Linear Complexity and Trace Representation of New Ding Generalized Cyclotomic Sequences with Period pq and Order Two

Jiang Ma,Haiyang Jiang,Xiumin Shen,Yanguo Jia,Wei Zhao

doi:10.3390/math9182285

Abstract

Linear complexity is an important property to measure the unpredictability of pseudo-random sequences. Trace representation is helpful for analyzing cryptography properties of pseudo-random sequences. In this paper, a class of new Ding generalized cyclotomic binary sequences of order two with period pq is constructed based on the new segmentation of Ding Helleseth generalized cyclotomy. Firstly, the linear complexity and minimal polynomial of the sequences are investigated. Then, their trace representation is given. It is proved that the sequences have larger linear complexity and can resist the attack of the Berlekamp–Massey algorithm. This paper also confirms that generalized cyclotomic sequences with good randomness may be obtained by modifying the characteristic set of generalized cyclotomy.

Highlights

Sequences with Period pq and OrderPseudo-random sequences are widely used in spread spectrum communication, multiple access communication, radar navigation, software testing, cryptography, and so on
By the Berlekamp–Massey algorithm [2], the linear complexity of a pseudo-random sequence must be greater than the half of its period
We presented the construction of a class of new balanced generalized cyclotomic binary sequences of order two with period pq based on the Ding’s new generalized cyclotomic classes (V0, V1 )

Summary

Introduction

Pseudo-random sequences are widely used in spread spectrum communication, multiple access communication, radar navigation, software testing, cryptography, and so on. Based on the Ding–Helleseth generalized cyclotomy of order two, Ding [6] constructed new generalized cyclotomic classes (V0 , V1 ). By use of these cyclotomic classes, Liu et al [7] constructed the generalized cyclotomic sequences, and calculated the linear complexity and autocorrelation values of the sequences. Bai et al [10] defined a class of balanced binary sequence based on the Ding–Helleseth generalized cyclotomy and calculated the linear complexity. We constructed a class of new balanced generalized cyclotomic sequences with an imbalance degree of 1 based on the Ding’s new generalized cyclotomic classes (V0 , V1 ), and discuss the linear complexity and trace representation of the sequences. According to the definition of the new sequences, their characteristic sets are different from those in [9,11,15], and they belong to different sequences

Preliminaries

The Construction of the New Ding Generalized Cyclotomic Binary Sequences

Linear Complexity and Minimal Polynomial of the New Sequences

Trace Representation of the New Sequences

Conclusions