On the Linear Complexity of Generalized Cyclotomic Sequences of Order Four Over Zpq

Pseudorandom sequences with certain properties are widely used in simulation, software testing, ranging systems, global positioning systems, code-division multiple-access systems, radar systems, spread-spectrum communication systems, and especially in stream ciphers. The cyclotomic sequences [1] have a number of attractive randomness properties. In this letter we first describe generalized cyclotomic sequence of order four with respect to pq (GCS4 for short), then we study important properties of this sequence such as its linear complexity and minimal polynomial. Let p and q (p < q) be two odd primes with gcd(p − 1, q − 1) = 4. Define N = pq, e = (p − 1)(q − 1)/4. The Chinese Remainder Theorem guarantees that there exists a common primitive root g of both p and q. Therefore,