Complexity Of Binary Sequences Research Articles

An Efficient Algorithm to Compute the Linear Complexity of Binary Sequences

Binary sequences are algebraic structures currently used as security elements in Internet of Things devices, sensor networks, e-commerce, and cryptography. In this work, a contribution to the evaluation of such sequences is introduced. In fact, we present a novel algorithm to compute a fundamental parameter for this kind of structure: the linear complexity, which is related to the predictability (or non-predictability) of the binary sequences. Our algorithm reduced the computation of the linear complexity to just the addition modulo two (XOR logic operation) of distinct terms of the sequence. The performance of this procedure was better than that of other algorithms found in the literature. In addition, the amount of required sequence to perform this computation was more realistic than in the rest of the algorithms analysed. Tables, figures, and numerical results complete the work.

On the $ k $-error linear complexity of binary sequences of periods $ p^n $ from new cyclotomy

<abstract><p>In this paper, we study the $ k $-error linear complexity of binary sequences with periods $ p^n $, which are derived from new generalized cyclotomic classes modulo a power of an odd prime $ p $. We establish a recursive relation and then estimate the $ k $-error linear complexity of the binary sequences with periods $ p^n $, the results extend the case $ p^2 $ that has been studied in an earlier work of Wu et al. at 2019. Our results show that the $ k $-error linear complexity of these sequences does not decrease dramatically for $ k < (p^{n}-p^{n-1})/2 $.</p></abstract>

Improvements on k-error linear complexity of q-ary sequences derived from Euler quotients

In this paper, we generalize the results about the k-error linear complexity of binary sequences derived from Euler quotients modulo on the q-ary sequences. We also continue the study of q-ary sequences derived from Euler quotients modulo 2p started by R. Mohammed et al. (LNCS, 11634, 2019) and investigate the k-error linear complexity of sequences derived from Euler quotients with a period 2pr . In particular, we generalize the results of Wu et al. (IEEE Access, 8, 2020) about k-error the binary sequences derived by Euler quotients modulo 2p on the same sequences with a period 2pr .

On k-Error Linear Complexity of Binary Sequences Derived From Euler Quotients Modulo 2p

We consider the $k$-error linear complexity of binary sequences derived from Eluer quotients modulo $2p$ ($p>3$ is an odd prime), recently introduced by J. Zhang and C. Zhao. We adopt certain decimal sequences to determine the values of $k$-error linear complexity for all $k>0$. Our results indicate that such sequences have good stability from the viewpoint of cryptography.

Linear complexity of binary sequences with interleaved structure

In this study, the minimal polynomials and the linear complexity of interleaved binary sequences are investigated. Both the linear complexity and the minimal polynomials of low correlation zone sequences constructed by Zhou et al. are completely determined. Besides, an open problem proposed by Li and Tang is discussed. At last, a sufficient condition and a necessary condition are presented about when the linear complexity of the interleaved sequences constructed by Tang et al. attains the maximum.

On the Linear Complexity of Binary Sequences of Period $4N$ With Optimal Autocorrelation Value/Magnitude

Three classes of binary sequences of period 4N with optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period N , i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4N and optimal autocorrelation value/magnitude are discussed in the splitting field of xN-1 . As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based on m-sequences instead of generalized GMW sequences.

Asymptotic analysis on the normalized k-error linear complexity of binary sequences

Linear complexity and k-error linear complexity are the important measures for sequences in stream ciphers. This paper discusses the asymptotic behavior of the normalized k-error linear complexity $${L_{n,k}(\underline{s})/n}$$ of random binary sequences $${\underline{s}}$$ , which is based on one of Niederreiter's open problems. For k = n ?, where 0 ? ? ? 1/2 is a fixed ratio, the lower and upper bounds on accumulation points of $${L_{n,k}(\underline{s})/n}$$ are derived, which holds with probability 1. On the other hand, for any fixed k it is shown that $${\lim_{n\rightarrow\infty} L_{n,k}(\underline{s})/n = 1/2}$$ holds with probability 1. The asymptotic bounds on the expected value of normalized k-error linear complexity of binary sequences are also presented.