

Scientific paper

This paper considers some random processes of the form X_{n+1}=TX_n+B_n (mod p) where B_n and X_n are random variables over (Z/pZ)^d and T is a fixed d x d integer matrix which is invertible over the complex numbers. For a particular distribution for B_n, this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det(T), order (log p)^2 steps suffice to make X_n close to uniformly distributed where X_0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p^b steps are needed for X_n to get close to uniform where b is a value which may depend on T and X_0 is the zero vector.

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