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Register a new userOn the 4-Adic Complexity of Quaternary Sequences with Ideal Autocorrelation
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Minghui Yang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper, we determine the 4-adic complexity of the balanced quaternary sequences of period $2p$ and $2(2^n-1)$ with ideal autocorrelation defined by Kim et al. (ISIT, pp. 282-285, 2009) and Jang et al. (ISIT, pp. 278-281, 2009), respectively. Our results show that the 4-adic complexity of the quaternary sequences defined in these two papers is large enough to resist the attack of the rational approximation algorithm.
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In cryptography, we hope a sequence over $mathbb{Z}_m$ with period $N$ having larger $m$-adic complexity. Compared with the binary case, the computation of 4-adic complexity of knowing quaternary sequences has not been well developed. In this paper, we determine the 4-adic complexity of the quaternary cyclotomic sequences with period 2$p$ defined in [6]. The main method we utilized is a quadratic Gauss sum $G_{p}$ valued in $mathbb{Z}_{4^N-1}$ which can be seen as a version of classical quadratic Gauss sum. Our results show that the 4-adic complexity of this class of quaternary cyclotomic sequences reaches the maximum if $5 mid p-2$ and close to the maximum otherwise.
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.
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.
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.
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