# 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 $$\frac{{u^{(p - 1)p^{r - 1} } - 1}} {{p^r }} \equiv a_0 (u) + a_1 (u)p + \cdots + a_{r - 1} (u)p^{r - 1} (\bmod p^r ), \gcd (u,p) = 1,$$ where 0 ⩽ a j (u) < p for 0 ⩽ jr−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⩾0 over $$\mathbb{F}_p$$ via introducing a new quotient. Then we determine the exact values of linear complexity of (a j (u)) u⩾0 and values of k-error linear complexity for binary sequences defined by (a j (u)) u⩾0.

## 摘要

### 创新点

1. (1)

通过定义新的商式, 讨论了欧拉商的权位序列的算术性质及最小周期;

2. (2)

确定了欧拉商的最高权位序列的线性复杂度的精确值;

3. (3)

利用最高权位序列定义伪随机二元序列, 确定这些序列的线性复杂度和 k-错线性复杂度的精确值。

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Correspondence to Zhihua Niu.

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