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Advances in Mathematics of Communications

February 2022 , Volume 16 , Issue 1

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Additive and linear conjucyclic codes over $ {\mathbb{F}}_4 $
Taher Abualrub and Steven T. Dougherty
2022, 16(1): 1-15 doi: 10.3934/amc.2020096 +[Abstract](1781) +[HTML](656) +[PDF](295.81KB)

Conjucyclic codes were first introduced by Calderbank, Rains, Shor and Sloane in [1] because of their applications in quantum error-correction. In this paper, we study linear and additive conjucyclic codes over the finite field \begin{document}$ {\mathbb{F}}_{4} $\end{document} and produce a duality for which the orthogonal, with respect to that duality, of conjucyclic codes is conjucyclic. Moreover, we show that this is not the case for other standard dualities. We show that additive conjucyclic codes are the only non-trivial conjucyclic codes over \begin{document}$ {\mathbb{F}}_{4} $\end{document} and we use a linear algebraic approach to classify these codes. We will also show that additive conjucyclic codes of length \begin{document}$ n $\end{document} over \begin{document}$ {\mathbb{F}}_{4} $\end{document} are isomorphic to binary cyclic codes of length \begin{document}$ 2n. $\end{document}

New quantum codes from constacyclic codes over the ring $ R_{k,m} $
Habibul Islam, Om Prakash and Ram Krishna Verma
2022, 16(1): 17-35 doi: 10.3934/amc.2020097 +[Abstract](1926) +[HTML](897) +[PDF](388.59KB)

For any odd prime \begin{document}$ p $\end{document}, we study constacyclic codes of length \begin{document}$ n $\end{document} over the finite commutative non-chain ring \begin{document}$ R_{k,m} = \mathbb{F}_{p^m}[u_1,u_2,\dots,u_k]/\langle u^2_i-1,u_iu_j-u_ju_i\rangle_{i\neq j = 1,2,\dots,k} $\end{document}, where \begin{document}$ m,k\geq 1 $\end{document} are integers. We determine the necessary and sufficient condition for these codes to contain their Euclidean duals. As an application, from the dual containing constacyclic codes, several MDS, new and better quantum codes compare to the best known codes in the literature are obtained.

Efficient fully CCA-secure predicate encryptions from pair encodings
Mridul Nandi and Tapas Pandit
2022, 16(1): 37-72 doi: 10.3934/amc.2020098 +[Abstract](1430) +[HTML](676) +[PDF](647.46KB)

Attrapadung (Eurocrypt 2014) proposed a generic framework for fully (adaptively) CPA-secure predicate encryption (PE) based on a new primitive, called pair encodings. Following the CCA conversions of Yamada et al. (PKC 2011, 2012) and Nandi et al. (ePrint Archive: 2015/457, AAECC 2018), one can have CCA-secure PE from CPA-secure PE if the primitive PE has either verifiability or delegation. These traditional approaches degrade the performance of the resultant CCA-secure PE scheme as compared to the primitive CPA-secure PE. As an alternative, we provide a direct fully secure CCA-construction of PE from the pair encoding scheme. This costs an extra computation of group element in encryption, three extra pairing computations and one re-randomization of key in decryption as compared to the CPA-construction of Attrapadung.

Recently, Blömer et al. (CT-RSA 2016) proposed a direct CCA-secure construction of predicate encryptions from pair encodings. Although they did not use the aforementioned traditional approaches, a sort of verifiability checking is still involved in the CCA-decryption. The number of pairing computations for this checking is nearly equal to the number of paring computations in CPA-decryption. Therefore, the performance of our direct CCA-secure PE is far better than Blömer et al.

Locally repairable codes with high availability based on generalised quadrangles
Michel Lavrauw and Geertrui Van de Voorde
2022, 16(1): 73-81 doi: 10.3934/amc.2020099 +[Abstract](1586) +[HTML](620) +[PDF](259.92KB)

Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in [3], and bounds on the repairability and availability were derived. In this paper, we determine the values of the repairability and availability of such LRC's for a large portion of the currently known generalised quadrangles. In order to do so, we determine the minimum weight of the codes of translation generalised quadrangles and characterise the codewords of minimum weight.

