Autocorrelation of New Generalized Cyclotomic Sequences of Period pn

Let n ≥ 2 be a positive integer and Z× n be the multiplicative group of the integer ring Zn. For a partition {Di|i = 0, 1, · · · , d − 1} of Z× n , if there exist elements g1, · · · , gd of Z× n satisfying Di = giD0 for all i where D0 is a multiplicative subgroup of Z× n , the Di are called generalized cyclotomic classes of order d. There have been lots of studies about cyclotomy with respect to p or p2 or pq where p and q are distinct odd primes [1]–[3]. In 1998, Ding and Helleseth [4] introduced the new generalized cyclotomy with respect to p1 1 · · · pt t and defined a balanced binary sequence based on their own generalized cyclotomy, where p1, · · · , pt are distinct odd primes and e1, · · · , et are positive integers. The linear complexity (LC) of new generalized cyclotomic sequence of order 2 with respect to p2 [2] and p3 [6] is known. Finally, the LC of those sequences of period pn is calculated by by Yan, Li and Xiao [7] and by Kim and Song [8] independently. While the linear complexity is an important measure of pseudo-random sequences for cryptographic application, autocorrelation is another important measure for their application to communication systems for various purposes [5]. In this paper, we compute autocorrelation of new generalized cyclotomic sequences of order 2 with respect to pn for arbitrary positive integer n. Legendre sequences (n = 1) [5], prime square sequences (n = 2) [2], and prime cube sequences (n = 3) [6]–[8] are some important subclasses of these sequences. For simplicity, we will call these sequences as new generalized cyclotomic sequences of period pn. It turned out that their autocorrelation property is not as