Linear complexity problems of level sequences of Euler quotients and their related binary 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 $$\frac{u^{(p-1)p^{r-1}}-1}{p^r}\equiv a_0(u)+a_1(u)p+\dots +a_{r-1}(u)p^{r-1}(\mathrm{mod}{p}^r),\gcd (u,p)=1,$$ 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 $(a_j(u){)}_{u\ge 0}$.

16 pages

References in corpus (3)

• Trace representation and linear complexity of binary sequences derived from Fermat quotients
• Fermat quotients: Exponential sums, value set and primitive roots
• On the $k$-error linear complexity of binary sequences derived from polynomial quotients

Cited by in corpus (1)

• Linear Complexity of A Family of Binary $p{q}^2$-periodic Sequences From Euler Quotients