Linear Complexity of Pseudorandom Sequences Derived from Polynomial Quotients: General Cases

We determine the linear complexity of binary sequences derived from the polynomial quotient modulo p defined by F(u) equivalent to f(u) - f(p)(u)/p (mod p), 0 <= F(u) <= p -1, u >= 0, where f(p)(u) equivalent to f(u) (mod p), for general polynomials f(x) epsilon Z[x}. The linear complexity equals to one of the following values [p(2) - p, p(2) - p + 1, p(2) - 1, p(2) - 1, p(2)} if if 2 is a primitive root modulo p(2), depending on p equivalent to 1 or 3 modulo 4 and the number of solutions of f'(u) equivalent to 0 (mod p), where f'(x) is the derivative of f(x). Furthermore, we extend the constructions to d-ary sequences for prime d vertical bar(p - 1) and d being a primitive root modulo p(2).