Dwork, Cynthia, Guy Rothblum, and Salil Vadhan. “Boosting and differential privacy.” In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS ‘10), 51-60. IEEE, 2010. Publisher's VersionAbstract
Boosting is a general method for improving the accuracy of learning algorithms. We use boosting to construct improved privacy-preserving synopses of an input database. These are data structures that yield, for a given set \(\mathcal{Q}\) of queries over an input database, reasonably accurate estimates of the responses to every query in \(\mathcal{Q}\), even when the number of queries is much larger than the number of rows in the database. Given a base synopsis generator that takes a distribution on \(\mathcal{Q}\) and produces a "weak" synopsis that yields "good" answers for a majority of the weight in \(\mathcal{Q}\), our Boosting for Queries algorithm obtains a synopsis that is good for all of \(\mathcal{Q}\). We ensure privacy for the rows of the database, but the boosting is performed on the queries. We also provide the first synopsis generators for arbitrary sets of arbitrary low-sensitivity queries, i.e., queries whose answers do not vary much under the addition or deletion of a single row. In the execution of our algorithm certain tasks, each incurring some privacy loss, are performed many times. To analyze the cumulative privacy loss, we obtain an \(O(\varepsilon^2)\) bound on the expected privacy loss from a single \(\varepsilon\)-differentially private mechanism. Combining this with evolution of confidence arguments from the literature, we get stronger bounds on the expected cumulative privacy loss due to multiple mechanisms, each of which provides \(\varepsilon\)-differential privacy or one of its relaxations, and each of which operates on (potentially) different, adaptively chosen, databases.
FOCS-IEEE 2010.pdf
Chung, Kai-Min, Yael Kalai, and Salil Vadhan. “Improved delegation of computation using fully homomorphic encryption.” In T. Rabin, editor, Advances in Cryptology—CRYPTO ‘10, Lecture Notes in Computer Science, 6223:483-501. Springer-Verlag, 2010. Publisher's VersionAbstract

Version HistoryFull version posted as Cryptology ePrint Archive Report 210/241.

Following Gennaro, Gentry, and Parno (Cryptology ePrint Archive 2009/547), we use fully homomorphic encryption to design improved schemes for delegating computation. In such schemes, a delegator outsources the computation of a function \({F}\) on many, dynamically chosen inputs \(x_i\) to a worker in such a way that it is infeasible for the worker to make the delegator accept a result other than \({F}(x_i)\). The “online stage” of the Gennaro et al. scheme is very efficient: the parties exchange two messages, the delegator runs in time poly\((\log{T})\), and the worker runs in time poly\((T)\), where \(T\) is the time complexity of \(F\). However, the “offline stage” (which depends on the function \(F\) but not the inputs to be delegated) is inefficient: the delegator runs in time poly\((T)\) and generates a public key of length poly\((T)\) that needs to be accessed by the worker during the online stage.

Our first construction eliminates the large public key from the Gennaro et al. scheme. The delegator still invests poly\((T)\) time in the offline stage, but does not need to communicate or publish anything. Our second construction reduces the work of the delegator in the offline stage to poly\((\log{T})\) at the price of a 4-message (offline) interaction with a poly\((T)\)-time worker (which need not be the same as the workers used in the online stage). Finally, we describe a “pipelined” implementation of the second construction that avoids the need to re-run the offline construction after errors are detected (assuming errors are not too frequent).

IACR2010.pdf CRYPTO2010.pdf
McGregor, Andrew, Ilya Mironov, Toniann Pitassi, Omer Reingold, Kunal Talwar, and Salil Vadhan. “The limits of two-party differential privacy.” In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS ‘10), 81-90. IEEE, 2010. Publisher's VersionAbstract

Version History and Errata: Subsequent version published in ECCC 2011. Proposition 8 and Part (b) of Theorem 13 in the FOCS version are incorrect, and are removed from the ECCC version.


We study differential privacy in a distributed setting where two parties would like to perform analysis of their joint data while preserving privacy for both datasets. Our results imply almost tight lower bounds on the accuracy of such data analyses, both for specific natural functions (such as Hamming distance) and in general. Our bounds expose a sharp contrast between the two-party setting and the simpler client-server setting (where privacy guarantees are one-sided). In addition, those bounds demonstrate a dramatic gap between the accuracy that can be obtained by differentially private data analysis versus the accuracy obtainable when privacy is relaxed to a computational variant of differential privacy. The first proof technique we develop demonstrates a connection between differential privacy and deterministic extraction from Santha-Vazirani sources. A second connection we expose indicates that the ability to approximate a function by a low-error differentially private protocol is strongly related to the ability to approximate it by a low communication protocol. (The connection goes in both directions).

