Abstract. Weighted threshold secret sharing was introduced by Shamir in his seminal work on secret sharing. In such settings, there is a set of users where each user is assigned a positive weight. A dealer wishes to distribute a secret among those users so that a subset of users may reconstruct the secret if and only if the sum of weights of its users exceeds a certain threshold. A secret sharing scheme is ideal if the size of the domain of shares of each user is the same as the size of the domain of possible secrets (this is the smallest possible size for the domain of shares). The family of subsets authorized to reconstruct the secret in a secret sharing scheme is called an access structure. An access structure is ideal if there exists an ideal secret sharing scheme that realizes it.It is known that some weighted threshold access structures are not ideal, while other nontrivial weighted threshold access structures do have an ideal scheme that realizes them. In this work we characterize all weighted threshold access structures that are ideal. We show that a weighted threshold access structure is ideal if and only if it is a hierarchical threshold access structure (as introduced by Simmons), or a tripartite access structure (these structures, that we introduce here, generalize the concept of bipartite access structures due to Padró and Sáez), or a composition of two ideal weighted threshold access structures that are defined on smaller sets of users. We further show that in all those cases the weighted threshold access structure may be realized by a linear ideal secret sharing scheme. The proof of our characterization relies heavily on the strong connection between ideal secret sharing schemes and matroids, as proved by Brickell and Davenport.

Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently non-monotone since addition is a non-monotone function. In this work we by-pass this addition step and construct a polynomial size logarithmic depth unbounded fan-in monotone circuit for every weighted threshold function, i.e., we show that weighted threshold functions are in mAC 1 . (To the best of our knowledge, prior to our work no polynomial monotone circuits were known for weighted threshold functions.) Our monotone circuits are applicable for the cryptographic tool of secret sharing schemes. Using general results for compiling monotone circuits (Yao, 1989) and monotone formulae (Benaloh and Leichter, 1990) into secret sharing schemes, we get secret sharing schemes for every weighted threshold access structure. Specifically, we get: (1) information-theoretic secret sharing schemes where the size of each share is quasi-polynomial in the number of users, and (2) computational secret sharing schemes where the size of each share is polynomial in the number of users.

Many approximation algorithms have been presented in the last decades for hard search problems. The focus of this paper is on cryptographic applications, where it is desired to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms -efficient algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information.For functions, [Feigenbaum et al., ICALP 2001] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to search problems is not straight forward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms -applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. Although our privacy requirement seems minimal, we show that for well studied problems, as vertex cover and maximum exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similar to [Halevi et al., STOC 2001], we define a relaxed notion of approximation algorithms that leak (little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for maximum exact 3SAT which leak little information.

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Abstract. We present communication efficient secure protocols for a variety of linear algebra problems. Our main building block is a protocol for computing Gaussian Elimination on encrypted data. As input for this protocol, Bob holds a k × k matrix M , encrypted with Alice's key. At the end of the protocol run, Bob holds an encryption of an upper-triangular matrix M such that the number of nonzero elements on the diagonal equals the rank of M . The communication complexity of our protocol is roughly O(k 2 ). Building on Oblivious Gaussian elimination, we present secure protocols for several problems: deciding the intersection of linear and affine subspaces, picking a random vector from the intersection, and obliviously solving a set of linear equations. Our protocols match known (insecure) communication complexity lower bounds, and improve the communication complexity of both Yao's garbled circuits and that of specific previously published protocols.