A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f–complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f–complexity. Finally, the cardinality of the smallest family achieving a prescibed f–complexity and multiplicity is estimated. * Research partially supported by Hungarian National Foundation for Scientific Research, Grant No. T029759 and MKM fund FKFP–0139/1997. 1991 AMS Mathematics Subject Classification number: primary 11K45, secondary 94A60, 68Q25 List of key words and phrases: pseudorandom, complexity, binary sequence, specification, Legendre symbol, covering hypergraphs. 1