On $k$-error linear complexity of pseudorandom binary sequences derived from Euler quotients

TL;DR

This paper studies the $k$-error linear complexity of pseudorandom binary sequences derived from Euler quotients modulo powers of an odd prime, providing recursive relations and exact values for certain cases.

Contribution

It establishes recursive formulas for the $k$-error linear complexity of these sequences for $ extgreater=3$, including exact values when $ =3$, advancing understanding of their cryptographic strength.

Findings

$k$-error linear complexity remains stable for certain $k$ values.

Recursive relations are derived for $ extgreater=3$.

Exact $k$-error linear complexity values are provided for $ =3$.

Abstract

We investigate the $k$-error linear complexity of pseudorandom binary sequences of period $p^{\mathfrak{r}}$ derived from the Euler quotients modulo $p^{\mathfrak{r}-1}$, a power of an odd prime $p$ for $\mathfrak{r}\geq 2$. When $\mathfrak{r}=2$, this is just the case of polynomial quotients (including Fermat quotients) modulo $p$, which has been studied in an earlier work of Chen, Niu and Wu. In this work, we establish a recursive relation on the $k$-error linear complexity of the sequences for the case of $\mathfrak{r}\geq 3$. We also state the exact values of the $k$-error linear complexity for the case of $\mathfrak{r}=3$. From the results, we can find that the $k$-error linear complexity of the sequences (of period $p^{\mathfrak{r}}$) does not decrease dramatically for $k<p^{\mathfrak{r}-2}(p-1)^2/2$.