Autocorrelation Values of Generalized Cyclotomic Sequences with Period pn+1

Recently Edemskiy proposed a method for computing the linear complexity of generalized cyclotomic binary sequences of period p n + 1 , where p = d R + 1 is an odd prime, d , R are two non-negative integers, and n > 0 is a positive integer. In this paper we determine the exact values of autocorrelation of these sequences of period p n + 1 ( n ≥ 0 ) with special subsets. The method is based on certain identities involving character sums. Our results on the autocorrelation values include those of Legendre sequences, prime-square sequences, and prime cube sequences.

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Autocorrelation and Linear Complexity of Binary Generalized Cyclotomic Sequences with Period pq

Journal of Mathematics ◽

10.1155/2021/5535887 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Yan Wang ◽

Liantao Yan ◽

Qing Tian ◽

Liping Ding

Keyword(s):

Autocorrelation Function ◽

Linear Complexity ◽

Minimal Polynomial ◽

Binary Sequences ◽

Cyclotomic Class ◽

Cyclotomic Sequences ◽

Autocorrelation Value

Ding constructed a new cyclotomic class V 0 , V 1 . Based on it, a construction of generalized cyclotomic binary sequences with period p q is described, and their autocorrelation value, linear complexity, and minimal polynomial are confirmed. The autocorrelation function C S w is 3-level if p ≡ 3 mod 4 , and C S w is 5-level if p ≡ 1 mod 4 . The linear complexity LC S > p q / 2 if p ≡ 1 mod 8 , p > q + 1 , or p ≡ 3 mod 4 or p ≡ − 3 mod 8 . The results show that these sequences have quite good cryptographic properties in the aspect of autocorrelation and linear complexity.

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2-Adic and Linear Complexities of a Class of Whiteman’s Generalized Cyclotomic Sequences of Order Four

International Journal of Foundations of Computer Science ◽

10.1142/s0129054119500205 ◽

2019 ◽

Vol 30 (05) ◽

pp. 759-779

Author(s):

Priti Kumari ◽

Pramod Kumar Kewat

Keyword(s):

Linear Complexity ◽

Binary Sequences ◽

Random Sequences ◽

Generalized Cyclotomic Classes ◽

Order Four ◽

Cyclotomic Sequences

Although for more than 20 years, Whiteman’s generalized cyclotomic sequences have been thought of as the most important pseudo-random sequences, but, there are only a few papers in which their 2-adic complexities have been discussed. In this paper, we construct a class of binary sequences of order four with odd length (product of two distinct odd primes) from Whiteman’s generalized cyclotomic classes. After that, we determine both 2-adic complexity and linear complexity of these sequences. Our results show that these complexities are greater than half of the period of the sequences, therefore, it may be good pseudo-random sequences.

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Linear Complexity of Some Generalized Cyclotomic Sequences

International Journal of Algebra and Computation ◽

10.1142/s021819679800020x ◽

1998 ◽

Vol 08 (04) ◽

pp. 431-442 ◽

Cited By ~ 19

Author(s):

Cunsheng Ding

Keyword(s):

Linear Complexity ◽

Linear Span ◽

Binary Sequences ◽

Cyclotomic Sequences

There are several kinds of cyclotomic sequences. They have a number of good random properties. In this paper we compute the linear complexity (linear span) of generalized cyclotomic binary sequences of order 2 with respect to p2.

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ON THE LINEAR COMPLEXITY OF SOME GENERALIZED CYCLOTOMIC SEQUENCES

International Journal of Algebra and Computation ◽

10.1142/s0218196704001827 ◽

2004 ◽

Vol 14 (04) ◽

pp. 431-439 ◽

Cited By ~ 9

Author(s):

YOUNG-HO PARK ◽

DEUKJO HONG ◽

HICHUN EUN

Keyword(s):

Linear Complexity ◽

Binary Sequences ◽

Cyclotomic Sequences

Cunsheng Ding computed the linear complexity of generalized cyclotomic binary sequences of order 2 with respect to p2, but made a mistake in his computation. These errors affect his main theorem seriously. We correct the errors and re-establish the main theorem.

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

Mathematics ◽

10.3390/math9182285 ◽

2021 ◽

Vol 9 (18) ◽

pp. 2285

Author(s):

Jiang Ma ◽

Wei Zhao ◽

Yanguo Jia ◽

Xiumin Shen ◽

Haiyang Jiang

Keyword(s):

Linear Complexity ◽

Minimal Polynomial ◽

Binary Sequences ◽

Characteristic Set ◽

Random Sequences ◽

Generalized Cyclotomy ◽

Trace Representation ◽

Cyclotomic Sequences

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.

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