arXiv subjects

Shakir Ali

Publications and source records attributed to Shakir Ali.

6 recordsLinked to original sources

arXiv2026

On the generalization of $g$-circulant MDS matrices

A matrix $M$ over the finite field $ \mathbb{F}_q $ is called \emph{maximum distance separable} (MDS) if all of its square submatrices are non-singular. These MDS matrices are very important in cryptography and coding theory because they provide strong data protection and help spread information efficiently. In this paper, we introduce a new type of matrix called a \emph{consta-$g$-circulant matrix}, which extends the idea of $g$-circulant matrices. These matrices come from a linear transformation defined by the polynomial $ h(x) = x^m - \lambda + \sum_{i=0}^{m-1} h_i x^i $ over $ \mathbb{F}_q $. We find the upper bound of such matrices exist and give conditions to check when they are invertible. This helps us know when they are MDS matrices. If the polynomial $ x^m - \lambda $ factors as $ x^m - \lambda = \prod_{i=1}^{t} f_i(x)^{e_i}, $ where each $ f_i(x) $ is irreducible, then the number of invertible consta-$g$-circulant matrices is $ N \cdot \prod_{i=1}^{t} \left( q^{\deg f_i} - 1 \right), $ where $r$ is the multiplicative order of $\lambda$, and $ N $ is the number of integers $ k $ such that $ 0 \leq k < \left\lfloor \frac{m - 1}{r} \right\rfloor + 1 \quad \text{and} \quad \gcd(1 + rk, m) = 1. $ This formula help us to reduce the number of cases to check whether such matrices is MDS. Moreover, we give complete characterization of $g$-circulant MDS matrices of order 3 and 4. Additionally, inspired by skew polynomial rings, we construct a new variant of $g$-circulant matrix. In the last, we provide some examples related to our findings.

cs.IT↗

arXiv2026

MDS matrices from skew polynomials with automorphisms and derivations

Maximum Distance Separable (MDS) matrices play a central role in coding theory and symmetric-key cryptography due to their optimal diffusion properties. In this paper, we present a construction of MDS matrices using skew polynomial rings $ \mathbb{F}_q[X;\theta,\delta] $, where $ \theta $ is an automorphism and $ \delta $ is a $ \theta$-derivation on $ \mathbb{F}_q $. We introduce the notion of $ \delta_{\theta} $-circulant matrices and study their structural properties. Necessary and sufficient conditions are derived under which these matrices are involutory and satisfy the MDS property. The resulting $\delta_\theta$-circulant matrix can be viewed as a generalization of classical constructions obtained in the absence of $\theta$-derivations. One of the main contribution of this work is the construction of quasi recursive MDS matrices. In the setting of the skew polynomial ring $\mathbb{F}_q[X;\theta]$, we construct quasi recursive MDS matrices associated with companion matrices. These matrices are shown to be involutory, yielding a strict improvement over the quasi-involutory constructions previously reported in the literature. Several illustrative results and examples are also provided.

cs.IT↗

arXiv2025

On the construction of Cauchy MDS matrices over Galois rings via nilpotent elements and Frobenius maps

Let $s,m$ be the positive integers and $p$ be any prime number. Next, let $GR(p^s,p^{sm})$ be a Galois ring of characteristic $p^s$ and cardinality $p^{sm}$. In the present paper, we explore the construction of Cauchy MDS matrices over Galois rings. Moreover, we introduce a new approach that considers nilpotent elements and Teichm\"uller set of Galois ring $GR(p^s,p^{sm})$ to reduce the number of entries in these matrices. Furthermore, we construct $p^{(s-1)m}(p^m-1)$ distinct functions with the help of Frobenius automorphisms. These functions preserve MDS property of matrices. Finally, we prove some results using automorphisms and isomorphisms of the Galois rings that can be used to generate new Cauchy MDS matrices.

cs.IT↗

arXiv2025

Quasi-recursive MDS Matrices over Galois Rings

Let $p$ be a prime and $s,m,n$ be positive integers. This paper studies quasi-recursive MDS matrices over Galois rings $GR(p^{s}, p^{sm})$ and proposes various direct construction methods for such matrices. The construction is based on skew polynomial rings $GR(p^{s}, p^{sm})[X;\sigma]$, whose rich factorization properties and enlarged class of polynomials are used to define companion matrices generating quasi-recursive MDS matrices. First, two criteria are established for characterizing polynomials that yield recursive MDS matrices, generalizing existing results, and then an additional criterion is derived in terms of the right roots of the associated Wedderburn polynomial. Using these criteria, methods are developed to construct skew polynomials that give rise to quasi-recursive MDS matrices over Galois rings. This framework extends known constructions to the non-commutative setting and significantly enlarges the family of available matrices, with potential applications to efficient diffusion layers in cryptographic primitives. The results are particularly relevant for practical implementations when $s = 1$ and $p = 2$, i.e., over the finite field $\mathbb{F}_{2^m}$, which is of central interest in real-world cryptographic applications.

cs.IT↗

arXiv2022

On centrally extended Jordan derivations and related maps in rings

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jordan derivation (resp. centrally extended Jordan $\ast$-derivation) $δ:R\to R$ such that \[ [δ(x),x]\in Z(R)~~(resp.~~[δ(x),x^{\ast}]\in Z(R))\text{~for~all~}x\in R, \] where $'\ast'$ is an involution on $R,$ then $R$ is an order in a central simple algebra of dimension at most 4 over its center.

math.RA↗

arXiv2015

Hypercube emulation of interconnection networks topologies

We address various topologies (de Bruijn, chordal ring, generalized Petersen, meshes) in various ways ( isometric embedding, embedding up to scale, embedding up to a distance) in a hypercube or a half-hypercube. Example of obtained embeddings: infinite series of hypercube embeddable Bubble Sort and Double Chordal Rings topologies, as well as of regular maps.

math.MG↗