Families of sequences with good family complexity and cross-correlation measure

In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.Comment: 13 pages. Comments are welcome

The cross-correlation measure for families of binary sequences

Large families of binary sequences of the same length are considered and a new measure, the cross-correlation measure of order $k$ is introduced to study the connection between the sequences belonging to the family. It is shown that this new measure is related to certain other important properties of families of binary sequences. Then the size of the cross-correlation measure is studied. Finally, the cross-correlation measures of two important families of pseudorandom binary sequences are estimated

LARGE FAMILIES OF PSEUDORANDOM SUBSETS FORMED BY POWER RESIDUES

International audienceIn an earlier paper the authors introduced the measures of pseudo-randomness of subsets of the set of the positive integers not exceeding N , and they also presented two examples for subsets possessing strong pseudorandom properties. One of these examples included permutation polynomials f (X) ∈ F p [X] and d-powers in F p. This construction is not of much practical use since very little is known on permutation polynomials and there are only very few of them. Here the construction is extended to a large class of polynomials which can be constructed easily, and it is shown that all the subsets belonging to the large family of subsets obtained in this way possess strong pseudorandom properties. The complexity of this large family is also studied