# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

May 2020 , Volume 14 , Issue 2

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2020, 14(2): 177-205 doi: 10.3934/amc.2020015 +[Abstract](2472) +[HTML](1126) +[PDF](397.63KB)
Abstract:

Proxy signature is a cryptographic primitive that allows an entity to delegate singing rights to another entity. Noticing the ad-hoc nature of security analysis prevalent in the existing literature, Boldyreva, Palacio and Warinschi proposed a formal security model for proxy signature. We revisit their proposed security definition in the context of the most natural construction of proxy signature – delegation-by-certificate. Our analysis indicates certain limitations of their definition that arise due to malleability of proxy signature as well as signature ownership in the context of standard signature. We propose a stronger definition of proxy signature to address these issues. However, we observe that the natural reductionist security argument of the delegation-by certificate proxy signature construction under this definition seems to require a rather unnatural security property for a standard signature.

2020, 14(2): 207-232 doi: 10.3934/amc.2020016 +[Abstract](2926) +[HTML](531) +[PDF](453.37KB)
Abstract:

Although currently several traceable (or linkable) ring signature schemes have been proposed, most of them are constructed on pairings. In this paper, we present an efficient traceable ring signature (TRS) scheme without pairings, which is based on the modified EDL signature (first proposed by D.Chaum et al. in Crypto 92). Compared with other ring signature schemes, the proposed scheme does not employ pairing computation and has some computational advantages, whose security can be reduced to the computational Diffie-Hellman (CDH) and decisional Diffie-Hellman (DDH) assumptions in the random oracle model. Also, the proposed scheme is similar to certificateless signature scheme, where user and key generating center make interaction to generate ring key. We give a formal security model for ring signature and prove that the proposed scheme has the properties of traceability and anonymity.

2020, 14(2): 233-245 doi: 10.3934/amc.2020017 +[Abstract](2281) +[HTML](447) +[PDF](294.34KB)
Abstract:

We use group algebra methods to study cyclic codes over finite chain rings and under some restrictive hypotheses, described in section 2, for codes of length \begin{document}$2p^n$\end{document}, \begin{document}$p$\end{document} a prime, we are able to compute the minimum weights of all possible cyclic codes of that length.

2020, 14(2): 247-263 doi: 10.3934/amc.2020018 +[Abstract](2092) +[HTML](517) +[PDF](386.84KB)
Abstract:

Based on the relation between resilient functions and large sets of orthogonal arrays, two classes of \begin{document}$q$\end{document}-variable 1-resilient rotation symmetric functions(RSFs) over \begin{document}${\mathbb F}_{p}$\end{document} are constructed. The first class of 1-resilient functions is obtained with the help of a Latin square with maximum cycle, and the second class of 1-resilient functions is constructed via switching the rotation symmetric orbits of the former class. Firstly, an efficient method to construct OA\begin{document}$(pq, q, p, 1)$\end{document} is presented via solving a linear equation system. Secondly, we propose some schemes to construct more \begin{document}$q$\end{document}-variable 1-resilient RSFs by modifying the \begin{document}$l$\end{document}-value support tables of the known \begin{document}$q$\end{document}-variable 1-resilient RSFs. In addition, two examples are given to demonstrate our constructions. It seems that the \begin{document}$q$\end{document}-variable 1-resilient RSFs constructed by our methods can not be constructed by earlier constructions.

2020, 14(2): 265-278 doi: 10.3934/amc.2020019 +[Abstract](2116) +[HTML](458) +[PDF](377.57KB)
Abstract:

A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from certain curves defined by equations with separated variables. The recovery of erasures is obtained by means of Lagrangian interpolation in general, and simply by one addition in some particular cases.

2020, 14(2): 279-299 doi: 10.3934/amc.2020020 +[Abstract](1823) +[HTML](855) +[PDF](430.21KB)
Abstract:

This paper is concerned with the construction of algebraic-geometric (AG) codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. In addition, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Multi-point codes with excellent parameters are found, among which, a presented code with parameters \begin{document}$[216,190,\geqslant 18]$\end{document} over \begin{document}$\mathbb{F}_{64}$\end{document} yields a new record.

2020, 14(2): 301-306 doi: 10.3934/amc.2020021 +[Abstract](2223) +[HTML](841) +[PDF](299.97KB)
Abstract:

McNie [8] is a code-based public key encryption scheme submitted to the NIST Post-Quantum Cryptography standardization [10] as a candidate. In this paper, we present Dual-Ouroboros, an improvement of McNie, which can be seen as a dual version of the Ouroboros-R protocol [1], another candidate to the NIST competition. This new improved protocol permits, first, to avoid an attack proposed by Gaborit [7] and second permits to benefit from a reduction security to a standard problem (as the original Ouroboros protocol).

