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

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 $2^{p-1}\not\equiv 1\mathrm{mod}{p}^2$.

14 pages, no figures

References in corpus (2)

• Fermat quotients: Exponential sums, value set and primitive roots
• The Arithmetic of Carmichael Quotients

Cited by in corpus (9)

• On the $k$-error linear complexity of binary sequences derived from polynomial quotients
• On $k$-error linear complexity of pseudorandom binary sequences derived from Euler quotients
• Linear Complexity of A Family of Binary $p{q}^2$-periodic Sequences From Euler Quotients
• Linear complexity problems of level sequences of Euler quotients and their related binary sequences
• Trace representation of pseudorandom binary sequences derived from Euler quotients
• Linear complexity of Legendre-polynomial quotients
• Polynomial quotients: Interpolation, value sets and Waring's problem
• On k-error linear complexity of binary sequences derived from Euler quotients modulo 2p
• On error linear complexity of new generalized cyclotomic binary sequences of period $p^2$