Suppose Alice would like to obtain the product of two $m\times m$ matrices i.e. $A$ and $B.$ Alice has $A,$ whereas Bob has $B.$
Since Alice does not want to reveal $A$ to Bob, she chooses a $m\times m$ random invertible matrix $R.$ She sends $RA$ to Bob over a secure channel.
Bob obtains $RA,$ and calculates $RAB,$ and sends it to Alice over a secure channel.
Alice obtains $AB$ by inverting $R$ i.e. $R^{-1}RAB$.
$R$ is only utilized once.
Any ideas on how to proceed with the security analysis of the above protocol?
Specifically is H(A|RA) = H(A) ?
If A can be any matrix, then this protocol is not secure. In particular, if RA=0, then Bob can be reasonably confident that A=0 (or at least that A is sparse).
If A is invertible (over a fixed finite field), then this protocol is information-theoretically secure. To see this, first note that, for any $A$, the ciphertext $RA$ is uniformly distributed. Furthermore, the value of $RA$ is independent of $A$. Therefore, for any prior $P$ over messages, we have $\Pr[A|RA] = \Pr[A \wedge RA] / \Pr[RA] = \Pr[A]\Pr[RA] / \Pr[RA] = \Pr[A]$.