Sequences of irreducible polynomials over odd prime fields via elliptic curve endomorphisms

In this paper we present and analyse a construction of irreducible polynomials over odd prime fields via the transforms which take any polynomial $f \in \mathbf{F}_p[x]$ of positive degree $n$ to $\left(\frac{x}{k} \right)^n \cdot f(k(x+x^{-1}))$, for some specific values of the odd prime $p$ and $k \in \mathbf{F}_p$.Comment: 9 pages. Exposition revised. References update

Sequences of irreducible polynomials without prescribed coefficients over odd prime fields

In this paper we construct infinite sequences of monic irreducible polynomials with coefficients in odd prime fields by means of a transformation introduced by Cohen in 1992. We make no assumptions on the coefficients of the first polynomial $f_0$ of the sequence, which belongs to \F_p [x], for some odd prime $p$, and has positive degree $n$. If $p^{2n}-1 = 2^{e_1} \cdot m$ for some odd integer $m$ and non-negative integer $e_1$, then, after an initial segment $f_0, ..., f_s$ with $s \leq e_1$, the degree of the polynomial $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.Comment: 10 pages. Fixed a typo in the reference

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