Families of sequences with good family complexity and cross-correlation measure
Kenan Doğan, Murat Şahin, Oğuz Yayla
TL;DR
The paper addresses constructing large families of pseudorandom sequences with high $f$-complexity and low cross-correlation for binary and $k$-symbol alphabets. It advances this goal by generalizing Legendre-symbol constructions and irreducible-polynomial-based families over finite fields, applying Weil-type bounds to control cross-correlation, and extending the $f$-complexity framework to $k$-ary alphabets and dual families. Key contributions include explicit bounds such as $\Phi_\ell(\mathcal{F}) \ll \cdots$ and $C(\mathcal{F}) \ge (\tfrac{1}{2}-o(1))\frac{\log(p/d^2)}{\log 2}$ (binary) and $C(\mathcal{F}) \ge (\tfrac{d}{2}-1)\log_2 p - \log_2((d-1)\log_2 p)$ (binary/large $d$), along with constructions yielding family sizes $|\mathcal{F}|$ on the order of $p^{d-1}/d$ and analogous $k$-ary results. These findings provide systematic methods to generate extensive pseudorandom sequence families with provable guarantees suitable for communications and cryptography.
Abstract
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.