Correlation measures of binary sequences derived from Euler quotients

TL;DR

This paper investigates the correlation properties of binary sequences derived from Euler quotients, revealing their high correlation measures which limit their cryptographic suitability.

Contribution

It introduces a new analysis approach using Dirichlet characters, Ramanujan sums, and Gauss sums to study correlation measures of these sequences.

Findings

4-order correlation measures are very large

Sequences may not be suitable for cryptography

Provides new analytical methods for correlation analysis

Abstract

Fermat-Euler quotients arose from the study of the first case of Fermat's Last Theorem, and have numerous applications in number theory. Recently they were studied from the cryptographic aspects by constructing many pseudorandom binary sequences, whose linear complexities and trace representations were calculated. In this work, we further study their correlation measures by using the approach based on Dirichlet characters, Ramanujan sums and Gauss sums. Our results show that the $4$-order correlation measures of these sequences are very large. Therefore they may not be suggested for cryptography.