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.