Distribution of recursive matrix pseudorandom number generator modulo prime powers

TL;DR

This paper investigates the distribution properties of vectors generated by a linear recurrence modulo prime powers, introducing new techniques based on Weyl sums and polynomial representations to analyze short segments.

Contribution

It develops a novel approach combining Weyl sum estimates and polynomial representations to study pseudorandomness of sequences modulo prime powers, especially over short segments.

Findings

Sequences exhibit pseudorandom distribution properties over short segments.

The method extends previous techniques by incorporating explicit polynomial representations.

Results apply to sequences generated by linear recurrences modulo prime powers.

Abstract

Given a matrix $A\in \mathrm{GL}_d(\mathbb{Z})$. We study the pseudorandomness of vectors $\mathbf{u}_n$ generated by a linear recurrent relation of the form $$ \mathbf{u}_{n+1} \equiv A \mathbf{u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots, $$ modulo $p^t$ with a fixed prime $p$ and sufficiently large integer $t \geq 1$. We study such sequences over very short segments of length which is not accessible via previously used methods. Our technique is based on the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford (2002). This is combined with some ideas from the work of I. E. Shparlinski (1978) which allows to construct polynomial representations of the coordinates of $\mathbf{u}_n$ and control the $p$-adic orders of their coefficients in polynomial representation.