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Designs, Codes and Cryptography

, Volume 40, Issue 3, pp 369–374

# On the k-error linear complexity over $$\mathbb{F}_p$$ of Legendre and Sidelnikov sequences

Article

## Abstract

We determine exact values for the k-error linear complexity L k over the finite field $$\mathbb{F}_{p}$$ of the Legendre sequence $$\mathcal{L}$$ of period p and the Sidelnikov sequence $$\mathcal{T}$$ of period p m  − 1. The results are
$$L_k(\mathcal{L}) =\left\{\begin{array}{ll} (p+1)/2, \quad 1 \le k \le (p-3)/2,\\ 0, \quad k\ge (p-1)/2, \end{array}\right.$$
$$L_k(\mathcal{T})\ge \min \left( \left( \frac{p+1}{2} \right)^{m}-1, \left \lceil \frac{p^m-1}{k+1} \right \rceil - \left(\frac{p+1}{2} \right)^{m} + 1 \right)$$
for 1 ≤ k ≤ (p m  − 3)/2 and $$L_k(\mathcal{T}) = 0$$ for k≥ (p m  − 1)/2. In particular, we prove
$$L_k(\mathcal{T}) = \left(\frac{p+1}{2} \right)^{m}-1,\quad 1\le k\le \frac{1}{2}\left(\frac{3}{2}\right)^{m}-1.$$

## Keywords

Linear complexity Legendre sequence Sidelnikov sequence Cryptography

## AMS Classification

94A55 11T71 94A60

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## Copyright information

© Springer Science + Business Media, LLC 2006

## Authors and Affiliations

1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria