Hadamard Modulo Prime Matrices and Their Application in Cryptography: A Survey of Some Recent Works

Abstract

The notion of Hadamard modulo prime (HMP) matrix inherits in basics that of classical real Hadamard matrix. Namely, by definition, HMP modulo odd prime p matrix \\(\\mathbf{H}\\) of size n, is a \\(n\\times n\\) non-singular over \\({\\mathbb {Z}}_{p}\\) matrix of \\(\\pm 1\\)’s satisfying the equality: \\(\\mathbf{H}{} \\mathbf{H}^{T} = n{(mod p)}\\) \\(\\mathbf{I}\\) where \\(\\mathbf{I}\\) is the identity matrix of same size. The HMP matrices have an attractive application in the modern cryptography due to the fact of their efficient employment in constructing of some all-or-nothing transform schemes. The present paper surveys some recent results on this kind of matrices by revealing their connections with coding theory, combinatorics, and elementary number theory.