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Keywords = generalized cyclotomy

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8 pages, 252 KB  
Article
A Construction of Optimal One-Coincidence Frequency-Hopping Sequences via Generalized Cyclotomy
by Minfeng Shao and Ying Miao
Entropy 2024, 26(11), 935; https://doi.org/10.3390/e26110935 - 31 Oct 2024
Cited by 3 | Viewed by 1145
Abstract
Frequency-hopping sequences (FHSs) with low Hamming correlation are essential for synchronization and multiple-access communication systems. In this paper, we propose a novel construction of FHSs using generalized cyclotomy. Our results reveal that the constructed FHSs exhibit a one-coincidence property, meaning that the smallest [...] Read more.
Frequency-hopping sequences (FHSs) with low Hamming correlation are essential for synchronization and multiple-access communication systems. In this paper, we propose a novel construction of FHSs using generalized cyclotomy. Our results reveal that the constructed FHSs exhibit a one-coincidence property, meaning that the smallest correlation between different FHSs, aside from the trivial case, is minimized. Additionally, the new sets of FHSs achieve an optimal size in relation to a known theoretical bound. Full article
(This article belongs to the Special Issue Advances in Information and Coding Theory, the Third Edition)
15 pages, 271 KB  
Article
On the Stability of the Linear Complexity of Some Generalized Cyclotomic Sequences of Order Two
by Chi Yan and Chengliang Tian
Mathematics 2024, 12(16), 2483; https://doi.org/10.3390/math12162483 - 11 Aug 2024
Viewed by 1339
Abstract
Linear complexity is an important pseudo-random measure of the key stream sequence in a stream cipher system. The 1-error linear complexity is used to measure the stability of the linear complexity, which means the minimal linear complexity of the new sequence by changing [...] Read more.
Linear complexity is an important pseudo-random measure of the key stream sequence in a stream cipher system. The 1-error linear complexity is used to measure the stability of the linear complexity, which means the minimal linear complexity of the new sequence by changing one bit of the original key stream sequence. This paper contributes to calculating the exact values of the linear complexity and 1-error linear complexity of the binary key stream sequence with two prime periods defined by Ding–Helleseth generalized cyclotomy. We provide a novel method to solve such problems by employing the discrete Fourier transform and the M–S polynomial of the sequence. Our results show that, by choosing appropriate parameters p and q, the linear complexity and 1-error linear complexity can be no less than half period, which shows that the linear complexity of this sequence not only meets the requirements of cryptography but also has good stability. Full article
(This article belongs to the Special Issue Coding Theory and the Impact of AI)
13 pages, 282 KB  
Article
Linear Complexity and Trace Representation of New Ding Generalized Cyclotomic Sequences with Period pq and Order Two
by Jiang Ma, Wei Zhao, Yanguo Jia, Xiumin Shen and Haiyang Jiang
Mathematics 2021, 9(18), 2285; https://doi.org/10.3390/math9182285 - 16 Sep 2021
Cited by 1 | Viewed by 2086
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 [...] Read more.
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. Full article
9 pages, 746 KB  
Article
Autocorrelation Values of Generalized Cyclotomic Sequences with Period pn+1
by Xiaolin Chen and Huaning Liu
Mathematics 2019, 7(10), 950; https://doi.org/10.3390/math7100950 - 12 Oct 2019
Viewed by 2040
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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