Person: Vadhan, Salil
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Vadhan
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Salil
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Vadhan, Salil
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Publication Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space(2017) Murtagh, Jack; Reingold, Omer; Sidford, Aaron; Vadhan, SalilWe give a deterministic O˜(log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using O(log n) space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using O(log3/2 n) space (Saks and Zhou, FOCS 1995 and JCSS 1999). Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC ‘04; Peng and Spielman, STOC ‘14) with ideas used to show that UNDIRECTED S-T CONNECTIVITY is in deterministic logspace (Reingold, STOC ‘05 and JACM ‘08; Rozenman and Vadhan, RANDOM ‘05).Publication Bridging the Gap between Computer Science and Legal Approaches to Privacy(Harvard Law School, 2018) Nissim, Kobbi; Bembenek, Aaron; Wood, Alexandra; Bun, Mark Mar; Gaboardi, Marco; Gasser, Urs; O'Brien, David; Vadhan, Salil; Steinke, ThomasThe analysis and release of statistical data about individuals and groups of individuals carries inherent privacy risks, and these risks have been conceptualized in different ways within the fields of law and computer science. For instance, many information privacy laws adopt notions of privacy risk that are sector- or context-specific, such as in the case of laws that protect from disclosure certain types of information contained within health, educational, or financial records. In addition, many privacy laws refer to specific techniques, such as deidentification, that are designed to address a subset of possible attacks on privacy. In doing so, many legal standards for privacy protection rely on individual organizations to make case-by-case determinations regarding concepts such as the identifiability of the types of information they hold. These regulatory approaches are intended to be flexible, allowing organizations to (1) implement a variety of specific privacy measures that are appropriate given their varying institutional policies and needs, (2) adapt to evolving best practices, and (3) address a range of privacy-related harms. However, in the absence of clear thresholds and detailed guidance on making case-specific determinations, flexibility in the interpretation and application of such standards also creates uncertainty for practitioners and often results in ad hoc, heuristic processes. This uncertainty may pose a barrier to the adoption of new technologies that depend on unambiguous privacy requirements. It can also lead organizations to implement measures that fall short of protecting against the full range of data privacy risks.Publication Interactive proofs of proximity: Delegating computation in sublinear time(ACM Press, 2013) Rothblum, Guy N.; Vadhan, Salil; Wigderson, AviWe study interactive proofs with sublinear-time verifiers. These proof systems can be used to ensure approximate correctness for the results of computations delegated to an untrusted server. Following the literature on property testing, we seek proof systems where with high probability the verifier accepts every input in the language, and rejects every input that is far from the language. The verifier's query complexity (and computation complexity), as well as the communication, should all be sublinear. We call such a proof system an Interactive Proof of Proximity (IPP). On the positive side, our main result is that all languages in NC have Interactive Proofs of Proximity with roughly √n query and communication and complexities, and polylog(n) communication rounds. This is achieved by identifying a natural language, membership in an affine subspace (for a structured class of subspaces), that is complete for constructing interactive proofs of proximity, and providing efficient protocols for it. In building an IPP for this complete language, we show a tradeoff between the query and communication complexity and the number of rounds. For example, we give a 2-round protocol with roughly \(n^{3/4}\) queries and communication. On the negative side, we show that there exist natural languages in \(NC^1\), for which the sum of queries and communication in any constant-round interactive proof of proximity must be polynomially related to n. In particular, for any 2-round protocol, the sum of queries and communication must be at least ~Ω(√n). Finally, we construct much better IPPs for specific functions, such as bipartiteness on random or well-mixing graphs, and the majority function. The query complexities of these protocols are provably better (by exponential or polynomial factors) than what is possible in the standard property testing model, i.e. without a prover.Publication Publicly verifiable proofs of sequential work(ACM Press, 2013) Mahmoody, Mohammad; Moran, Tal; Vadhan, SalilWe construct a publicly verifiable protocol for proving computational work based on collision-resistant hash functions and a new plausible complexity assumption regarding the existence of "inherently sequential" hash functions. Our protocol is based on a novel construction of time-lock puzzles. Given a sampled "puzzle" P getsr Dn, where $n$ is the security parameter and Dn is the distribution of the puzzles, a corresponding "solution" can be generated using N evaluations of the sequential hash function, where N>n is another parameter, while any feasible adversarial strategy