Distribution of recursive matrix pseudorandom number generator modulo prime powers

Mérai, László; Shparlinski, Igor

doi:10.1090/mcom/3895

Distribution of recursive matrix pseudorandom number generator modulo prime powers
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by László Mérai and Igor E. Shparlinski;

Math. Comp. 93 (2024), 1355-1370

DOI: https://doi.org/10.1090/mcom/3895

Published electronically: October 25, 2023

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Abstract:

Given a matrix $A\in \mathrm {GL}_d(\mathbb {Z})$. We study the pseudorandomness of vectors $\mathbf {u}_n$ generated by a linear recurrence relation of the form \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*} modulo $p^t$ with a fixed prime $p$ and sufficiently large integer $t \geqslant 1$. We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of $\mathbf {u}_n$ and control the $p$-adic orders of their coefficients in polynomial representations.

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