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Advances in Mathematics of Communications

February 2020 , Volume 14 , Issue 1

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Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems
Chunlei Xie and Yujuan Sun
2020, 14(1): 1-9 doi: 10.3934/amc.2020001 +[Abstract](264) +[HTML](150) +[PDF](314.52KB)
Abstract:

Orthogonal sequences can be assigned to a regular tessellation of hexagonal cells, typical for synchronised code-division multiple-access (S-CDMA) systems. In this paper, we first construct a new class of orthogonal sequences with increasing the number of users per cell to be \begin{document}$ 2^{m-2} $\end{document} for even number \begin{document}$ m\geq 4 $\end{document} (where \begin{document}$ 2^m $\end{document} is the length of the sequences). In addition, based on the above construction we construct a family of orthogonal sequences with zero correlation zone property which can be applied to the quasi-synchronous CDMA (QS-CMDA) spread spectrum systems.

New self-dual and formally self-dual codes from group ring constructions
Steven T. Dougherty, Joe Gildea, Abidin Kaya and Bahattin Yildiz
2020, 14(1): 11-22 doi: 10.3934/amc.2020002 +[Abstract](311) +[HTML](145) +[PDF](329.37KB)
Abstract:

In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings \begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document} and \begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document}, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters \begin{document}$ [72,36,14] $\end{document}.

A construction of bent functions with optimal algebraic degree and large symmetric group
Wenying Zhang, Zhaohui Xing and Keqin Feng
2020, 14(1): 23-33 doi: 10.3934/amc.2020003 +[Abstract](236) +[HTML](116) +[PDF](328.89KB)
Abstract:

As maximal, nonlinear Boolean functions, bent functions have many theoretical and practical applications in combinatorics, coding theory, and cryptography. In this paper, we present a construction of bent function \begin{document}$ f_{a,S} $\end{document} with \begin{document}$ n = 2m $\end{document} variables for any nonzero vector \begin{document}$ a\in \mathbb{F}_{2}^{m} $\end{document} and subset \begin{document}$ S $\end{document} of \begin{document}$ \mathbb{F}_{2}^{m} $\end{document} satisfying \begin{document}$ a+S = S $\end{document}. We give a simple expression of the dual bent function of \begin{document}$ f_{a,S} $\end{document} and prove that \begin{document}$ f_{a,S} $\end{document} has optimal algebraic degree \begin{document}$ m $\end{document} if and only if \begin{document}$ |S|\equiv 2 (\bmod 4) $\end{document}. This construction provides a series of bent functions with optimal algebraic degree and large symmetric group if \begin{document}$ a $\end{document} and \begin{document}$ S $\end{document} are chosen properly. We also give some examples of those bent functions \begin{document}$ f_{a,S} $\end{document} and their dual bent functions.

Some generalizations of good integers and their applications in the study of self-dual negacyclic codes
Somphong Jitman, Supawadee Prugsapitak and Madhu Raka
2020, 14(1): 35-51 doi: 10.3934/amc.2020004 +[Abstract](272) +[HTML](109) +[PDF](350.46KB)
Abstract:

Good integers introduced in 1997 form an interesting family of integers that has been continuously studied due to their rich number theoretical properties and wide applications. In this paper, we have focused on classes of \begin{document}$ 2^\beta $\end{document}-good integers, \begin{document}$ 2^\beta $\end{document}-oddly-good integers, and \begin{document}$ 2^\beta $\end{document}-evenly-good integers which are generalizations of good integers. Properties of such integers have been given as well as their applications in characterizing and enumerating self-dual negacyclic codes over finite fields. An alternative proof for the characterization of the existence of a self-dual negacyclic code over finite fields has been given in terms of such generalized good integers. A general enumeration formula for the number of self-dual negacyclic codes of length \begin{document}$ n $\end{document} over finite fields has been established. For some specific lengths, explicit formulas have been provided as well. Some known results on self-dual negacyclic codes over finite fields can be formalized and viewed as special cases of this work.

Skew constacyclic codes over the local Frobenius non-chain rings of order 16
Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty and Esengül Saltürk
2020, 14(1): 53-67 doi: 10.3934/amc.2020005 +[Abstract](289) +[HTML](116) +[PDF](347.53KB)
Abstract:

We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.

A complete classification of partial MDS (maximally recoverable) codes with one global parity
Anna-Lena Horlemann-Trautmann and Alessandro Neri
2020, 14(1): 69-88 doi: 10.3934/amc.2020006 +[Abstract](197) +[HTML](105) +[PDF](337.38KB)
Abstract:

We generalize the definition of partial MDS codes to locality blocks of various length and show that these codes are maximally recoverable. Then we focus on partial MDS codes with exactly one global parity. We derive a general construction for such codes by describing a suitable parity check matrix. Then we give a construction of generator matrices of such codes. Afterwards we show that all partial MDS codes with one global parity have a generator matrix (or parity check matrix) of this form. This gives a complete classification and hence also a sufficient and necessary condition on the underlying field size for the existence of such codes. This condition is related to the classical MDS conjecture. Moreover, we investigate the decoding of such codes and give some comments on partial MDS codes with more than one global parity.

