Christian Mauduit

Large families of pseudorandom sequences of k symbols and their complexity: part II

In earlier papers we introduced the measures of pseudorandomness of finite binary sequences [13], introduced the notion of f–complexity of families of binary sequences, constructed large families of binary sequences with strong PR (= pseudorandom) properties [6], [12], and we showed that one of the earlier constructions can be modified to obtain families with high f–complexity [4]. In another paper [14] we extended the study of pseudorandomness from binary sequences to sequences on k symbols (“letters”). In [14] we also constructed one “good” pseudorandom sequence of a given length on k symbols. However, in the applications we need not only a few good sequences but large families of them, and in certain applications (cryptography) the complexity of the family of these sequences is more important than its size. In this paper our goal is to construct “many” “good” PR sequences on k symbols, to extend the notion of f–complexity to the k symbol case and to study this extended f–complexity concept.

Finite and infinite pseudorandom binary words

We present a notion of pseudorandomness for finite binary words based on the measure of the well distribution in arithmetic progressions and of the correlations. We give several examples and we focuss our interest on two arithmetical constructions connected to the Legendre symbol and the Liouville function.

An algorithm for the word entropy

For any infinite word

Fonctions digitales le long des nombres premiers

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Spectral theory and geometric representation of substitutions

on a finite alphabet

Polynomial dynamical systems associated with substitutions

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Somme des chiffres et répartition dans les classes de congruence pour les palindromes ellipséphiques

, the complexity function

Piecewise linear transformations of the unit interval and Cantor sets

p_w

Substitutions and symbolic dynamical systems

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Some open problems

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