for generating valid solutions must take at least as much time as Ω(N) serial evaluations of the hash function after receiving $P$. Thus, valid solutions constitute a "proof" that Ω(N) parallel time elapsed since p was received. Solutions can be publicly and efficiently verified in time poly(n) ⋅ polylog(N). Applications of these "time-lock puzzles" include noninteractive timestamping of documents (where the distribution over the possible documents corresponds to the puzzle distribution Dn) and universally verifiable CPU benchmarks. Our construction is secure in the standard model under complexity assumptions (collision-resistant hash functions and inherently sequential hash functions), and makes black-box use of the underlying primitives. Consequently, the corresponding construction in the random oracle model is secure unconditionally. Moreover, as it is a public-coin protocol, it can be made non-interactive in the random oracle model using the Fiat-Shamir Heuristic. Our construction makes a novel use of "depth-robust" directed acyclic graphs---ones whose depth remains large even after removing a constant fraction of vertices---which were previously studied for the purpose of complexity lower-bounds. The construction bypasses a recent lower-bound of Mahmoody, Moran, and Vadhan (CRYPTO '11) for time-lock puzzles in the random oracle model, which showed that it is impossible to have time-lock puzzles like ours in the random oracle model if the puzzle generator also computes a solution together with the puzzle.Publication Deterministic Public-Key Encryption for Adaptively Chosen Plaintext Distributions(Springer Berlin Heidelberg, 2013) Raghunathan, Ananth; Segev, Gil; Vadhan, SalilBellare, Boldyreva, and O’Neill (CRYPTO ’07) initiated the study of deterministic public-key encryption as an alternative in scenarios where randomized encryption has inherent drawbacks. The resulting line of research has so far guaranteed security only for adversarially-chosen plaintext distributions that are independent of the public key used by the scheme. In most scenarios, however, it is typically not realistic to assume that adversaries do not take the public key into account when attacking a scheme. We show that it is possible to guarantee meaningful security even for plaintext distributions that depend on the public key. We extend the previously proposed notions of security, allowing adversaries to adaptively choose plaintext distributions after seeing the public key, in an interactive manner. The only restrictions we make are that: (1) plaintext distributions are unpredictable (as is essential in deterministic public-key encryption), and (2) the number of plaintext distributions from which each adversary is allowed to adaptively choose is upper bounded by \(2^ p\) , where p can be any predetermined polynomial in the security parameter. For example, with p = 0 we capture plaintext distributions that are independent of the public key, and with p = O(s logs) we capture, in particular, all plaintext distributions that are samplable by circuits of size s. Within our framework we present both constructions in the random-oracle model based on any public-key encryption scheme, and constructions in the standard model based on lossy trapdoor functions (thus, based on a variety of number-theoretic assumptions). Previously known constructions heavily relied on the independence between the plaintext distributions and the public key for the purposes of randomness extraction. In our setting, however, randomness extraction becomes significantly more challenging once the plaintext distributions and the public key are no longer independent. Our approach is inspired by research on randomness extraction from seed-dependent distributions. Underlying our approach is a new generalization of a method for such randomness extraction, originally introduced by Trevisan and Vadhan (FOCS ’00) and Dodis (PhD Thesis, MIT, ’00).Publication A Uniform Min-Max Theorem with Applications in Cryptography(Springer Berlin Heidelberg, 2013) Vadhan, Salil; Zheng, Colin JiaWe present a new, more constructive proof of von Neumann’s Min-Max Theorem for two-player zero-sum game — specifically, an algorithm that builds a near-optimal mixed strategy for the second player from several best-responses of the second player to mixed strategies of the first player. The algorithm extends previous work of Freund and Schapire (Games and Economic Behavior ’99) with the advantage that the algorithm runs in poly(n) time even when a pure strategy for the first player is a distribution chosen from a set of distributions over {0, 1}\(^n\) . This extension enables a number of additional applications in cryptography and complexity theory, often yielding uniform security versions of results that were previously only proved for nonuniform security (due to use of the non-constructive Min-Max Theorem). We describe several applications, including a more modular and improved uniform version of Impagliazzo’s Hardcore Theorem (FOCS ’95), showing impossibility of constructing succinct non-interactive arguments (SNARGs) via black-box reductions under uniform hardness assumptions (using techniques from Gentry and Wichs (STOC ’11) for the nonuniform setting), and efficiently simulating high entropy distributions within any sufficiently nice convex set (extending a result of Trevisan, Tulsiani and Vadhan (CCC ’09)).Publication Pseudorandomness for Regular Branching Programs via Fourier Analysis(Springer Berlin Heidelberg, 2013) Reingold, Omer; Steinke, Thomas Alexander; Vadhan, SalilWe present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is \(O(log^2 n)\), where n is the length of the branching program. The previous best seed length known for this model was \(n^{ 1/2 + o(1)}\), which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of \(s^{ 1/2 + o(1)}\) for arbitrary branching programs of size s). Our techniques also give seed length \(n^{ 1/2 + o(1)}\) for general oblivious, read-once branching programs of width \(2^{n^{o(1)}}\), which is incomparable to the results of Impagliazzo et al. Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width w has Fourier mass at most \((2w^ 2) ^k\) at level k, independent of the length of the program.Publication Differential Privacy with Imperfect Randomness(Springer Verlag, 2012) Dodis, Yevgeniy; López-Alt, Adriana; Mironov, Ilya; Vadhan, SalilIn this work we revisit the question of basing cryptogra- phy on imperfect randomness. Bosley and Dodis (TCC’07) showed that if a source of randomness R is “good enough” to generate a secret key capable of encrypting k bits, then one can deterministically extract nearly k almost uniform bits from R, suggesting that traditional privacy notions (namely, indistinguishability of encryption) requires an “extractable” source of randomness. Other, even stronger impossibility results are known for achieving privacy under specific “non-extractable” sources of randomness, such as the γ-Santha-Vazirani (SV) source, where each next bit has fresh entropy, but is allowed to have a small bias γ < 1 (possibly depending on prior bits). We ask whether similar negative results also hold for a more recent notion of privacy called differential privacy (Dwork et al., TCC’06), concentrating, in particular, on achieving differential privacy with the Santha-Vazirani source. We show that the answer is no. Specifically, we give a differentially private mechanism for approximating arbitrary “low sensitivity” functions that works even with randomness coming from a γ-Santha-Vazirani source, for any γ < 1. This provides a somewhat surprising “separation” between traditional privacy and differential privacy with respect to imperfect randomness. Interestingly, the design of our mechanism is quite different from the traditional “additive-noise” mechanisms (e.g., Laplace mechanism) successfully utilized to achieve differential privacy with perfect randomness. Indeed, we show that any (non-trivial) “SV-robust” mechanism for our problem requires a demanding property called consistent sampling, which is strictly stronger than differential privacy, and cannot be satisfied by any additive-noise mechanism.Publication Randomness Condensers for Efficiently Samplable, Seed-Dependent Sources(Springer Verlag, 2012) Dodis, Yevgeniy; Ristenpart, Thomas; Vadhan, SalilWe initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the con- denser. That is, we seek functions Cond : {0, 1}n ×{0, 1}d → {0, 1}m such that if we choose a random seed S ← {0,1}d, and a source X = A(S) is generated by a randomized circuit A of size t such that X has min- entropy at least k given S, then Cond(X;S) should have min-entropy at least some k′ given S. The distinction from the standard notion of ran- domness condensers is that the source X may be correlated with the seed S (but is restricted to be efficiently samplable). Randomness extractors of this type (corresponding to the special case where k′ = m) have been implicitly studied in the past (by Trevisan and Vadhan, FOCS ‘00). We show that: – Unlike extractors, we can have randomness condensers for samplable, seed-dependent sources whose computational complexity is smaller than the size t of the adversarial sampling algorithm A. Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy k′ = m − O(log t), i.e. logarithmic entropy deficiency. – Randomness condensers suffice for key derivation in many crypto- graphic applications: when an adversary has negligible success proba- bility (or negligible “squared advantage” [3]) for a uniformly random key, we can use instead a key generated by a condenser whose output has logarithmic entropy deficiency. – Randomness condensers for seed-dependent samplable sources that are robust to side information generated by the sampling algorithm imply soundness of the Fiat-Shamir Heuristic when applied to any constant-round, public-coin interactive proof system.Publication Faster Algorithms for Privately Releasing Marginals(Springer Verlag, 2012) Thaler, Justin; Ullman, Jonathan; Vadhan, SalilWe study the problem of releasing k-way marginals of a database D ∈ {0,1}d)n, while preserving differential privacy. The an- swer to a k-way marginal query is the fraction of D’s records x ∈ {0, 1}d with a given value in each of a given set of up to k columns. Marginal queries enable a rich class of statistical analyses of a dataset, and de- signing efficient algorithms for privately releasing marginal queries has been identified as an important open problem in private data analysis (cf. Barak et. al., PODS ’07). We give an algorithm that runs in time dO( k) and releases a private summary capable of answering any k-way marginal query with at most ±.01 error on every query as long as n ≥ dO( k). To our knowledge, ours is the first algorithm capable of privately releasing marginal queries with non-trivial worst-case accuracy guarantees in time substantially smaller than the number of k-way marginal queries, which is dΘ(k) (for k ≪ d).