Linear complexity of quaternary sequences of length pq with low autocorrelation

The linear complexity L, of a sequence is an important parameter in its evaluation as a key stream cipher for cryptographic applications. Ideally, good sequences combine the autocorrelation properties of a random sequence with high linear complexity. Cyclotomic and generalized cyclotomic sequences are important pseudorandom sequences in stream ciphers due to their good pseudorandom cryptographic properties and large linear complexity [1]. The study of these sequences properties is the subject of many articles, in most of which the binary cases are considered. Particularly, their linear complexity was explored in [2]-[6] (see also references therein). Quaternary sequences are also the most important sequences in view of many practical applications, see for example [7, 8]. In their paper [9] Z. Yang and P. Ke constructed new quaternary sequences over Zpq of length pq with low autocorrelation by using inverse Gray mapping and generalized cyclotomic sequences over Zp and Zq. On the one hand, it is possible to calculate the linear complexity of quaternary sequences over the finite ring Z4. On the other hand, we can consider sequences defined over F4 (the finite field of 4 elements) with respect to the quaternary sequences by using the Gray map. These two approaches will usually lead to different values for the linear complexity because the arithmetic of F4 is not the same as that of the finite ring Z4. Our goal is to explore the linear complexity of quaternary sequences from [9] for both alternatives. For this we generalize the method of computing the linear complexity of binary sequences with period pq proposed in [6]. Let us briefly recall the definition of sequences from [9]. Let p and q be two different odd primes. According to the Chinese Remainder Theorem ZN ∼= Zp × Zq relatively to isomorphism f(t) = ( t1 , t2 ), where t1 = t (mod p) , t2 = t (mod q). Define P = {p, 2p, ..., (q − 1)p}, Q = {q, 2q, ..., (p− 1)q}. Let φ[a, b] be the inverse Gray mapping defined by φ[0, 0] = 0, φ[0, 1] = 1, φ[1, 1] = 2, φ[1, 0] = 3. By ( x y ) we denote the Legendre symbol. Z. Yang