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arXiv · 1410.2182

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

Zhihua Niu·Zhixiong Chen·Xiaoni Du
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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 $0\le a_j(u) 1$. 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 $(a_j(u))_{u\ge 0}$.

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Zhihua Niu, Zhixiong Chen, Xiaoni Du. 2014-09-20. Linear complexity problems of level sequences of Euler quotients and their related binary sequences. https://doi.org/10.1007/s11432-015-5305-y

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