# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

May 2019 , Volume 13 , Issue 2

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2019, 13(2): 213-220 doi: 10.3934/amc.2019014 +[Abstract](4450) +[HTML](517) +[PDF](295.67KB)
Abstract:

In this paper, we consider additive circulant graph codes which are self-dual additive \begin{document}$\mathbb{F}_4$\end{document}-codes. We classify all additive circulant graph codes of length \begin{document}$n = 30, 31$\end{document} and \begin{document}$34 \le n \le 40$\end{document} having the largest minimum weight. We also classify bordered circulant graph codes of lengths up to 40 having the largest minimum weight.

2019, 13(2): 221-233 doi: 10.3934/amc.2019015 +[Abstract](4795) +[HTML](430) +[PDF](480.46KB)
Abstract:

In this paper, we present a comparison study on three RLWE key exchange protocols: one from Ding et al. in 2012 (DING12) and two from Alkim et al. in 2016 (NewHope and NewHope-Simple). We compare and analyze protocol construction, notion of designing and realizing key exchange, signal computation, error reconciliation and cost of these three protocols. We show that NewHope and NewHope-Simple share very similar notion as DING12 in the sense that NewHope series also send small additional bits with small size (i.e. signal) to assist error reconciliation, where this idea was first practically proposed in DING12. We believe that DING12 is the first work that presented complete LWE & RLWE-based key exchange constructions. The idea of sending additional information in order to realize error reconciliation and key exchange in NewHope and NewHope-Simple remain the same as DING12, despite concrete approaches to compute signal and reconcile error are not the same.

2019, 13(2): 235-251 doi: 10.3934/amc.2019016 +[Abstract](4132) +[HTML](365) +[PDF](428.21KB)
Abstract:

We introduce a new property for mixing layers which guarantees protection against algebraic attacks based on the imprimitivity of the group generated by the round functions. Mixing layers satisfying this property are called non-type-preserving. Our main result is to characterize such mixing layers by providing a list of necessary and sufficient conditions on the structure of their underlying binary matrices. Then we show how several families of linear maps are non-type-preserving, including the mixing layers of AES, GOST and PRESENT. Finally we prove that the group generated by the round functions of an SPN cipher with addition modulo \begin{document}$2^n$\end{document} as key mixing function is primitive if its mixing layer satisfies this property.

2019, 13(2): 253-265 doi: 10.3934/amc.2019017 +[Abstract](4604) +[HTML](364) +[PDF](378.05KB)
Abstract:

In this paper, for any even integer \begin{document}$n = 2m\ge4$\end{document}, a new construction of \begin{document}$n$\end{document}-variable rotation symmetric bent function with maximal algebraic degree \begin{document}$m$\end{document} is given as

whose dual function is

where \begin{document}$\overline{x_{i}} = x_{i}\oplus 1$\end{document} and the subscript of \begin{document}$x$\end{document} is modulo \begin{document}$n$\end{document}.

2019, 13(2): 267-280 doi: 10.3934/amc.2019018 +[Abstract](5105) +[HTML](456) +[PDF](417.6KB)
Abstract:

Due to their important applications in theory and practice, linear complementary dual (LCD) codes and self-orthogonal codes have received much attention in the last decade. The objective of this paper is to extend a recent construction of binary LCD codes and self-orthogonal codes to the general \begin{document}$p$\end{document}-ary case, where \begin{document}$p$\end{document} is an odd prime. Based on the extended construction, several classes of \begin{document}$p$\end{document}-ary linear codes are obtained. The characterizations of these linear codes to be LCD or self-orthogonal are derived. The duals of these linear codes are also studied. It turns out that the proposed linear codes are optimal in many cases in the sense that their parameters meet certain bounds on linear codes. The weight distributions of these linear codes are settled.

2019, 13(2): 281-296 doi: 10.3934/amc.2019019 +[Abstract](5357) +[HTML](533) +[PDF](410.75KB)
Abstract:

This paper presents some new constructions of LCD, isodual, self-dual and LCD-isodual codes based on the structure of repeated-root constacyclic codes. We first characterize repeated-root constacyclic codes in terms of their generator polynomials and lengths. Then we provide simple conditions on the existence of repeated-root codes which are either self-dual negacyclic or LCD cyclic and negacyclic. This leads to the construction of LCD, self-dual, isodual, and LCD-isodual cyclic and negacyclic codes.

2019, 13(2): 297-312 doi: 10.3934/amc.2019020 +[Abstract](4013) +[HTML](324) +[PDF](422.14KB)
Abstract:

Let \begin{document}$W = \{w_1, w_2, \cdots, w_r\}$\end{document} be a set of \begin{document}$r$\end{document} integers greater than 1, \begin{document}$\Lambda_a = (\lambda_a^{(1)}, \lambda_a^{(2)}, \cdots, \lambda_a^{(r)})$\end{document} be an \begin{document}$r$\end{document}-tuple of positive integers, \begin{document}$\lambda_c$\end{document} be a positive integer, and \begin{document}$Q = (q_1, q_2, \cdots, q_r)$\end{document} be an \begin{document}$r$\end{document}-tuple of positive rational numbers whose sum is 1. Variable-weight optical orthogonal code (\begin{document}$(n, W, \Lambda_a, \lambda_c, Q)$\end{document}-OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service requirements. In this paper, tight upper bounds on the maximum code size of \begin{document}$(n, \{3, 4, 5\}, \Lambda_a, 1, Q)$\end{document}-OOCs are obtained, and infinite classes of optimal \begin{document}$(n, \{3, 4, 5\}, \Lambda_a, 1, Q)$\end{document}-OOCs are constructed.

2019, 13(2): 313-328 doi: 10.3934/amc.2019021 +[Abstract](3871) +[HTML](349) +[PDF](389.76KB)
Abstract:

Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.

2019, 13(2): 329-341 doi: 10.3934/amc.2019022 +[Abstract](3913) +[HTML](389) +[PDF](412.45KB)
Abstract:

Kitayama proposed a novel OCDMA (called spatial CDMA) for parallel transmission of 2-D images through multicore fiber. Optical orthogonal signature pattern codes (OOSPCs) play an important role in this CDMA network. Multiple-weight (MW) optical orthogonal signature pattern code (OOSPC) was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirements. Some results had been done on optimal balanced \begin{document}$(m, n, \{3, 4\}, 1)$\end{document}-OOSPCs. In this paper, it is proved that there exist optimal balanced \begin{document}$(2u, 16v, \{4, 5\}, 1)$\end{document}-OOSPCs for odd integers \begin{document}$u\geq 1$\end{document}, \begin{document}$v\geq 1$\end{document}.

2019, 13(2): 343-371 doi: 10.3934/amc.2019023 +[Abstract](4410) +[HTML](461) +[PDF](3517.4KB)
Abstract:

The public-key operation in multivariate encryption and signature schemes evaluates \begin{document}$m$\end{document} quadratic polynomials in \begin{document}$n$\end{document} variables. In this paper we analyze how fast this simple operation can be made. We optimize it for different finite fields on modern architectures. We provide an objective and inherent efficiency measure of our implementations, by comparing their performance with the peak performance of the CPU. In order to provide a fair comparison for different parameter sets, we also analyze the expected security based on the algebraic attack taking into consideration the hybrid approach. We compare the attack's efficiency for different finite fields and establish trends. We detail the role that the field equations play in the attack. We then provide a broad picture of efficiency of MQ-public-key operation against security.

2020 Impact Factor: 0.935
5 Year Impact Factor: 0.976
2020 CiteScore: 1.5