Multivariate permutation polynomial systems and nonlinear pseudorandom number generators

Abstract

In this paper we study a class of dynamical systems generated by iterations of multivariate permutation polynomial systems which lead to polynomial growth of the degrees of these iterations. Using these estimates and the same techniques studied previously for inversive generators, we bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates "on average" over all initial values v ∈ F m+1 p than in the general case and thus can be of use for pseudorandom number generation. p is an interesting and challenging question. We remark that in the case of the so-called inversive generator rather stronger estimates "on average" are available (see [16])

Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators

“…It is easy to see that the proof of Theorem 6 also works for the sequence {u n } defined by (11) and (12). In fact it even shortens a little as some transformations become redundant.…”

“…, F m } induces a permutation of F m p we can obtain rather strong estimates of the discrepancy "on average" over the initial values. First we need the following estimate (which is also a simple unification of several previously known results, see [8,11,14]). …”

“…, m, grow exponentially with k (which is always the case for iterations of nonlinear univariate polynomials). On the other hand, it has been shown in recent works [11,12,13,14,16,17] that there are rich families of multivariate polynomial systems with much slower degree growth and that such families lead to better pseudorandom number generators.…”

“…In particular, the polynomial systems constructed in [11,13,14,16] are of algebraic entropy zero. The degree growth of this class of systems is polynomial in the number of iterations and therefore, it satisfies a linear recurrence.…”

“…to study the distribution of sequences (11). Also, to generate that sequence, repeated applications of A −1 are not necessary, which reduces the time to generate the sequence.…”