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Register a new userDetermination of 2-Adic Complexity of Generalized Binary Sequences of Order 2
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Minghui Yang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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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 $n$.
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 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.
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.
Via interleaving Ding-Helleseth-Lam sequences, a class of binary sequences of period $4p$ with optimal autocorrelation magnitude was constructed in cite{W. Su}. Later, Fan showed that the linear complexity of this class of sequences is quite good cite{C. Fan}. Recently, Sun et al. determined the upper and lower bounds of the 2-adic complexity of such sequences cite{Y. Sun3}. We determine the exact value of the 2-adic complexity of this class of sequences. The results show that the 2-adic complexity of this class of binary sequences is close to the maximum.
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