Trace representation and linear complexity of binary sequences derived from Fermat quotients

TL;DR

This paper analyzes binary sequences derived from Fermat quotients, providing trace representations and linear complexity results, including new findings for Legendre-Fermat quotient sequences under specific modular conditions.

Contribution

It introduces trace representations for two families of sequences from Fermat quotients and establishes linear complexity results, including new insights for Legendre-Fermat sequences.

Findings

Trace representations of the sequences are derived.

Linear complexity of binary threshold sequences is reaffirmed.

New linear complexity results are obtained for Legendre-Fermat quotient sequences.

Abstract

We describe the trace representations of two families of binary sequences derived from Fermat quotients modulo an odd prime $p$ (one is the binary threshold sequences, the other is the Legendre-Fermat quotient sequences) via determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of pre-images modulo $p^2$ of each fixed value of Fermat quotients. From the defining pairs, we can obtain an earlier result of linear complexity for the binary threshold sequences and a new result of linear complexity for the Legendre-Fermat quotient sequences under the assumption of $2^{p-1}\not\equiv 1 \bmod {p^2}$.