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

欧拉商的权位序列及其二元序列的线性复杂度问题

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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Niu, Z., Chen, Z. & Du, X. Linear complexity problems of level sequences of Euler quotients and their related binary sequences. Sci. China Inf. Sci. 59, 32106 (2016). https://doi.org/10.1007/s11432-015-5305-y

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Keywords

  • cryptography
  • Euler quotients
  • Fermat quotients
  • pseudorandom sequences
  • binary sequences
  • linear complexity
  • k-error linear complexity

关键词

  • 密码学
  • 欧拉商
  • 费马商
  • 伪随机序列
  • 二元序列
  • 线性复杂度
  • k-错线性复杂度