Designs, Codes and Cryptography

, Volume 40, Issue 3, pp 369–374 | Cite as

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

  • Hassan Aly
  • Arne WinterhofEmail author


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
$$L_k(\mathcal{T}) = \left(\frac{p+1}{2} \right)^{m}-1,\quad 1\le k\le \frac{1}{2}\left(\frac{3}{2}\right)^{m}-1.$$


Linear complexity Legendre sequence Sidelnikov sequence Cryptography 

AMS Classification

94A55 11T71 94A60 


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© 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

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