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Atul Gaur

Publications and source records attributed to Atul Gaur.

8 recordsLinked to original sources
arXiv2025

New Insights into Involutory and Orthogonal MDS Matrices

MDS matrices play a critical role in the design of diffusion layers for block ciphers and hash functions due to their optimal branch number. Involutory and orthogonal MDS matrices offer additional benefits by allowing identical or nearly identical circuitry for both encryption and decryption, leading to equivalent implementation costs for both processes. These properties have been further generalized through the notions of semi-involutory and semi-orthogonal matrices. While much of the existing literature focuses on identifying efficiently implementable MDS candidates or proposing new constructions for MDS matrices of various orders, this work takes a different direction. Rather than introducing novel constructions or prioritizing implementation efficiency, we investigate structural relationships between the generalized variants and their conventional counterparts. Specifically, we establish nontrivial interconnections between semi-involutory and involutory matrices, as well as between semi-orthogonal and orthogonal matrices. Exploiting these relationships, we show that the number of semi-involutory MDS matrices can be directly derived from the number of involutory MDS matrices, and vice versa. A similar correspondence holds for semi-orthogonal and orthogonal MDS matrices. We also examine the intersection of these classes and show that the number of $3 \times 3$ MDS matrices that are both semi-involutory and semi-orthogonal coincides with the number of semi-involutory MDS matrices over $\mathbb{F}_{2^m}$. Furthermore, we derive the general structure of orthogonal matrices of arbitrary order $n$ over $\mathbb{F}_{2^m}$. Finally, leveraging the aforementioned interconnections, we present an alternative and direct derivation of the explicit formulae for counting $3 \times 3$ semi-involutory MDS matrices and $3 \times 3$ semi-orthogonal MDS matrices.

cs.CR↗
arXiv2024

A New Algorithm for Computing Branch Number of Non-Singular Matrices over Finite Fields

The notion of branch numbers of a linear transformation is crucial for both linear and differential cryptanalysis. The number of non-zero elements in a state difference or linear mask directly correlates with the active S-Boxes. The differential or linear branch number indicates the minimum number of active S-Boxes in two consecutive rounds of an SPN cipher, specifically for differential or linear cryptanalysis, respectively. This paper presents a new algorithm for computing the branch number of non-singular matrices over finite fields. The algorithm is based on the existing classical method but demonstrates improved computational complexity compared to its predecessor. We conduct a comparative study of the proposed algorithm and the classical approach, providing an analytical estimation of the algorithm's complexity. Our analysis reveals that the computational complexity of our algorithm is the square root of that of the classical approach.

cs.CR↗
arXiv2024

A Systematic Construction Approach for All $4\times 4$ Involutory MDS Matrices

Maximum distance separable (MDS) matrices play a crucial role not only in coding theory but also in the design of block ciphers and hash functions. Of particular interest are involutory MDS matrices, which facilitate the use of a single circuit for both encryption and decryption in hardware implementations. In this article, we present several characterizations of involutory MDS matrices of even order. Additionally, we introduce a new matrix form for obtaining all involutory MDS matrices of even order and compare it with other matrix forms available in the literature. We then propose a technique to systematically construct all $4 \times 4$ involutory MDS matrices over a finite field $\mathbb{F}_{2^m}$. This method significantly reduces the search space by focusing on involutory MDS class representative matrices, leading to the generation of all such matrices within a substantially smaller set compared to considering all $4 \times 4$ involutory matrices. Specifically, our approach involves searching for these representative matrices within a set of cardinality $(2^m-1)^5$. Through this method, we provide an explicit enumeration of the total number of $4 \times 4$ involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=3,4,\ldots,8$.

cs.CR↗
arXiv2024

Construction of all MDS and involutory MDS matrices

In this paper, we propose two algorithms for a hybrid construction of all $n\times n$ MDS and involutory MDS matrices over a finite field $\mathbb{F}_{p^m}$, respectively. The proposed algorithms effectively narrow down the search space to identify $(n-1) \times (n-1)$ MDS matrices, facilitating the generation of all $n \times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. To the best of our knowledge, existing literature lacks methods for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. In our approach, we introduce a representative matrix form for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. The determination of these representative MDS matrices involves searching through all $(n-1)\times (n-1)$ MDS matrices over $\mathbb{F}_{p^m}$. Our contributions extend to proving that the count of all $3\times 3$ MDS matrices over $\mathbb{F}_{2^m}$ is precisely $(2^m-1)^5(2^m-2)(2^m-3)(2^{2m}-9\cdot 2^m+21)$. Furthermore, we explicitly provide the count of all $4\times 4$ MDS and involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=2, 3, 4$.

cs.CR↗
arXiv2020

Maximal non valuative domains

The notion of maximal non valuative domain is introduced and characterized. An integral domain R is called a maximal non valuative domain if R is not a valuative domain but every proper overring of R is a valuative domain. Maximal non valuative domains have at most four maximal ideals. Various properties of maximal non valuative domains are discussed. Conditions are given under which pseudo-valuation domains and maximal non pseudo-valuation domains are maximal non valuative domains.

math.AC↗
arXiv2020

On minimal ring extensions

Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a, b\in R$, the extension $\Delta(R)[(a,b)] \subset R\times R$ is a minimal ring extension if and only if the ideal $ $ is a maximal ideal of $R$. A complete classification of maximal subrings of $R(+)R$ is also given. The minimal ring extension of a von Neumann regular ring $R$ is either a von Neumann regular ring or the idealization $R(+)R/\mathfrak{m}$ where $\mathfrak{m}\in \text{Max}(R)$. If $R\subset T$ is a minimal ring extension and $T$ is an integral domain, then $(R:T) = 0$ if and only if $R$ is a field and $T$ is a minimal field extension of $R$, or $R_J$ is a valuation ring of altitude one and $T_{J}$ is its quotient field.

math.AC↗
arXiv2020

A note on pairs of rings with same prime ideals

We study the ring extensions R \subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples are also discussed to strengthen the results.

math.AC↗