Linear complexity problems of level sequences of Euler quotients and their related binary sequences

TL;DR

This paper investigates the arithmetic properties and linear complexity of level sequences derived from Euler quotients modulo prime powers, providing exact and error linear complexity values for related binary sequences.

Contribution

It introduces a new quotient to analyze level sequences of Euler quotients and determines their linear and k-error linear complexities.

Findings

Exact linear complexity values for the sequences.

Determination of k-error linear complexity.

Analysis of arithmetic properties of level sequences.

Abstract

The Euler quotient modulo an odd-prime power $p^r~(r>1)$ can be uniquely decomposed as a $p$-adic number of the form $$ \frac{u^{(p-1)p^{r-1}} -1}{p^r}\equiv a_0(u)+a_1(u)p+\ldots+a_{r-1}(u)p^{r-1} \pmod {p^r},~ \gcd(u,p)=1, $$ where $0\le a_j(u)<p$ for $0\le j\le r-1$ and we set all $a_j(u)=0$ if $\gcd(u,p)>1$. We firstly study certain arithmetic properties of the level sequences $(a_j(u))_{u\ge 0}$ over $\mathbb{F}_p$ via introducing a new quotient. Then we determine the exact values of linear complexity of $(a_j(u))_{u\ge 0}$ and values of $k$-error linear complexity for binary sequences defined by $(a_j(u))_{u\ge 0}$.