Large Families of Pseudorandom Sequences of k Symbols and Their Complexity – Part I

Abstract

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.