Differential privacy is a notion that has emerged in the community of statistical databases, as a response to the problem of protecting the privacy of the database's participants when performing statistical queries. The idea is that a randomized query satisfies differential privacy if the likelihood of obtaining a certain answer for a database x is not too different from the likelihood of obtaining the same answer on adjacent databases, i.e. databases which differ from x for only one individual. Information flow is an area of Security concerned with the problem of controlling the leakage of confidential information in programs and protocols. Nowadays, one of the most established approaches to quantify and to reason about leakage is based on the Renyi min entropy version of information theory. In this paper, we analyze critically the notion of differential privacy in light of the conceptual framework provided by the Renyi min information theory. We show that there is a close relation between differential privacy and leakage, due to the graph symmetries induced by the adjacency relation. Furthermore, we consider the utility of the randomized answer, which measures its expected degree of accuracy. We focus on certain kinds of utility functions called "binary", which have a close correspondence with the Renyi min mutual information. Again, it turns out that there can be a tight correspondence between differential privacy and utility, depending on the symmetries induced by the adjacency relation and by the query. Depending on these symmetries we can also build an optimal-utility randomization mechanism while preserving the required level of differential privacy. Our main contribution is a study of the kind of structures that can be induced by the adjacency relation and the query, and how to use them to derive bounds on the leakage and achieve the optimal utility.
As far as we know, the first work to investigate the relation between differential privacy and information-theoretic leakage was @cite_16 . In this work, a channel is relative to a given database @math , and the channel inputs are all possible databases adjacent to @math . Two bounds on leakage were presented, one for teh R 'enyi min entropy, and one for Shannon entropy. Our bound in Proposition is an improvement with respect to the (R 'enyi min entropy) bound in @cite_16 .
@cite_16
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Recent research in quantitative theories for information-hiding topics, such as Anonymity and Secure Information Flow, tend to converge towards the idea of modeling the system as a noisy channel in the information-theoretic sense. The notion of information leakage, or vulnerability of the system, has been related in some approaches to the concept of mutual information of the channel. A recent work of Smith has shown, however, that if the attack consists in one single try, then the mutual information and other concepts based on Shannon entropy are not suitable, and he has proposed to use Renyi's min-entropy instead. In this paper, we consider and compare two different possibilities of defining the leakage, based on the Bayes risk, a concept related to Renyi min-entropy. Differential privacy is a notion of privacy that has become very popular in the database community. Roughly, the idea is that a randomized query mechanism provides sufficient privacy protection if the ratio between the probabilities of two different entries to originate a certain answer is bound by e^e. In the fields of anonymity and information flow there is a similar concern for controlling information leakage, i.e. limiting the possibility of inferring the secret information from the observables. In recent years, researchers have proposed to quantify the leakage in terms of the information-theoretic notion of mutual information. There are two main approaches that fall in this category: One based on Shannon entropy, and one based on Renyi's min entropy. The latter has connection with the so-called Bayes risk, which expresses the probability of guessing the secret. In this paper, we show how to model the query system in terms of an information-theoretic channel, and we compare the notion of differential privacy with that of mutual information. We show that the notion of differential privacy is strictly stronger, in the sense that it implies a bound on the mutual information, but not viceversa. This note describes an information theory problem that arose from some analysis of quantum key distribution protocols. The problem seems very natural and is very easy to state but has not to our knowledge been addressed before in the information theory literature: suppose that we have a random bit string y of length n and we reveal k bits at random positions, preserving the order but without revealing the positions, how much information about y is revealed? We show that while the cardinality of the set of compatible y strings depends only on n and k, the amount of leakage does depend on the exact revealed x string. We observe that the maximal leakage, measured as decrease in the Shannon entropy of the space of possible bit strings, corresponds to the x string being all zeros or all ones and that the minimum leakage corresponds to the alternating x strings. We derive a formula for the maximum leakage minimal entropy in terms of n and k. We discuss the relevance of other measures of information, in particular min-entropy, in a cryptographic context. Finally, we describe