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.
Highlights
It is well-known that most of the pseudorandom number generators used in Monte Carlo methods and cryptography are based on the iteration of rational functions, see [9, 18, 19, 22]
We discuss the properties of pseudorandom number generators based on the iteration of several special systems of rational functions that lead to better generators
We present general results of this kind which we apply to certain specific examples which lead to new families of pseudorandom number generators
Summary
The degree growth of this class of systems is polynomial in the number of iterations and it satisfies a linear recurrence This is in full agreement with [1, Conjecture 1], which asserts that the generating function of the degree sequence Dn(F) is rational, that is,. Another class of algebraic dynamical systems which can be useful for designing good pseudorandom number generators is the class of polynomial systems such that their iterations have certain sparsity with respect to some variables (and with strictly positive algebraic entropy) Such are the polynomial systems constructed in [15], for which we have Fi(k) = (Xi − hi)eki Gi + hi for some integers ei, elements hi ∈ Fp and polynomials Gi ∈ Fp[Xi+1, . The letters, m, n, r, s in lower case, always denote integer numbers
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