Abstract. A secret-sharing scheme is a method by which a dealer distributes shares to parties such that only authorized subsets of parties can reconstruct the secret. Secret-sharing schemes are important tools in cryptography and they are used as a building box in many secure protocols, e.g., general protocol for multiparty computation, Byzantine agreement, threshold cryptography, access control, attribute-based encryption, and generalized oblivious transfer. In this survey, we will describe the most important constructions of secret-sharing schemes, explaining the connections between secret-sharing schemes and monotone formulae and monotone span programs. The main problem with known secret-sharing schemes is the large share size: it is exponential in the number of parties. We conjecture that this is unavoidable. We will discuss the known lower bounds on the share size. These lower bounds are fairly weak and there is a big gap between the lower and upper bounds. For linear secret-sharing schemes, which is a class of schemes based on linear algebra that contains most known schemes, super-polynomial lower bounds on the share size are known. We will describe the proofs of these lower bounds. We will also present two results connecting secret-sharing schemes for a Hamiltonian access structure to the NP vs. coNP problem and to a major open problem in cryptography -constructing oblivious-transfer protocols from one-way functions.
Abstract. Learning is a task that generalizes many of the analyses that are applied to collections of data, and in particular, collections of sensitive individual information. Hence, it is natural to ask what can be learned while preserving individual privacy. [Kasiviswanathan, Lee, Nissim, Raskhodnikova, and Smith; FOCS 2008] initiated such a discussion. They formalized the notion of private learning, as a combination of PAC learning and differential privacy, and investigated what concept classes can be learned privately. Somewhat surprisingly, they showed that, ignoring time complexity, every PAC learning task could be performed privately with polynomially many samples, and in many natural cases this could even be done in polynomial time. While these results seem to equate non-private and private learning, there is still a significant gap: the sample complexity of (non-private) PAC learning is crisply characterized in terms of the VC-dimension of the concept class, whereas this relationship is lost in the constructions of private learners, which exhibit, generally, a higher sample complexity. Looking into this gap, we examine several private learning tasks and give tight bounds on their sample complexity. In particular, we show strong separations between sample complexities of proper and improper private learners (such separation does not exist for non-private learners), and between sample complexities of efficient and inefficient proper private learners. Our results show that VC-dimension is not the right measure for characterizing the sample complexity of proper private learning. We also examine the task of private data release (as initiated by STOC 2008]), and give new lower bounds on the sample complexity. Our results show that the logarithmic dependence on size of the instance space is essential for private data release.
We compare the sample complexity of private learning [Kasiviswanathan et al. 2008] and sanitization [Blum et al. 2008] under pure ǫ-differential privacy [Dwork et al. TCC 2006] and approximate (ǫ, δ)-differential privacy [Dwork et al. Eurocrypt 2006]. We show that the sample complexity of these tasks under approximate differential privacy can be significantly lower than that under pure differential privacy.We define a family of optimization problems, which we call Quasi-Concave Promise Problems, that generalizes some of our considered tasks. We observe that a quasi-concave promise problem can be privately approximated using a solution to a smaller instance of a quasi-concave promise problem. This allows us to construct an efficient recursive algorithm solving such problems privately. Specifically, we construct private learners for point functions, threshold functions, and axis-aligned rectangles in high dimension. Similarly, we construct sanitizers for point functions and threshold functions.We also examine the sample complexity of label-private learners, a relaxation of private learning where the learner is required to only protect the privacy of the labels in the sample. We show that the VC dimension completely characterizes the sample complexity of such learners, that is, the sample complexity of learning with label privacy is equal (up to constants) to learning without privacy.
Abstract. We examine the combination of two directions in the field of privacy concerning computations over distributed private inputs -secure function evaluation (SFE) and differential privacy. While in both the goal is to privately evaluate some function of the individual inputs, the privacy requirements are significantly different. The general feasibility results for SFE suggest a natural paradigm for implementing differentially private analyses distributively: First choose what to compute, i.e., a differentially private analysis; Then decide how to compute it, i.e., construct an SFE protocol for this analysis. We initiate an examination whether there are advantages to a paradigm where both decisions are made simultaneously. In particular, we investigate under which accuracy requirements it is beneficial to adapt this paradigm for computing a collection of functions including Binary Sum, Gap Threshold, and Approximate Median queries. Our results yield new separations between the local and global models of computations for private data analysis.
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