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.
We study information leakage through guesswork, the minimum expected number of guesses required to guess a random variable. In particular, we define maximal guesswork leakage as the multiplicative decrease, upon observing
Y, of the guesswork of a randomized function of
X, maximized over all such randomized functions. We also study a pointwise form of the leakage which captures the leakage due to the release of a single realization of
Y. We also study these two notions of leakage with oblivious (or memoryless) guessing. We obtain closed-form expressions for all these leakage measures, with the exception of one. Specifically, we are able to obtain closed-form expression for maximal guesswork leakage for the binary erasure source only; deriving expressions for arbitrary sources appears challenging. Some of the consequences of our results are - a connection between guesswork and differential privacy and a new operational interpretation to maximal
α-leakage in terms of guesswork.
An alternative measure of information leakage for quantum encoding of classical data is defined. An adversary can access a single copy of the state of a quantum system that encodes some classical data and is interested in correctly guessing a general randomized or deterministic function of the data (e.g., a specific feature or attribute of the data in quantum machine learning) that is unknown to the security analyst. The resulting measure of information leakage, referred to as maximal quantum leakage, is the multiplicative increase of the probability of correctly guessing any function of the classical data upon observing measurements of the quantum state. Maximal quantum leakage is shown to satisfy the postprocessing inequality (i.e., applying a quantum channel reduces information leakage) and independence property (i.e., leakage is zero if the quantum state is independent of the classical data), which are fundamental properties required for privacy and security analysis. It also bounds accessible information. Effects of global and local depolarizing noise models on the maximal quantum leakage are established.
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.
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.
We investigate the problem of guessing a discrete random variable
Y under a privacy constraint dictated by another correlated discrete random variable
X, where both guessing efficiency and privacy are assessed in terms of the probability of correct guessing. We define
h(PXY,ε) as the maximum probability of correctly guessing
Y given an auxiliary random variable
Z, where the maximization is taken over all
PZ∣Y ensuring that the probability of correctly guessing
X given
Z does not exceed
ε. We show that the map
ε↦h(PXY,ε) is strictly increasing, concave, and piecewise linear, which allows us to derive a closed form expression for
h(PXY,ε) when
X and
Y are connected via a binary-input binary-output channel. For
(Xn,Yn) being pairs of independent and identically distributed binary random vectors, we similarly define
hn(PXnYn,ε) under the assumption that
Zn is also a binary vector. Then we obtain a closed form expression for
hn(PXnYn,ε) for sufficiently large, but nontrivial values of
ε.
Maximal quantum leakage quantifies privacy against adversaries with arbitrary intentions. In this work, we prove that computing this leakage is equivalent to minimum-error quantum state discrimination with equal priors. This establishes a computable operational interpretation, addressing the previous difficulty in computing maximal quantum leakage. We further analyze the impact of collective measurements on multiple copies of a state, demonstrating that leakage increases monotonically with the number of copies, which leads to explicitly characterizing the maximal leakage in the asymptotic limit. Extending this framework to quantum channels, we develop an iterative algorithm for the jointly designing of input states and measurements. Numerical examples involving collective measurements and the maximal channel leakage demonstrate our theoretical findings.