a simulation tool to explore these results. The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R∘N)(ρ), where R is a recovery map with the property that σ=(R∘N)(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction. A tunable measure for information leakage called maximal a-leakage is introduced. This measure quantifies the maximal gain of an adversary in refining a tilted version of its prior belief of any (potentially random) function of a dataset conditioning on a disclosed dataset. The choice of @math determines the specific adversarial action ranging from refining a belief for @math to guessing the best posterior for @math , and for these extremal values this measure simplifies to mutual information (MI) and maximal leakage (MaxL), respectively. For all other @math this measure is shown to be the Arimoto channel capacity. Several properties of this measure are proven including: (i) quasi-convexity in the mapping between the original and disclosed datasets; (ii) data processing inequalities; and (iii) a composition property. A full version of this paper is in [1]. There is growing interest in quantitative theories of information flow in a variety of contexts, such as secure information flow, anonymity protocols, and side-channel analysis. Such theories offer an attractive way to relax the standard noninterference properties, letting us tolerate "small" leaks that are necessary in practice. The emerging consensus is that quantitative information flow should be founded on the concepts of Shannon entropy and mutual information . But a useful theory of quantitative information flow must provide appropriate security guarantees: if the theory says that an attack leaks x bits of secret information, then x should be useful in calculating bounds on the resulting threat. In this paper, we focus on the threat that an attack will allow the secret to be guessed correctly in one try. With respect to this threat model, we argue that the consensus definitions actually fail to give good security guarantees--the problem is that a random variable can have arbitrarily large Shannon entropy even if it is highly vulnerable to being guessed. We then explore an alternative foundation based on a concept of vulnerability (closely related to Bayes risk ) and which measures uncertainty using Renyi's min-entropy , rather than Shannon entropy. Consider two parties Alice and Bob, who hold private inputs x and y, and wish to compute a function f(x, y) privately in the information theoretic sense; that is, each party should learn nothing beyond f(x, y). However, the communication channel available to them is noisy. This means that the channel can introduce errors in the transmission between the two parties. Moreover, the channel is adversarial in the sense that it knows the protocol that Alice and Bob are running, and maliciously introduces errors to disrupt the communication, subject to some bound on the total number of errors. A fundamental question in this setting is to design a protocol that remains private in the presence of large number of errors. If Alice and Bob are only interested in computing f(x, y) correctly, and not privately, then quite robust protocols are known that can tolerate a constant fraction of errors. However, none of these solutions is applicable in the setting of privacy, as they inherently leak information about the parties' inputs. This leads to the question whether we can simultaneously achieve privacy and error-resilience against a constant fraction of errors. We show that privacy and error-resilience are contradictory goals. In particular, we show that for every constant c > 0, there exists a function f which is privately computable in the error-less setting, but for which no private and correct protocol is resilient against a c-fraction of errors. The same impossibility holds also for sub-constant noise rate, e.g., when c is exponentially small (as a function of the input size). This paper explores what kinds of information two parties must communicate in order to correct errors which occur in a shared secret string W. Any bits they communicate must leak a significant amount of information about W --- that is, from the adversary's point of view, the entropy of W will drop significantly. Nevertheless, we construct schemes with which Alice and Bob can prevent an adversary from learning any useful information about W. Specifically, if the entropy of W is sufficiently high, then there is no function f(W) which the adversary can learn from the error-correction information with significant probability.This leads to several new results: (a) the design of noise-tolerant "perfectly one-way" hash functions in the sense of [7], which in turn leads to obfuscation of proximity queries for high entropy secrets W; (b) private fuzzy extractors [11], which allow one to extract uniformly random bits from noisy and nonuniform data W, while also insuring that no sensitive information about W is leaked; and (c) noise tolerance and stateless key re-use in the Bounded Storage Model, resolving the main open problem of Ding [10].The heart of our constructions is the design of strong randomness extractors with the property that the source W can be recovered from the extracted randomness and any string W' which is close to W. The tradeoff between privacy and utility is studied for small datasets using tools from fixed error asymptotics in information theory. The problem is formulated as determining the privacy mechanism (random mapping) which minimizes the mutual information (a metric for privacy leakage) between the private features of the original dataset and a released version, subject to a distortion constraint between the public features and the released version. An excess probability bound is used to constrain the distortion, thus limiting the random variation in distortion due to the finite length. Bounds are derived for the following variants of the problem: (1) whether the mechanism is memoryless (local privacy) or not (global privacy), (2) whether the privacy mechanism has direct access to the private data or not. It is shown that these settings yield different performance in the first order: for global privacy, the first-order leakage decreases with the excess probability, whereas for local privacy it remains constant. The derived bounds also provide tight performance results up to second order for local privacy, as well as bounds on the second order term for global privacy. Theories of quantitative information flow have seen growing interest recently, in view of the fundamental importance of controlling the leakage of confidential information, together with the pragmatic necessity of tolerating intuitively "small" leaks. Given such a theory, it is crucial to develop automated techniques for calculating the leakage in a system. In this paper, we address this question in the context of deterministic imperative programs and under the recently-proposed min-entropy measure of information leakage, which measures leakage in terms of the confidential information's vulnerability to being guessed in one try by an adversary. In this context, calculating the maximum leakage of a program reduces to counting the number of feasible outputs that it can produce. We approach this task by determining patterns among pairs of bits in the output, for instance by determining that two bits must be unequal. By counting the number of solutions to the two-bit patterns, we obtain an upper bound on the number of feasible outputs and hence on the leakage. We explore the effectiveness of our approach on a number of case studies, in terms of both efficiency and accuracy. In this work, we study information leakage in timing side channels that arise in the context of shared event schedulers. Consider two processes, one of them an innocuous process (referred to as Alice) and the other a malicious one (referred to as Bob), using a common scheduler to process their jobs. Based on when his jobs get processed, Bob wishes to learn about the pattern (size and timing) of jobs of Alice. Depending on the context, knowledge of this pattern could have serious implications on Alice's privacy and security. For instance, shared routers can reveal traffic patterns, shared memory access can reveal cloud usage patterns, and suchlike. We present a formal framework to study the information leakage in shared resource schedulers using the pattern estimation error as a performance metric. In this framework, a uniform upper bound is derived to benchmark different scheduling policies. The first-come-first-serve scheduling policy is analyzed, and shown to leak significant information when the scheduler is loaded heavily. To mitigate the timing information leakage, we propose an “Accumulate-and-Serve” policy which trades in privacy for a higher delay. The policy is analyzed under the proposed framework and is shown to leak minimum information to the attacker, and is shown to have comparatively lower delay than a fixed scheduler that preemptively assigns service times irrespective of traffic patterns. There is a clear intuitive connection between the notion of leakage of information in a program and concepts from information theory. This intuition has not been satisfactorily pinned down, until now. In particular, previous information-theoretic models of programs are imprecise, due to their overly conservative treatment of looping constructs. In this paper we provide the first precise information-theoretic semantics of looping constructs. Our semantics describes both the amount and rate of leakage; if either is small enough, then a program might be deemed "secure". Using the semantics we provide an investigation and classification of bounded and unbounded covert channels. Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where the data providers and data analysts want to maximize the utility of statistical inferences performed on the released data, we study the fundamental tradeoff between local differential privacy and information theoretic utility functions. We introduce a family of extremal privatization mechanisms, which we call staircase mechanisms, and prove that it contains the optimal privatization mechanism that maximizes utility. We further show that for all information theoretic utility functions studied in this paper, maximizing utility is equivalent to solving a linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the data size. To account for this, we show that two simple staircase mechanisms, the binary and randomized response mechanisms, are universally optimal in the high and low privacy regimes, respectively, and well approximate the intermediate regime. We derive lower bounds to the second-order coding rates for the wiretap channel. The decoding error probability and the information leakage measured in terms of the variational distance secrecy metric are fixed at some constants e r and e s respectively. We leverage on the connection between wiretap