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Register a new userAbout the k-Error Linear Complexity over $mathbb{F}_p$ of sequences of length 2$p$ with optimal three-level autocorrelation
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Added by
Vladimir Edemskiy
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Authors
Vladimir Edemskiy
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We investigate the $k$-error linear complexity over $mathbb{F}_p$ of binary sequences of length $2p$ with optimal three-level autocorrelation. These balanced sequences are constructed by cyclotomic classes of order four using a method presented by Ding et al.
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Let $mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $mathcal{R}=mathbb{F}_p+umathbb{F}_p+u^2mathbb{F}_p+u^{3}mathbb{F}_{p}+cdots+u^{k}mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.
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.
We study the $k$-error linear complexity of subsequences of the $d$-ary Sidelnikov sequences over the prime field $mathbb{F}_{d}$. A general lower bound for the $k$-error linear complexity is given. For several special periods, we show that these sequences have large $k$-error linear complexity.
In this paper, the linear complexity over $mathbf{GF}(r)$ of generalized cyclotomic quaternary sequences with period $2pq$ is determined, where $ r $ is an odd prime such that $r ge 5$ and $r otin lbrace p,qrbrace$. The minimal value of the linear complexity is equal to $tfrac{5pq+p+q+1}{4}$ which is greater than the half of the period $2pq$. According to the Berlekamp-Massey algorithm, these sequences are viewed as enough good for the use in cryptography. We show also that if the character of the extension field $mathbf{GF}(r^{m})$, $r$, is chosen so that $bigl(tfrac{r}{p}bigr) = bigl(tfrac{r}{q}bigr) = -1$, $r mid 3pq-1$, and $r mid 2pq-4$, then the linear complexity can reach the maximal value equal to the length of the sequences.
The autocorrelation and the linear complexity of a key stream sequence in a stream cipher are important cryptographic properties. Many sequences with these good properties have interleaved structure, three classes of binary sequences of period $4N$ with optimal autocorrelation values have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period $N$. In this paper, we use the interleaving technique to construct a binary sequence with the optimal autocorrelation of period $2N$, then we calculate its autocorrelation values and its distribution, and give a lower bound of linear complexity. Results show that these sequences have low autocorrelation and the linear complexity satisfies the requirements of cryptography.
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