# Jacobi Sums and Correlations of Sidelnikov Sequences

## TL;DR

This paper derives explicit formulas for the cross-correlation of Sidelnikov sequences over finite fields using Jacobi sums, enabling easier computation and analysis of their correlation properties.

## Contribution

It introduces a Jacobi sum-based method to compute cross-correlation values of Sidelnikov sequences, including explicit formulas in special cases.

## Key findings

- Cross-correlation values expressed via Jacobi sums.
- Explicit formulas derived for specific modular conditions.
- Facilitates efficient correlation analysis of Sidelnikov sequences.

## Abstract

We consider the problem of determining the cross-correlation values of the sequences in the families comprised of constant multiples of $M$-ary Sidelnikov sequences over $\mathbb{F}_q$, where $q$ is a power of an odd prime $p$. We show that the cross-correlation values of pairs of sequences from such a family can be expressed in terms of certain Jacobi sums. This insight facilitates the computation of the cross-correlation values of these sequence pairs so long as $\phi(M)^{\phi(M)} \leq q.$ We are also able to use our Jacobi sum expression to deduce explicit formulae for the cross-correlation distribution of a family of this type in the special case that there exists an integer $x$ such that $p^x \equiv -1 \pmod{M}.$