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|Yensuring 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 {(Xi, Yi)}ni=1being pairs of independent and identically distributed binary random vectors, we similarly define h_n(PX n Y n, ε) under the assumption that Znis also a binary vector. Then we obtain a closed form expression for h_n(PX n Y n, ε) for sufficiently large, but nontrivial values of ε.
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|Yensuring 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 {(Xi, Yi)}ni=1being pairs of independent and identically distributed binary random vectors, we similarly define h_n(PX n Y n, ε) under the assumption that Znis also a binary vector. Then we obtain a closed form expression for h_n(PX n Y n, ε) for sufficiently large, but nontrivial values of ε.
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.
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 observingY, of the guesswork of a randomized function ofX, 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 ofY. 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 maximalα-leakage in terms of guesswork.
Let x, y, z be finite sets, X,Y random variables uniformly distributed over x×y, f a function from x×y to Z and 0≤ε&le1. A person PX knows X and a person PY knows Y and they want to exchange X and Y. An eavesdropper who knows their protocol listens to their communication in order to obtain information about f(X, Y). PX and PY want to ensure that for every value (x,y) of (X,Y) the eavesdropper's a priori and a posteriori probabilities of {f(X,Y)=j} are ε-close for all j. Therefore, they encrypt some of the transmitted bits. The problem is to find a protocol that minimizes the number of bits encrypted in the worst case. Two kinds of protocols are considered: deterministic and randomized.
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.
This paper is concerned with secret-key agreement by public discussion: two parties Alice and Bob and an adversary Eve have access to independent realizations of random variables X , Y , and Z, respectively, with joint distribution PXY Z . The secret key rate S(X ; Y jjZ) has been defined as the maximal rate at which Alice and Bob can generate a secret key by communication over an insecure, but authenticated channel such that Eve's information about this key is arbitrarily small. We define a new conditional mutual information measure, the intrinsic conditional mutual information between X and Y when given Z, denoted by I(X ; Y#Z), which is an upper bound on S(X ; Y jjZ). The special scenarios where X , Y , and Z are generated by sending a binary random variable R, for example a signal broadcast by a satellite, over independent channels, or where Z is generated by sending X and Y over erasure channels, are analyzed. In the first scenario it can be shown that the secret key rate is str...
We define "variable privacy" as the use of non-perfect protocols with parameters controlled by Alice. Variable privacy enables Alice to choose the amount of information leaked to Bob, in situations where information revelation bears a privacy cost and also provides a benefit. We propose a framework for the study of variable privacy, using a security perspective to obtain a privacy measure of the binary symmetric randomization protocol (flipping a bit with probability 1-#). We define an attack as any sequence of protocol instances that decreases estimation error on a bit beyond that possible with a single instance. Viewing the protocol as a communication channel for the data to be protected, we show that channel codes - i.e. error-correcting and error-detecting codes - are particularly e#cient attacks. In particular, they can be more e#cient than the repeated query attack.