Aperiodic pseudorandom number generators based on infinite words

In this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators (PRNGs) with good statistical behavior. We introduce the well distributed occurrences (WELLDOC) combinatorial property for infinite words, which guarantees absence of the lattice structure defect in related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WELLDOC property if, for each factor w of u, positive integer m, and vector v in (Z_d)^m, there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. (The Parikh vector of a finite word v over an alphabet A has its i-th component equal to the number of occurrences of the i-th letter of A in v.) We prove that Sturmian words, and more generally Arnoux–Rauzy words and some morphic images of them, have the WELLDOC property. Using the TestU01 and PractRand statistical tests, we moreover show that not only the lattice structure is absent, but also other important properties of PRNGs are improved when linear congruential generators are combined using infinite words having the WELLDOC property

On implemented graph based generator of cryptographically strong pseudorandom sequences of multivariate nature

Classical Multivariate Cryptography (MP) is searching for special families of functions of kind ^nF=T_1FTT_2 on the vector space V= (F_q)^n where F is a quadratic or cubical polynomial map of the space to itself, T_1 and T^2 are affine transformations and T is the piece of information such that the knowledge of the triple T_1, T_2, T allows the computation of reimage x of given nF(x) in polynomial time O(n^ᾳ). Traditionally F is given by the list of coefficients C(^nF) of its monomial terms ordered lexicographically. We consider the Inverse Problem of MP of finding T_1, T_2, T for F given in its standard form. The solution of inverse problem is harder than finding the procedure to compute the reimage of ^nF in time O(n^ᾳ). For general quadratic or cubic maps nF this is NP hard problem. In the case of special family some arguments on its inclusion to class NP has to be given

Derivated sequences of complementary symmetric Rote sequences

Complementary symmetric Rote sequences are binary sequences which have factor complexity $\mathcal{C}(n) = 2n$ for all integers $n \geq 1$ and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the prefixes of any complementary symmetric Rote sequence $\mathbf{v}$ which is associated with a standard Sturmian sequence $\mathbf{u}$. We show that any non-empty prefix of $\mathbf{v}$ has three return words. We prove that any derivated sequence of $\mathbf{v}$ is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that $\mathbf{v}$ is primitive substitutive if and only if $\mathbf{u}$ is primitive substitutive. Moreover, if the sequence $\mathbf{u}$ is a fixed point of a primitive morphism, then all derivated sequences of $\mathbf{v}$ are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms

Aperiodic pseudorandom number generators based on infinite words

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