On the linear complexity of binary threshold sequences derived from Fermat quotients

We determine the linear complexity of a family of p2-periodic binary threshold sequences derived fromFermat quotientsmodulo an odd prime p, where p satisfies 2p−1 ≡ 1 (mod p2). The linear complexity equals p2 − p or p2 − 1, depending whether p ≡ 1 or 3 (mod 4). Our research extends the results from previous work on the linear complexity of the corresponding binary threshold sequences when 2 is a primitive root modulo p2. Moreover, we present a partial result on their linear complexities for primes p with 2p−1 ≡ 1 (mod p2). However such so called Wieferich primes are very rare.