Character Values of the Sidelnikov-Lempel-Cohn-Eastman Sequences

TL;DR

This paper investigates the linear complexity of Sidelnikov-Lempel-Cohn-Eastman sequences, which are binary sequences with optimal autocorrelation, by analyzing character values through Jacobi sums to gain new insights.

Contribution

It introduces a novel approach using Jacobi sums to analyze character values, advancing understanding of the sequences' linear complexity.

Findings

Expressed character values in terms of Jacobi sums.

Applied known evaluations of Gauss and Jacobi sums.

Provided new insights into the linear complexity of SLCE sequences.

Abstract

Binary sequences with good autocorrelation properties and large linear complexity are useful in stream cipher cryptography. The Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences have nearly optimal autocorrelation. However, the problem of determining the linear complexity of the SLCE sequences is still open. Our approach is to exploit the fact that character values associated with the SLCE sequences can be expressed in terms of a certain type of Jacobi sum. By making use of known evaluations of Gauss and Jacobi sums in the "pure" and "small index" cases, we are able to obtain new insight into the linear complexity of the SLCE sequences.