Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Abstract

A matrix is MDS or super-regular if and only if every square submatrices of it are nonsingular. MDS matrices provide perfect diffusion in block ciphers and hash functions. In this paper we provide a brief survey on cryptographically significant MDS matrices-a first to the best of our knowledge. In addition to providing a summary of existing results, we make several contributions. We exhibit some deep and nontrivial interconnections between different constructions of MDS matrices. For example, we prove that all known Vandermonde constructions are basically equivalent to Cauchy constructions. We prove some folklore results which are used in MDS matrix literature. Wherever possible, we provide some simpler alternative proofs. We do not discuss efficiency issues or hardware implementations; however, the theory accumulated and discussed here should provide an easy guide towards efficient implementations.

On the Construction of New Lightweight Involutory MDS Matrices in Generalized Subfield Form

“…In the direct construction methods, the algebraic properties (which are provably MDS) are used to find MDS matrices. The two basic direct construction methods are Cauchy matrices and Vandermonde matrices (37), but these matrices are generally not efficient for low-cost implementations (16). On the contrary, in Cauchy matrices and Vandermonde matrices, the new construction method (a matrix form) given in (38) generates all 3×3 involutory MDS matrices (with low-cost implementations) over F 2 m .…”

Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices

“…The known methods for obtaining MDS matrices over F q , are exhaustive search [15,20], construction by Cauchy and Vandermonde matrices [18,25,26], using recursive methods [8,9,32] and applying some methods from coding theory [1,5]. A brief survey of the various methods on the construction of MDS matrices is given in [12]. Recently, by applying heuristic algorithms that are used in Reed-Solomon encoders [23], some new forms of lightweight MDS matrices have been proposed [17,19].…”

Lifted MDS Codes over Finite Fields

“…There are some approaches for constructing MDS matrices such that Vandermonde matrix, circulant matrix, Cauchy matrix, Toeplitz matrices etc. [2,3,10,[44][45][46][47]. All of them compute and improve their method over the defined field in the papers.…”

“…In Theorem 1 we satisfy this preservation to the upper finite field by the characteristic of the field. Let M be a matrix dh-matrix M = diag(w 244 , w 28 , w 326 , w 294 , w 239 , w 76 , w 212 , w 84 ) over F 7 3 . By Example 1, Distance has been preserved.…”

“…0 w 221 w 223 w 288 w 253 w 239 0 1 0 w 323 w 211 w 333 w 184 w 113 0 0 1 w 25 w 198 w 206 MDS code over F 73 . Another example for same code over F 7 : Let M be a matrix dh-matrix M = diag(w 108 , w 191 , w 261 , w 312 , w 95 , w 249 , w 278 , w 47 ) over F 7 3 .…”