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

Abstract

Abstract. We present several general results that show how algebraic dynamical systems with a slow degree growth and also rational automorphisms can be used to construct stronger pseudorandom number generators. We then give several concrete constructions that illustrate the applicability of these general results.

Reductions modulo primes of systems of polynomial equations and algebraic dynamical systems

“…When this growth is slower than "generic", one can expect stronger bounds. For example, this is true for the following family of systems which stems from that introduced in [OS10], see also [GOS14,OS12].…”

“…The polynomial systems of the form (7.1) have been generalised in various directions, including their rational function analogues [GOS14,HP07]. It is expected that similar improvements hold for all these systems as well.…”

Reduction of polynomial dynamical systems modulo primes

“…When this growth is slower than generic, one can expect stronger bounds. Although for a typical system an exponential degree growth is expected, there are rich families of multivariate polynomial systems with a much slower degree growth (see [4,5,6,7]). For example, for triangular system of polynomials, it has been shown in [6] that degrees of the iterations of the polynomials in triangular system grow very slowly.…”

The Algebraic Theory of Fractional Jumps

“…Generating sequences of pseudorandom numbers is of great importance in applied areas and especially in cryptography and for Monte Carlo methods (for example to compute integrals over the reals). The task of generating streams of pseudorandom numbers is closely related to the study of dynamical systems over finite fields, which have been of great interest recently [11,12,13,14,20,21,18,19]. More in general, for an interesting survey on open problems in arithmetic dynamics see [4].…”