Linear complexity of Legendre‐polynomial quotients

Abstract

We continue to investigate binary sequence (f u is the Legendre symbol and we restrict 0 p = 1. In an earlier work, the linear complexity of (f u ) was determined for w = p − 1 under the assumption of 2 p−1 ≡ 1 (mod p 2 ). In this work, we give possible values on the linear complexity of (f u ) for all 1 ≤ w < p − 1 under the same conditions. We also state that the case of larger w(≥ p) can be reduced to that of 0 ≤ w ≤ p − 1.

Linear Complexity of New Binary Sequence Derived From Polynomial Quotients Modulo p in General Case and Their Generalizations

“…Many number theoretic and cryptographic questions have been studied for polynomial quotients sequences [2,[7][8][9][10]. It needs to be pointed out that most studies have focused on special cases of f (x) but rarely considered the general cases [7][8][9][10][11][12][13]. There are two families of binary sequences derived from polynomial quotients in general case to be discussed from the view point of cryptography [2].…”