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Register a new userBinary sequences with length n and nonlinear complexity not less than n/2
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Added by
Zibi Xiao
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all binary sequences with length $n$ and nonlinear complexity $c_{n}geq n/2$, where $n$ is an integer larger than $2$. Furthermore, a formula is established to calculate the exact number of these sequences. The distribution of nonlinear complexity for these sequences is thus completely determined.
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The generalized binary sequences of order 2 have been used to construct good binary cyclic codes [4]. The linear complexity of these sequences has been computed in [2]. The autocorrelation values of such sequences have been determined in [1] and [3]. Some lower bounds of 2-adic complexity for such sequences have been presented in [5] and [7]. In this paper we determine the exact value of 2-adic complexity for such sequences. Particularly, we improve the lower bounds presented in [5] and [7] and the condition for the 2-adic complexity reaching the maximum value.
The autocorrelation values of two classes of binary sequences are shown to be good in [6]. We study the 2-adic complexity of these sequences. Our results show that the 2-adic complexity of such sequences is large enough to resist the attack of the rational approximation algorithm.
We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, Gauss periods and quadratic Gauss sum on finite field $mathbb{F}_q$ and valued in $mathbb{Z}_{2^N-1}$ where $q equiv 5pmod 8$ is a prime number and $N=2q$ is the period of the DHM sequences.
The generalized cyclotomic binary sequences $S=S(a, b, c)$ with period $n=pq$ have good autocorrelation property where $(a, b, c)in {0, 1}^3$ and $p, q$ are distinct odd primes. For some cases, the sequences $S$ have ideal or optimal autocorrelation. In this paper we determine the autocorrelation distribution and 2-adic complexity of the sequences $S=S(a, b, c)$ for all $(a, b, c)in {0, 1}^3$ in a unified way by using group ring language and a version of quadratic Gauss sums valued in group ring $R=mathbb{Z}[Gamma]$ where $Gamma$ is a cyclic group of order $n$.
A class of binary sequences with period $2p$ is constructed using generalized cyclotomic classes, and their linear complexity, minimal polynomial over ${mathbb{F}_{{q}}}$ as well as 2-adic complexity are determined using Gauss period and group ring theory. The results show that the linear complexity of these sequences attains the maximum when $pequiv pm 1(bmod~8)$ and is equal to {$p$+1} when $pequiv pm 3(bmod~8)$ over extension field. Moreover, the 2-adic complexity of these sequences is maximum. According to Berlekamp-Massey(B-M) algorithm and the rational approximation algorithm(RAA), these sequences have quite good cryptographyic properties in the aspect of linear complexity and 2-adic complexity.
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