FOCS2010.pdf ECCC2011.pdf
Guruswami, Venkatesan, Christopher Umans, and Salil Vadhan. “Unbalanced expanders and randomness extractors from Parvaresh–Vardy codes.” Journal of the ACM 56, no. 4 (2009): 1–34. Publisher's VersionAbstract

Version History: Preliminary versions of this article appeared as Technical Report TR06-134 in Electronic Colloquium on Computational Complexity, 2006, and in Proceedings of the 22nd Annual IEEE Conference on Computional Complexity (CCC '07), pp. 96–108. Preliminary version recipient of Best Paper Award at CCC '07.

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma et al. [2007] required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy [2005].

Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. [2003] and improving upon it when the error parameter is small (e.g., 1/poly(n)).

JACM2009.pdf CCC2007.pdf ECCC2006.pdf ECCC - Rev. 2008.pdf
Haitner, Iftach, Minh Nguyen, Shien Jin Ong, Omer Reingold, and Salil Vadhan. “Statistically hiding commitments and statistical zero-knowledge arguments from any one-way function.” SIAM Journal on Computing 39, no. 3 (2009): 1153-1218. Publisher's VersionAbstract

Version HistorySpecial Issue on STOC ‘07. Merge of papers from FOCS ‘06 and STOC ‘07. Received SIAM Outstanding Paper Prize 2011.

We give a construction of statistically hiding commitment schemes (those in which the hiding property holds against even computationally unbounded adversaries) under the minimal complexity assumption that one-way functions exist. Consequently, one-way functions suffice to give statistical zero-knowledge arguments for any NP statement (whereby even a computationally unbounded adversarial verifier learns nothing other than the fact that the assertion being proven is true, and no polynomial-time adversarial prover can convince the verifier of a false statement). These results resolve an open question posed by Naor et al. [J. Cryptology, 11 (1998), pp. 87–108].

Ong, Shien Jin, David Parkes, Alon Rosen, and Salil Vadhan. “Fairness with an honest minority and a rational majority.” In O. Reingold, editor, Proceedings of the Fourth Theory of Cryptography Conference (TCC ‘09), Lecture Notes in Computer Science, 5444:36-53. Springer-Verlag, 2009. Publisher's VersionAbstract

Version HistoryPreliminary version posted as Cryptology ePrint Archive Report 2008/097, March 2008.

We provide a simple protocol for secret reconstruction in any threshold secret sharing scheme, and prove that it is fair when executed with many rational parties together with a small minority of honest parties. That is, all parties will learn the secret with high probability when the honest parties follow the protocol and the rational parties act in their own self-interest (as captured by a set-Nash analogue of trembling hand perfect equilibrium). The protocol only requires a standard (synchronous) broadcast channel, tolerates both early stopping and incorrectly computed messages, and only requires 2 rounds of communication.

Previous protocols for this problem in the cryptographic or economic models have either required an honest majority, used strong communication channels that enable simultaneous exchange of information, or settled for approximate notions of security/equilibria. They all also required a nonconstant number of rounds of communication.

TCC2009.pdf ePrint2008.pdf
Dodis, Yevgeniy, Salil Vadhan, and Daniel Wichs. “Proofs of retrievability via hardness amplification.” In O. Reingold, editor, Proceedings of the Fourth Theory of Cryptography Conference (TCC ‘09), Lecture Notes in Computer Science, 5444:109-127. Springer-Verlag, 2009. Publisher's VersionAbstract

Version History: Originally presented at Theory of Cryptography Conference (TCC) 2009. Full version published in Cryptology ePrint Archive (attached as ePrint2009).

Proofs of Retrievability (PoR), introduced by Juels and Kaliski [JK07], allow the client to store a file F on an untrusted server, and later run an efficient audit protocol in which the server proves that it (still) possesses the client’s data. Constructions of PoR schemes attempt to minimize the client and server storage, the communication complexity of an audit, and even the number of file-blocks accessed by the server during the audit. In this work, we identify several different variants of the problem (such as bounded-use vs. unbounded-use, knowledge-soundness vs. information-soundness), and giving nearly optimal PoR schemes for each of these variants. Our constructions either improve (and generalize) the prior PoR constructions, or give the first known PoR schemes with the required properties. In particular, we

  • Formally prove the security of an (optimized) variant of the bounded-use scheme of Juels and Kaliski [JK07], without making any simplifying assumptions on the behavior of the adversary.
  • Build the first unbounded-use PoR scheme where the communication complexity is linear in the security parameter and which does not rely on Random Oracles, resolving an open question of Shacham and Waters [SW08].
  • Build the first bounded-use scheme with information-theoretic security.

