Maximal Correlation Secrecy

Abstract

This paper shows that the Hirschfeld-Gebelein-Rényi maximal correlation between the message and the ciphertext provides good secrecy guarantees for cryptosystems that use short keys. We first establish a bound on the eavesdropper's advantage in guessing functions of the message in terms of maximal correlation and the Rényi entropy of the message. This result implies that maximal correlation is stronger than the notion of entropic security introduced by Russell and Wang. We then show that a small maximal correlation ρ can be achieved via a randomly generated cipher with key length ≈ 2 log(1/ρ), independent of the message length, and by a stream cipher with key length 2 log(1/ρ) + log n + 2 for a message of length n. We establish a converse showing that these ciphers are close to optimal. This is in contrast to entropic security for which there is a gap between the lower and upper bounds. Finally, we show that a small maximal correlation implies secrecy with respect to several mutual information based criteria but is not necessarily implied by them. Hence, maximal correlation is a stronger and more practically relevant measure of secrecy than mutual information.

Information Extraction Under Privacy Constraints

“…We remark that using maximal correlation as a privacy measure is by no means new as it appears in other works, see e.g., [33], [30] and [12] for different utility functions. We do not put any likelihood constraints on the privacy filters as opposed to the definition of LDP.…”

Principal Inertia Components and Applications

“…The results presented in this paper are closely connected to the study of hypercontractivity coefficients and strong data processing results, such as in [22,23,33,34,57]. PIC-based analysis were used in the context of security in [58,59]. Extremal properties of privacy were also investigated in [60,61], and in particular [62] builds upon some of the results introduced here.…”

A Correlation Measure Based on Vector-Valued $L_p$ -Norms

“…Here we suggest the use of V α (A; B) or I α (A; B) as a replacement of mutual information for measuring secrecy. 7 The point is that, by Proposition 8 when V α (A; B) is small, again ρ AB and ρ A ⊗ ρ B are close in trace distance. Moreover, our decoupling theorems in the previous section can be used to prove more effective exponentially small bounds on V α .…”

“…To measure the security of a communication system, one has to quantify the amount of information leaked to an eavesdropper. While the common security metric for measuring the leakage is mutual information (see e.g., see [3]) or the total variation distance [2,4], there have been few recent works that motivate and define other measures of correlation to quantify leakage [5][6][7][8][9][10][11][12]. Herein, we suggest the use of our metric instead of mutual information because it is a stronger metric and has a better rate-security tradeoff curve.…”