Invertible binary matrix with maximum number of $2$-by-$2$ invertible submatrices

TL;DR

This paper determines the maximum proportion of invertible 2-by-2 submatrices in large binary matrices, confirming the limit conjecture and establishing it as exactly 0.5.

Contribution

It completely solves the problem for t=2 by proving the limit of the maximum proportion is 0.5, confirming the conjecture.

Findings

The limit of the maximum proportion of invertible 2-by-2 submatrices is 0.5.

The conjectured bounds 0.492 and 0.625 are refined to an exact limit.

The result advances understanding of binary matrices in cryptographic applications.

Abstract

The problem is related to all-or-nothing transforms (AONT) suggested by Rivest as a preprocessing for encrypting data with a block cipher. Since then there have been various applications of AONTs in cryptography and security. D'Arco, Esfahani and Stinson posed the problem on the constructions of binary matrices for which the desired properties of an AONT hold with the maximum probability. That is, for given integers $t\le s$, what is the maximum number of $t$-by-$t$ invertible submatrices in a binary matrix of order $s$? For the case $t=2$, let $R_2(s)$ denote the maximal proportion of 2-by-2 invertible submatrices. D'Arco, Esfahani and Stinson conjectured that the limit is between 0.492 and 0.625. We completely solve the case $t=2$ by showing that $\lim_{s\rightarrow\infty}R_2(s)=0.5$.