The main insight of our work comes from a simple connection between PoR schemes and the notion of hardness amplification, extensively studied in complexity theory. In particular, our im- provements come from first abstracting a purely information-theoretic notion of PoR codes, and then building nearly optimal PoR codes using state-of-the-art tools from coding and complexity theory.

ePrint2009.pdf TCC2009.pdf
Dwork, Cynthia, Moni Naor, Omer Reingold, Guy Rothblum, and Salil Vadhan. “On the complexity of differentially private data release: Efficient algorithms and hardness results.” In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC ‘09), 381-390. ACM, 2009. Publisher's VersionAbstract
We consider private data analysis in the setting in which a trusted and trustworthy curator, having obtained a large data set containing private information, releases to the public a "sanitization" of the data set that simultaneously protects the privacy of the individual contributors of data and offers utility to the data analyst. The sanitization may be in the form of an arbitrary data structure, accompanied by a computational procedure for determining approximate answers to queries on the original data set, or it may be a "synthetic data set" consisting of data items drawn from the same universe as items in the original data set; queries are carried out as if the synthetic data set were the actual input. In either case the process is non-interactive; once the sanitization has been released the original data and the curator play no further role. For the task of sanitizing with a synthetic dataset output, we map the boundary between computational feasibility and infeasibility with respect to a variety of utility measures. For the (potentially easier) task of sanitizing with unrestricted output format, we show a tight qualitative and quantitative connection between hardness of sanitizing and the existence of traitor tracing schemes.
Haitner, Iftach, Omer Reingold, Salil Vadhan, and Hoeteck Wee. “Inaccessible entropy.” In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC ‘09), 611-620. ACM, 2009. Publisher's VersionAbstract

We put forth a new computational notion of entropy, which measures the (in)feasibility of sampling high entropy strings that are consistent with a given protocol. Specifically, we say that the \(i\)’th round of a protocol \((\mathsf{A,B})\) has accessible entropy at most \(k\), if no polynomial-time strategy \(\mathsf{A}^*\) can generate messages for \(\mathsf{A}\) such that the entropy of its message in the \(i\)’th round has entropy greater than \(k\) when conditioned both on prior messages of the protocol and on prior coin tosses of \(\mathsf{A}^*\). We say that the protocol has inaccessible entropy if the total accessible entropy (summed over the rounds) is noticeably smaller than the real entropy of \(\mathsf{A}\)’s messages, conditioned only on prior messages (but not the coin tosses of \(\mathsf{A}\)). As applications of this notion, we

  • Give a much simpler and more efficient construction of statistically hiding commitment schemes from arbitrary one- way functions.

  • Prove that constant-round statistically hiding commitments are necessary for constructing constant-round zero-knowledge proof systems for NP that remain secure under parallel composition (assuming the existence of one-way functions).

Trevisan, Luca, Madhur Tulsiani, and Salil Vadhan. “Regularity, boosting, and efficiently simulating every high-entropy distribution.” In Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC ‘09), 126-136. IEEE, 2009. Publisher's VersionAbstract

Version HistoryPreliminary version posted as ECCC TR08-103.

We show that every bounded function \({g} \) : \(\{0,1\}^n \to [0,1]\) admits an efficiently computable "simulator" function \({h} \) : \(\{0,1\}^n \to [0,1]\) such that every fixed polynomial size circuit has approximately the same correlation with \({g}\) as with \({h}\). If g describes (up to scaling) a high min-entropy distribution \(D\), then \({h}\) can be used to efficiently sample a distribution \(D'\) of the same min-entropy that is indistinguishable from \(D\) by circuits of fixed polynomial size. We state and prove our result in a more abstract setting, in which we allow arbitrary finite domains instead of \(\{0,1\}^n\), and arbitrary families of distinguishers, instead of fixed polynomial size circuits. Our result implies (a) the weak Szemeredi regularity Lemma of Frieze and Kannan (b) a constructive version of the dense model theorem of Green, Tao and Ziegler with better quantitative parameters (polynomial rather than exponential in the distinguishing probability), and (c) the Impagliazzo hardcore set Lemma. It appears to be the general result underlying the known connections between "regularity" results in graph theory, "decomposition" results in additive combinatorics, and the hardcore Lemma in complexity theory. We present two proofs of our result, one in the spirit of Nisan's proof of the hardcore Lemma via duality of linear programming, and one similar to Impagliazzo's "boosting" proof. A third proof by iterative partitioning, which gives the complexity of the sampler to be exponential in the distinguishing probability, is also implicit in the Green-Tao-Ziegler proofs of the dense model theorem.

