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IEEE Computer Security Foundations Workshop (CSFW)

Measuring Information Leakage Using Generalized Gain Functions

Mário S. Alvim, Kostas Chatzikokolakis, Catuscia Palamidessi A, Geoffrey Smith

Published Jun 1, 2012• 251 citations
Publisher

Abstract

This paper introduces g-leakage, a rich generalization of the min-entropy model of quantitative information flow. In g-leakage, the benefit that an adversary derives from a certain guess about a secret is specified using a gain function g. Gain functions allow a wide variety of operational scenarios to be modeled, including those where the adversary benefits from guessing a value close to the secret, guessing a part of the secret, guessing a property of the secret, or guessing the secret within some number of tries. We prove important properties of g-leakage, including bounds between min-capacity, g-capacity, and Shannon capacity. We also show a deep connection between a strong leakage ordering on two channels, C1and C2, and the possibility of factoring C1into C2C3, for some C3. Based on this connection, we propose a generalization of the Lattice of Information from deterministic to probabilistic channels.

DOI
10.1109/csf.2012.26

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This paper introduces g-leakage, a rich generalization of the min-entropy model of quantitative information flow. In g-leakage, the benefit that an adversary derives from a certain guess about a secret is specified using a gain function g. Gain functions allow a wide variety of operational scenarios to be modeled, including those where the adversary benefits from guessing a value close to the secret, guessing a part of the secret, guessing a property of the secret, or guessing the secret within some number of tries. We prove important properties of g-leakage, including bounds between min-capacity, g-capacity, and Shannon capacity. We also show a deep connection between a strong leakage ordering on two channels, C1and C2, and the possibility of factoring C1into C2C3, for some C3. Based on this connection, we propose a generalization of the Lattice of Information from deterministic to probabilistic channels.
Quantitative theories of information flow are of growing interest, due to the fundamental importance of protecting confidential information from improper disclosure, together with the unavoidability of "small" leaks in practical systems. But while it is tempting to measure leakage using classic information-theoretic concepts like Shannon entropy and mutual information, these turn out not to provide very satisfactory security guarantees. As a result, several researchers have developed an alternative theory based on Renyi's min-entropy. In this theory, uncertainty is measured in terms of a random variable's vulnerability to being guessed in one try by an adversary, note that this is the complement of the Bayes Risk. In this paper, we survey the main theory of min-entropy leakage in deterministic and probabilistic systems, including comparisons with mutual information leakage, results on min-capacity, results on channels in cascade, and techniques for calculating min-entropy leakage in systems.
Information flow is the branch of security that studies the leakage of information due to correlation between secrets and observables. Since in general such correlation cannot be avoided completely, it is important to quantify the leakage. The most followed approaches to defining appropriate measures are those based on information theory. In particular, one of the most successful approaches is the recently proposed ggg-leakage framework, which encompasses most of the information-theoretic ones. A problem with ggg-leakage, however, is that it is defined in terms of a minimization problem, which, in the case of large systems, can be computationally rather heavy. In this paper we study the case in which the channel associated to the system can be decomposed into simpler channels, which typically happens when the observables consist of multiple components. Our main contribution is the derivation of bounds on the (multiplicative version of) ggg-leakage of the whole system in terms of the ggg-leakages of its components. We also consider the particular cases of min-entropy leakage and of parallel channels, generalizing and systematizing results from the literature. We demonstrate the effectiveness of our method and evaluate the precision of our bounds using examples.
We study the asymptotic behaviour of (a) information leakage and (b) adversary's error probability in information hiding systems modelled as noisy channels. Specifically, we assume the attacker can make a single guess after observing n independent executions of the system, throughout which the secret information is kept fixed. We show that the asymptotic behaviour of quantities (a) and (b) can be determined in a simple way from the channel matrix. Moreover, simple and tight bounds on them as functions of n show that the convergence is exponential. We also discuss feasible methods to evaluate the rate of convergence. Our results cover both the Bayesian case, where an a priori probability distribution on the secrets is assumed known to the attacker, and the maximum-likelihood case, where the attacker does not know such distribution. In the Bayesian case, we identify the distributions that maximize leakage. We consider both the min-entropy setting studied by Smith and the additive form recently proposed by Braun et al. and show the two forms do agree asymptotically. Next, we extend these results to a more sophisticated eavesdropping scenario, where the attacker can perform a (noisy) observation at each state of the computation and the systems are modelled as hidden Markov models.
Perfect secrecy describes cases where an adversary cannot learn anything about the secret beyond its prior distribution. A classical result by Shannon shows that a necessary condition for perfect secrecy is that the adversary should not be able to eliminate any of the possible secrets. In this paper we answer the following fundamental question: What is the lowest leakage of information that can be achieved when some of the secrets have to be eliminated? We address this question by deriving the minimum leakage in closed-form, and explicitly providing "universally optimal" randomized strategies, in the sense that they guarantee the minimum leakage irrespective of the measure of entropy used to quantify the leakage. We then introduce a generalization of Rényi family of asymmetric measures of leakage which generalizes the g-leakage and show that a slight modification of our strategies are optimal with respect to an important class of such measures. Subsequently, we show that our schemes constitute the Nash Equilibria of closely related two-person zero sum games. This game perspective provides implicit solutions for a wider set of structural constraints and asymmetric entropies. Finally we demonstrate how this work can also be seen as designing a universally optimal channel given a specified prior.
We propose an operational measure of information leakage in a non-stochastic setting to formalize privacy against a brute-force guessing adversary. We use uncertain variables, non-probabilistic counterparts of random variables, to construct a guessing framework in which an adversary is interested in determining private information based on uncertain reports. We consider brute-force trial-and-error guessing in which an adversary can potentially check all the possibilities of the private information that are compatible with the available outputs to find the actual private realization. The ratio of the worst-case number of guesses for the adversary in the presence of the output and in the absence of it captures the reduction in the adversary’s guessing complexity and is thus used as a measure of private information leakage. We investigate the relationship between the newly-developed measure of information leakage with maximin information and stochastic maximal leakage that are shown to arise in one-shot guessing.
Given two discrete random variables X and Y, an operational approach is undertaken to quantify the “leakage” of information from X to Y. The resulting measure ℒ(X→Y ) is called maximal leakage, and is defined as the multiplicative increase, upon observing Y, of the probability of correctly guessing a randomized function of X, maximized over all such randomized functions. It is shown to be equal to the Sibson mutual information of order infinity, giving the latter operational significance. Its resulting properties are consistent with an axiomatic view of a leakage measure; for example, it satisfies the data processing inequality, it is asymmetric, and it is additive over independent pairs of random variables. Moreover, it is shown that the definition is robust in several respects: allowing for several guesses or requiring the guess to be only within a certain distance of the true function value does not change the resulting measure.