Maximal Correlation Secrecy

This paper shows that the Hirschfeld-Gebelein-R\'enyi maximal correlation between the message and the ciphertext provides good secrecy guarantees for ciphers that use short keys. We show that a maximal correlation $0< \rho < 1$ can be achieved via a randomly generated cipher with key length of around $2 \log(1/\rho)$ for small $\rho$, independent of the message length. It can also be achieved by a stream cipher with key length of $2\log(1/\rho) + \log n+2$ for a message of length $n$. We provide a converse result showing that the maximal correlations of these randomly generated ciphers are close to optimal. We then show that any cipher with a small maximal correlation achieves a variant of semantic security with computationally unbounded adversary. These results clearly demonstrate that maximal correlation is a stronger and more practically relevant measure of secrecy than mutual information.