On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators

Ostafe, Alina; Shparlinski, Igor E.

doi:10.1090/s0025-5718-09-02271-6

Cited by 25 publications

(50 citation statements)

References 46 publications

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“…, 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.…”

Section: It Is Certainly Natural To Expect That the Degrees Of The Itmentioning

confidence: 99%

“…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.…”

Section: It Is Certainly Natural To Expect That the Degrees Of The Itmentioning

confidence: 99%

“…As in [13], we can describe explicitly the iterations of the rational functions F i as follows. We define…”

Section: 4mentioning

confidence: 99%

“…The initial part of the argument is essentially a repetition with some minor modification of the standard approach, see [7,10,13], so we suppress some details. We consider the sum S I (a; N ) defined by (4) and for a sufficiently large integer K ≥ 1, we have…”

Section: 4mentioning

confidence: 99%

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

2013

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Section: It Is Certainly Natural To Expect That the Degrees Of The Itmentioning

confidence: 99%

Section: It Is Certainly Natural To Expect That the Degrees Of The Itmentioning

confidence: 99%

“…As in [13], we can describe explicitly the iterations of the rational functions F i as follows. We define…”

Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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

2013

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“…A particularly interesting and fruitful bridge to travel is the one linking dynamical systems concepts with number-theoretic ideas, and indeed several well established subfields in pure mathematics lie at the interface between number theory and dynamics, namely arithmetic dynamics, dynamics over finite fields, or symbolic dynamics to cite some examples (see [1][2][3][4] for just a few relevant references).…”

Section: Introductionmentioning

confidence: 99%

On a Dynamical Approach to Some Prime Number Sequences

Lacasa

Luque

Gomez

et al. 2018

Entropy

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Abstract:We show how the cross-disciplinary transfer of techniques from dynamical systems theory to number theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo k are maximally chaotic and, while lacking forbidden patterns, unexpectedly display a non-trivial spectrum of Renyi entropies which suggest that every block of size m > 1, while admissible, occurs with different probability. This non-uniform distribution of blocks for m > 1 contrasts Dirichlet's theorem that guarantees equiprobability for m = 1. We then explore in a similar fashion the sequence of prime gap residues. We numerically find that this sequence is again chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as forbidden patterns emerge for every block of size m > 1. We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy-Littlewood k-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the chaos game to these symbolic sequences, and comparing the Iterated Function System (IFS) attractors found for the experimental sequences with appropriate null models.

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On Multivariate Algorithms of Digital Signatures on Secure El Gamal‐Type Mode

USTIMENKO

2024

Computational Methods and Mathematical Modeling in Cyberphysics and Engineering Applications 1

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On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators

Cited by 25 publications

References 46 publications

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

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

On a Dynamical Approach to Some Prime Number Sequences

On Multivariate Algorithms of Digital Signatures on Secure El Gamal‐Type Mode

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