2020, 14(2): 307-318 doi: 10.3934/amc.2020022 +[Abstract](1856) +[HTML](429) +[PDF](281.59KB)
Abstract:

In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. Recently, a series of paper has been published for analysis of expansion complexity and for testing sequences in terms of this new measure of randomness. In this paper, we continue this analysis. First we study the expansion complexity in terms of the Gröbner basis of the underlying polynomial ideal. Next, we prove bounds on the expansion complexity for random sequences. Finally, we study the expansion complexity of sequences defined by differential equations, including the inversive generator.

2020, 14(2): 319-332 doi: 10.3934/amc.2020023 +[Abstract](2052) +[HTML](542) +[PDF](328.54KB)
Abstract:

We study \begin{document}$G$\end{document}-codes over the ring \begin{document}${\mathbb{Z}}_4$\end{document}, which are codes that are held invariant by the action of an arbitrary group \begin{document}$G$\end{document}. We view these codes as ideals in a group ring and we study the rank and kernel of these codes. We use the rank and kernel to study the image of these codes under the Gray map. We study the specific case when the group is the dihedral group and the dicyclic group. Finally, we study quaternary self-dual dihedral and dicyclic codes, tabulating the many good self-dual quaternary codes obtained via these constructions, including the octacode.

2020, 14(2): 333-357 doi: 10.3934/amc.2020024 +[Abstract](2154) +[HTML](520) +[PDF](556.2KB)
Abstract:

A major issue of locally repairable codes is their robustness. If a local repair group is not able to perform the repair process, this will result in increasing the repair cost. Therefore, it is critical for a locally repairable code to have multiple repair groups. In this paper we consider robust locally repairable coding schemes which guarantee that there exist multiple distinct (not necessarily disjoint) alternative local repair groups for any single failure such that the failed node can still be repaired locally even if some of the repair groups are not available. We use linear programming techniques to establish upper bounds on the size of these codes. We also provide two examples of robust locally repairable codes that are optimal regarding our linear programming bound. Furthermore, we address the update efficiency problem of the distributed data storage networks. Any modification on the stored data will result in updating the content of the storage nodes. Therefore, it is essential to minimise the number of nodes which need to be updated by any change in the stored data. We characterise the update-efficient storage code properties and establish the necessary conditions of existence update-efficient locally repairable storage codes.

2020, 14(2): 359-378 doi: 10.3934/amc.2020025 +[Abstract](1877) +[HTML](425) +[PDF](356.64KB)
Abstract:

Let \begin{document}$p$\end{document} be a prime different from 3, and \begin{document}$\ell$\end{document} be an odd prime different from 3 and \begin{document}$p$\end{document}. In terms of generator polynomials, structures of constacyclic codes and their duals of length \begin{document}$3\ell^mp^s$\end{document} over \begin{document}$\mathbb{F}_q$\end{document} are established, where \begin{document}$q$\end{document} is a power of \begin{document}$p$\end{document}. We discuss the enumeration of all cyclic codes of length \begin{document}$3\cdot2^s\ell^m$\end{document}, that generalizes the construction of [15] (2016), which is the special case of \begin{document}$m = 1$\end{document}. In addition, as an application, the characterization and enumeration of all linear complementary dual cyclic codes of length \begin{document}$6\ell^mp^s$\end{document} over \begin{document}$\mathbb{F}_q$\end{document} are obtained.

2020, 14(2): 379-395 doi: 10.3934/amc.2020026 +[Abstract](1755) +[HTML](459) +[PDF](341.09KB)
Abstract:

In this paper, we introduce additive Toeplitz codes over \begin{document}$\mathbb{F}_{4}$\end{document}. The additive Toeplitz codes are a generalization of additive circulant codes over \begin{document}$\mathbb{F}_{4}$\end{document}. We find many optimal additive Toeplitz codes (OATC) over \begin{document}$\mathbb{F}_{4}$\end{document}. These optimal codes also contain optimal non-circulant codes, so we find new additive codes in this manner. We provide some theorems to partially classify OATC. Then, we give a new algorithm that fully classifies OATC by combining these theorems with Gaborit's algorithm. We classify OATC over \begin{document}$\mathbb{F}_{4}$\end{document} of length up to \begin{document}$13$\end{document}. We obtain \begin{document}$2$\end{document} inequivalent optimal additive toeplitz codes (IOATC) that are non-circulant codes of length \begin{document}$5$\end{document}, \begin{document}$92$\end{document} of length \begin{document}$8$\end{document}, \begin{document}$2068$\end{document} of length \begin{document}$9$\end{document}, and \begin{document}$39$\end{document} of length \begin{document}$11$\end{document}. Moreover, we improve an idea related to quadratic residue codes to construct optimal and near-optimal additive Toeplitz codes over \begin{document}$\mathbb{F}_{4}$\end{document} of length prime \begin{document}$p$\end{document}. We obtain many optimal and near-optimal additive Toeplitz codes for some primes \begin{document}$p$\end{document} from this construction.

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