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Register a new userLinear Complexity and Autocorrelation of two Classes of New Interleaved Sequences of Period $2N$
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Tongjiang Yan
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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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Binary periodic sequences with good autocorrelation property have many applications in many aspects of communication. In past decades many series of such binary sequences have been constructed. In the application of cryptography, such binary sequences are required to have larger linear complexity. Tang and Ding cite{X. Tang} presented a method to construct a series of binary sequences with period 4$n$ having optimal autocorrelation. Such sequences are interleaved by two arbitrary binary sequences with period $nequiv 3pmod 4$ and ideal autocorrelation. In this paper we present a general formula on the linear complexity of such interleaved sequences. Particularly, we show that the linear complexity of such sequences with period 4$n$ is not bigger than $2n+2$. Interleaving by several types of known binary sequences with ideal autocorrelation ($m$-sequences, Legendre, twin-prime and Halls sequences), we present many series of such sequences having the maximum value $2n+2$ of linear complexity which gives an answer of a problem raised by N. Li and X. Tang cite{N. Li}. Finally, in the conclusion section we show that it can be seen easily that the 2-adic complexity of all such interleaved sequences reaches the maximum value $log_{2}(2^{4n}-1)$.
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 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.
We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set ${1,2}^mathbb{N}$ of directive sequences. For a given set $mathcal{C}$ of two substitutions, we show that there exists a $mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $mu$-almost every $mathcal{C}$-adic sequence is balanced, where $mu$ is any shift-invariant ergodic Borel probability measure on ${1,2}^mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $mu$ is negative.
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.
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