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

• Authors: Hassan Aly;Arne Winterhof
• Venue: Designs, Codes and Cryptography
• Year: 2006

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 驴 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 驴 k 驴 (p m 驴 3)/2 and $$L_k(\mathcal{T}) = 0$$ for k驴 (p m 驴 1)/2. In particular, we prove $$L_k(\mathcal{T}) = \left(\frac{p+1}{2} \right)^{m}-1,\quad 1\le k\le \frac{1}{2}\left(\frac{3}{2}\right)^{m}-1.$$