Some Notes on Generalized Cyclotomic Sequences of Length pq

Abstract

We review the constructions of two main kinds of generalized cyclotomic binary sequences with length pq (the product with two distinct primes). One is the White-generalized cyclotomic sequences, the other is the Ding-Helleseth(DH, for short)-generalized cyclotomic sequences. We present some new pseudo-random properties of DH-generalized cyclotomic sequences using the theory of character sums instead of the theory of cyclotomy, which is a conventional method for investigating generalized cyclotomic sequences.

Cyclic codes from the second class two-prime Whiteman’s generalized cyclotomic sequence with order 6

“…, where (j, h) ∈ {(0, 1), (0, 2), (1, 2), (1, 3), (2, 3), (2,4), (3,4), (3,5), (4, 0), (4, 5), (5, 0), (5, 1)}.…”

Pseudo-Randomness of Certain Sequences of k Symbols with Length pq

“…The (Whiteman) generalized cyclotomic classes were applied to constructing binary sequences in numerous references, see, e.g., [8][9][10][11][12][13][14][15]. In this article, we will extend the constructions of binary sequences to sequences of k symbols.…”

“…The sequence E (1) N is related to the Whitemangeneralized cyclotomic classes over Z N . Ding and Helleseth introduced a new generalized cyclotomic classes [23] , which were called the Ding-Hellesethgeneralized cyclotomic classes in [11].…”

Linear complexity and autocorrelation values of a polyphase generalized cyclotomic sequence of length pq

“…[10], were defined by , where j ¼ 0,1,:::,d # -1. In [17], the (Ding-Helleseth) generalized cyclotomic classes were used to define a polyphase generalized cyclotomic sequence R # ¼ fr í 1 ,r í 2 ,:::g over F d# by where d # is a prime divisor of d = gcd(p -1, q -1), A(i) and B(i) take different values in terms of i belonging to different sub-classes of P and Q 0 , respectively: see [17] for details.…”

“…It exhibits high linear complexity and possesses ideal periodic and aperiodic autocorrelation functions, which make it significant for cryptographic applications, see for example [1][2][3][4][5][6][7]. Many related generalized binary cyclotomic sequences of length pq are also investigated in the literature due to defining different generalized cyclotomic classes [8][9][10][11][12][13][14][15][16]. This article contributes a method to construct generalized polyphase cyclotomic sequences of length pq.…”