Binary sequences derived from differences of consecutive quadratic residues
Arne Winterhof and Zibi Xiao
2022, 16(1): 83-93 doi: 10.3934/amc.2020100 +[Abstract](1457) +[HTML](617) +[PDF](296.55KB)

For a prime \begin{document}$ p\ge 5 $\end{document} let \begin{document}$ q_0,q_1,\ldots,q_{(p-3)/2} $\end{document} be the quadratic residues modulo \begin{document}$ p $\end{document} in increasing order. We study two \begin{document}$ (p-3)/2 $\end{document}-periodic binary sequences \begin{document}$ (d_n) $\end{document} and \begin{document}$ (t_n) $\end{document} defined by \begin{document}$ d_n = q_n+q_{n+1}\bmod 2 $\end{document} and \begin{document}$ t_n = 1 $\end{document} if \begin{document}$ q_{n+1} = q_n+1 $\end{document} and \begin{document}$ t_n = 0 $\end{document} otherwise, \begin{document}$ n = 0,1,\ldots,(p-5)/2 $\end{document}. For both sequences we find some sufficient conditions for attaining the maximal linear complexity \begin{document}$ (p-3)/2 $\end{document}.

Studying the linear complexity of \begin{document}$ (d_n) $\end{document} was motivated by heuristics of Caragiu et al. However, \begin{document}$ (d_n) $\end{document} is not balanced and we show that a period of \begin{document}$ (d_n) $\end{document} contains about \begin{document}$ 1/3 $\end{document} zeros and \begin{document}$ 2/3 $\end{document} ones if \begin{document}$ p $\end{document} is sufficiently large. In contrast, \begin{document}$ (t_n) $\end{document} is not only essentially balanced but also all longer patterns of length \begin{document}$ s $\end{document} appear essentially equally often in the vector sequence \begin{document}$ (t_n,t_{n+1},\ldots,t_{n+s-1}) $\end{document}, \begin{document}$ n = 0,1,\ldots,(p-5)/2 $\end{document}, for any fixed \begin{document}$ s $\end{document} and sufficiently large \begin{document}$ p $\end{document}.

On the diffusion of the Improved Generalized Feistel
Tsonka Baicheva and Svetlana Topalova
2022, 16(1): 95-114 doi: 10.3934/amc.2020102 +[Abstract](1295) +[HTML](562) +[PDF](396.6KB)

We consider the Improved Generalized Feistel Structure (IGFS) suggested by Suzaki and Minematsu (LNCS, 2010). It is a generalization of the classical Feistel cipher. The message is divided into \begin{document}$ k $\end{document} subblocks, a Feistel transformation is applied to each pair of successive subblocks, and then a permutation of the subblocks follows. This permutation affects the diffusion property of the cipher. IGFS with relatively big \begin{document}$ k $\end{document} and good diffusion are of particular interest for light weight applications.

Suzaki and Minematsu (LNCS, 2010) study the case when one and the same permutation is applied at each round, while we consider IGFS with possibly different permutations at the different rounds. In this case we present permutation sequences yielding IGFS with the best known by now diffusion for all even \begin{document}$ k\le 2048 $\end{document}. For \begin{document}$ k\le 16 $\end{document} they are found by a computer-aided search, while for \begin{document}$ 18\le k\le 2048 $\end{document} we first consider several recursive constructions of a permutation sequence for \begin{document}$ k $\end{document} subblocks from two permutation sequences for \begin{document}$ k_a< k $\end{document} and \begin{document}$ k_b< k $\end{document} subblocks respectively. Using computer, we apply these constructions to obtain permutation sequences with good diffusion for each even \begin{document}$ k\le 2048 $\end{document}. Finally we obtain infinite families of permutation sequences for \begin{document}$ k>2048 $\end{document}.

