  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
$\lim_{s\rightarrow\infty}R_2(s)=0.5$.