New doubly even self-dual codes having minimum weight 20
Masaaki Harada
2020, 14(1): 89-96 doi: 10.3934/amc.2020007 +[Abstract](200) +[HTML](110) +[PDF](262.01KB)
Abstract:

In this note, we construct new doubly even self-dual codes having minimum weight 20 for lengths 112,120 and 128. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length 112.

Two classes of differentially 4-uniform permutations over $ \mathbb{F}_{2^{n}} $ with $ n $ even
Guangkui Xu and Longjiang Qu
2020, 14(1): 97-110 doi: 10.3934/amc.2020008 +[Abstract](205) +[HTML](112) +[PDF](311.43KB)
Abstract:

A construction of differentially 4-uniform permutations by modifying the values of the inverse function on a union of some cosets of a multiplication subgroup of \begin{document}$ \mathbb{F}_{2^n}^* $\end{document} was given by Peng et al. in [15]. In this paper, we extend their results to differentially 4-uniform permutations whose values are different from the values of the inverse function on some subsets of the unit circle of \begin{document}$ \mathbb{F}_{2^n} $\end{document} or on the multiplication group of some subfield of \begin{document}$ \mathbb{F}_{2^n} $\end{document}. Moreover, it has been checked by the Magma software that some permutations in the resulted differentially 4-uniform permutations are CCZ-inequivalent to the known functions for small \begin{document}$ n $\end{document}.

A note on the fast algebraic immunity and its consequences on modified majority functions
Deng Tang
2020, 14(1): 111-125 doi: 10.3934/amc.2020009 +[Abstract](181) +[HTML](115) +[PDF](341.74KB)
Abstract:

Boolean functions used as nonlinear filters and/or combiners in LFSR-based stream ciphers should satisfy several desired cryptographic properties simultaneously, to withstand all known cryptographic attacks. In the past decade, the algebraic and fast algebraic immunities are the most infusive criteria on the design of cryptographic Boolean functions, due to the high efficiency of the algebraic and fast algebraic attacks on stream ciphers. Up to now, Boolean functions with optimal algebraic immunity have been built in several ways, but there are not many known results on their fast algebraic immunities. In this paper, we first derive a relation on the fast algebraic immunity between a Boolean function f and it’s modifications f + s, which shows that if f has low fast algebraic immunity and s has low algebraic immunity then f + s may also have low fast algebraic immunity in general. Thanks to this relation, we obtain some upper bounds on the fast algebraic immunity of several known classes of modified majority functions.

Highly nonlinear (vectorial) Boolean functions that are symmetric under some permutations
SelÇuk Kavut and Seher Tutdere
2020, 14(1): 127-136 doi: 10.3934/amc.2020010 +[Abstract](215) +[HTML](118) +[PDF](303.85KB)
Abstract:

We first give a brief survey of the results on highly nonlinear single-output Boolean functions and bijective S-boxes that are symmetric under some permutations. After that, we perform a heuristic search for the symmetric (and involution) S-boxes which are bijective in dimension 8 and identify corresponding permutations yielding rich classes in terms of cryptographically desirable properties.

Certified lattice reduction
Thomas Espitau and Antoine Joux
2020, 14(1): 137-159 doi: 10.3934/amc.2020011 +[Abstract](682) +[HTML](121) +[PDF](681.01KB)
Abstract:

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovász reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants---where the rational arithmetic required by Gram–Schmidt orthogonalization is replaced by floating-point arithmetic---are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields.

Letters for Post-Quantum Cryptography (PQC)

Letters for post-quantum cryptography standard evaluation
Jintai Ding, Sihem Mesnager and Lih-Chung Wang
2020, 14(1): i-i doi: 10.3934/amc.2020012 +[Abstract](333) +[HTML](137) +[PDF](212.1KB)
Abstract:
New mission and opportunity for mathematics researchers: Cryptography in the quantum era
Lidong Chen and Dustin Moody
2020, 14(1): 161-169 doi: 10.3934/amc.2020013 +[Abstract](260) +[HTML](373) +[PDF](221.27KB)
Abstract:

This article introduces the NIST post-quantum cryptography standardization process. We highlight the challenges, discuss the mathematical problems in the proposed post-quantum cryptographic algorithms and the opportunities for mathematics researchers to contribute.

Giophantus distinguishing attack is a low dimensional learning with errors problem
Jintai Ding, Joshua Deaton and Kurt Schmidt
2020, 14(1): 171-175 doi: 10.3934/amc.2020014 +[Abstract](196) +[HTML](116) +[PDF](226.84KB)
Abstract:

In this paper, we attack the recent NIST submission Giophantus, a public key encryption scheme. We find that the complicated structure of Giophantus's ciphertexts leaks information via a correspondence from a low dimensional lattice. This allows us to distinguish encrypted data from random data by the LLL algorithm. This is a more efficient attack than previous proposed attacks.

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