channel coding and channel resolvability to derive tighter secrecy bounds than those available in the literature. We then use central limit theorem-style analysis to evaluate these bounds for the discrete memoryless wiretap channel with cost constraints and the Gaussian wiretap channel. This paper investigates the relation between three different notions of privacy: identifiability, differential privacy, and mutual-information privacy. Under a unified privacy-distortion framework, where the distortion is defined to be the expected Hamming distance between the input and output databases, we establish some fundamental connections between these three privacy notions. Given a maximum allowable distortion @math , we define the privacy-distortion functions @math , @math , and @math to be the smallest (most private best) identifiability level, differential privacy level, and mutual information between the input and the output, respectively. We characterize @math and @math , and prove that @math for @math within certain range, where @math is a constant determined by the prior distribution of the original database @math , and diminishes to zero when @math is uniformly distributed. Furthermore, we show that @math and @math can be achieved by the same mechanism for @math within certain range, i.e., there is a mechanism that simultaneously minimizes the identifiability level and achieves the best mutual-information privacy. Based on these two connections, we prove that this mutual-information optimal mechanism satisfies @math -differential privacy with @math . The results in this paper reveal some consistency between two worst case notions of privacy, namely, identifiability and differential privacy, and an average notion of privacy, mutual-information privacy. A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether a given database row is included. The goal of this paper is to formulate and provide strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a symmetric and monotone loss function). Our main result is the following: for each fixed count query and differential privacy level, there is a geometric mechanism @math ---a discrete variant of the simple and well-studi... A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Privacy can be rigorously quantified using the framework of differential privacy , which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism @math -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user @math , no matter what its side information and preferences, derives as much utility from @math as from interacting with a differentially private mechanism @math that is optimally tailored to @math . We compare the disclosure risk criterion of e-differential privacy with a criterion based on probabilities that intruders uncover actual values given the released data. To do so, we generate fully synthetic data that satisfy e-differential privacy at different levels of e, make assumptions about the information available to intruders, and compute posterior probabilities of uncovering true values. The simulation results suggest that the two paradigms are not easily reconciled, since differential privacy is agnostic to the specific values in the observed data whereas probabilistic disclosure risk measures depend greatly on them. The results also suggest, perhaps surprisingly, that probabilistic disclosure risk measures can be small even when e is large. Motivated by these findings, we present an alternative disclosure risk assessment approach that integrates some of the strong confidentiality protection features in e-differential privacy with the interpretability and data-specific nature of probabilistic disclosure risk measures. We propose a general statistical inference framework to capture the privacy threat incurred by a user that releases data to a passive but curious adversary, given utility constraints. We show that applying this general framework to the setting where the adversary uses the self-information cost function naturally leads to a non-asymptotic information-theoretic approach for characterizing the best achievable privacy subject to utility constraints. Based on these results we introduce two privacy metrics, namely average information leakage and maximum information leakage. We prove that under both metrics the resulting design problem of finding the optimal mapping from the user's data to a privacy-preserving output can be cast as a modified rate-distortion problem which, in turn, can be formulated as a convex program. Finally, we compare our framework with differential privacy. A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism M* -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user u, no matter what its side information and preferences, derives as much utility from M* as from interacting with a differentially private mechanism Mu that is optimally tailored to u. More precisely, for every user u there is an optimal mechanism Mu for it that factors into a user-independent part (the geometric mechanism M*) followed by user-specific post-processing that can be delegated to the user itself. The first part of our proof of this result characterizes the optimal differentially private mechanism for a fixed but arbitrary user in terms of a certain basic feasible solution to a linear program with constraints that encode differential privacy. The second part shows that all of the relevant vertices of this polytope (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range.