Distribution of recursive matrix pseudorandom number generator modulo prime powers

Abstract

Given a matrix A ∈ G L d ( Z ) A\in \mathrm {GL}_d(\mathbb {Z}) . We study the pseudorandomness of vectors u n \mathbf {u}_n generated by a linear recurrence relation of the form u n + 1 ≡ A u n ( mod p t ) , n = 0 , 1 , … , \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*} modulo p t p^t with a fixed prime p p and sufficiently large integer t ⩾ 1 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 u n \mathbf {u}_n and control the p p -adic orders of their coefficients in polynomial representations.