A geometric characterization of minimal codes and their asymptotic performance
Gianira N. Alfarano, Martino Borello and Alessandro Neri
2022, 16(1): 115-133 doi: 10.3934/amc.2020104 +[Abstract](1968) +[HTML](634) +[PDF](345.06KB)

In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes as cutting blocking sets.

Orbit codes from forms on vector spaces over a finite field
Angela Aguglia, Antonio Cossidente, Giuseppe Marino, Francesco Pavese and Alessandro Siciliano
2022, 16(1): 135-155 doi: 10.3934/amc.2020105 +[Abstract](1499) +[HTML](607) +[PDF](480.37KB)

In this paper we construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an \begin{document}$ n $\end{document}-dimensional vector space over the finite field \begin{document}$ {\mathbb F_{q}} $\end{document}. All these codes admit the general linear group \begin{document}$ {{{{\rm{GL}}}}}(n,q) $\end{document} as a transitive automorphism group.

Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros
Xiaoni Du, Rong Wang, Chunming Tang and Qi Wang
2022, 16(1): 157-168 doi: 10.3934/amc.2020106 +[Abstract](1365) +[HTML](658) +[PDF](349.76KB)

Combinatorial \begin{document}$ t $\end{document}-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a \begin{document}$ t $\end{document}-design. Till now only a small amount of work on constructing \begin{document}$ t $\end{document}-designs from codes has been done. In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the triple-error correcting binary BCH codes, and the other related to the cyclic codes with parameters satisfying the generalized Kasami case, respectively. We then obtain infinite families of \begin{document}$ 2 $\end{document}-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters. In particular, the codes derived from the dual of binary BCH codes hold five \begin{document}$ 3 $\end{document}-designs when \begin{document}$ m = 4 $\end{document}.

Optimal antiblocking systems of information sets for the binary codes related to triangular graphs
Hans-Joachim Kroll, Sayed-Ghahreman Taherian and Rita Vincenti
2022, 16(1): 169-183 doi: 10.3934/amc.2020107 +[Abstract](1344) +[HTML](641) +[PDF](332.75KB)

We present AI-systems for the binary codes obtained from the adjacency relation of the triangular graphs \begin{document}$ T(n) $\end{document} for any \begin{document}$ n\ge 5 $\end{document}. These AI-systems are optimal and have for \begin{document}$ n $\end{document} odd the full error-correcting capability.

${\sf {FAST}}$: Disk encryption and beyond
Debrup Chakraborty, Sebati Ghosh, Cuauhtemoc Mancillas López and Palash Sarkar
2022, 16(1): 185-230 doi: 10.3934/amc.2020108 +[Abstract](1711) +[HTML](604) +[PDF](611.43KB)

This work introduces ${\sf {FAST}}$ which is a new family of tweakable enciphering schemes. Several instantiations of ${\sf {FAST}}$ are described. These are targeted towards two goals, the specific task of disk encryption and a more general scheme suitable for a wide variety of practical applications. A major contribution of this work is to present detailed and careful software implementations of all of these instantiations. For disk encryption, the results from the implementations show that ${\sf {FAST}}$ compares very favourably to the IEEE disk encryption standards XCB and EME2 as well as the more recent proposal AEZ. ${\sf {FAST}}$ is built using a fixed input length pseudo-random function and an appropriate hash function. It uses a single-block key, is parallelisable and can be instantiated using only the encryption function of a block cipher. The hash function can be instantiated using either the Horner's rule based usual polynomial hashing or hashing based on the more efficient Bernstein-Rabin-Winograd polynomials. Security of ${\sf {FAST}}$ has been rigorously analysed using the standard provable security approach and concrete security bounds have been derived. Based on our implementation results, we put forward ${\sf {FAST}}$ as a serious candidate for standardisation and deployment.

2020 Impact Factor: 0.935
5 Year Impact Factor: 0.976
2021 CiteScore: 1.8




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