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

We determine the linear complexity of a family of p (2)-periodic binary threshold sequences derived from Fermat quotients modulo an odd prime p, where p satisfies . The linear complexity equals p (2) - p or p (2) - 1, depending whether 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 p (2). Moreover, we present a partial result on their linear complexities for primes p with . However such so called Wieferich primes are very rare.