On the linear complexity of Legendre-Sidelnikov sequences

Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s n = 1, n p = −1, 0, otherwise, n ≥ 0, where. p denotes the Legendre symbol. • Sidelnikov Sequence Let q be an odd prime power, g a primitive element of F q , and let η denote the quadratic character of F Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s n = 1, n p = −1, 0, otherwise, n ≥ 0, where. p denotes the Legendre symbol. • Sidelnikov Sequence Let q be an odd prime power, g a primitive element of F q , and let η denote the quadratic character of F • We consider the n-periodic binary sequence (s i) : s i =      1, if (i mod n) ∈ P, 0, if (i mod n) ∈ Q * , 1− " i p " η(g i +1) 2 , if (i mod n) ∈ R, i ≥ 0, where p is an odd prime and q is the power of an odd prime such that gcd(p, q − 1) = 1. • This new sequence is balanced if p = q. (−1) l − 1 + l p 1 + (−1) (p−1)/2 −η(−g l + 1) (1 + (−1) (p−1)/2+(q−1)/2+l) , l ∈ R, q − 1 |l. • This new sequence is balanced if p = q. (−1) l − 1 + l p 1 + (−1) (p−1)/2 −η(−g l + 1) (1 + (−1) (p−1)/2+(q−1)/2+l) , l ∈ R, q − 1 |l. The linear complexity L(S) over F 2 of a binary sequence (s i) is the shortest length L of a linear recurrence relation over F 2