CCC2009.pdf ECCC2008.pdf
Mironov, Ilya, Omkant Pandey, Omer Reingold, and Salil Vadhan. “Computational differential privacy.” In S. Halevi, editor, Advances in Cryptology—CRYPTO ‘09, Lecture Notes in Computer Science, 5677:126-142. Springer-Verlag, 2009. Publisher's VersionAbstract

The definition of differential privacy has recently emerged as a leading standard of privacy guarantees for algorithms on statistical databases. We offer several relaxations of the definition which require privacy guarantees to hold only against efficient—i.e., computationally-bounded—adversaries. We establish various relationships among these notions, and in doing so, we observe their close connection with the theory of pseudodense sets by Reingold et al.[1]. We extend the dense model theorem of Reingold et al. to demonstrate equivalence between two definitions (indistinguishability-and simulatability-based) of computational differential privacy.

Our computational analogues of differential privacy seem to allow for more accurate constructions than the standard information-theoretic analogues. In particular, in the context of private approximation of the distance between two vectors, we present a differentially-private protocol for computing the approximation, and contrast it with a substantially more accurate protocol that is only computationally differentially private.

Lovett, Shachar, Omer Reingold, Luca Trevisan, and Salil Vadhan. “Pseudorandom bit generators that fool modular sums.” In Proceedings of the 13th International Workshop on Randomization and Computation (RANDOM ‘09), Lecture Notes in Computer Science, 5687:615-630. Springer-Verlag, 2009. Publisher's VersionAbstract

We consider the following problem: for given \(n, M,\) produce a sequence \(X_1, X_2, . . . , X_n\) of bits that fools every linear test modulo \(M\). We present two constructions of generators for such sequences. For every constant prime power \(M\), the first construction has seed length \(O_M(\log(n/\epsilon))\), which is optimal up to the hidden constant. (A similar construction was independently discovered by Meka and Zuckerman [MZ]). The second construction works for every \(M, n,\) and has seed length \(O(\log n + \log(M/\epsilon) \log( M \log(1/\epsilon)))\).

The problem we study is a generalization of the problem of constructing small bias distributions [NN], which are solutions to the \(M=2\) case. We note that even for the case \(M=3\) the best previously known con- structions were generators fooling general bounded-space computations, and required \(O(\log^2 n)\) seed length.

For our first construction, we show how to employ recently constructed generators for sequences of elements of \(\mathbb{Z}_M\) that fool small-degree polynomials modulo \(M\). The most interesting technical component of our second construction is a variant of the derandomized graph squaring operation of [RV]. Our generalization handles a product of two distinct graphs with distinct bounds on their expansion. This is then used to produce pseudorandom walks where each step is taken on a different regular directed graph (rather than pseudorandom walks on a single regular directed graph as in [RTV, RV]).

Sanghvi, Saurabh, and Salil Vadhan. “The round complexity of two-party random selection.” SIAM Journal on Computing: Special Issue on STOC '05 38, no. 2 (2008): 523-550. Publisher's VersionAbstract

Version History. Preliminary versions of this work appeared in the first author's undergraduate thesis and in the conference paper (STOC '05).

We study the round complexity of two-party protocols for generating a random \(n\)-bit string such that the output is guaranteed to have bounded “bias,” even if one of the two parties deviates from the protocol (possibly using unlimited computational resources). Specifically, we require that the output’s statistical difference from the uniform distribution on \(\{0, 1\}^n\) is bounded by a constant less than 1. We present a protocol for the above problem that has \(2\log^* n + O(1)\) rounds, improving a previous 2\(n\)-round protocol of Goldreich, Goldwasser, and Linial (FOCS ’91). Like the GGL Protocol, our protocol actually provides a stronger guarantee, ensuring that the output lands in any set \(T ⊆ \{0, 1\}^n\) of density \(μ\) with probability at most \(O( \sqrt{μ + δ})\), where \(δ\) may be an arbitrarily small constant. We then prove a nearly matching lower bound, showing that any protocol guaranteeing bounded statistical difference requires at least \(\log^* n−\log^* \log^* n−O(1)\) rounds. We also prove several results for the case when the output’s bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set \(T ⊆ \{0, 1\}^n\)

Chailloux, André, Dragos Florin Ciocan, Iordanis Kerenidis, and Salil Vadhan. “Interactive and noninteractive zero knowledge are equivalent in the help model.” In Proceedings of the Third Theory of Cryptography Conference (TCC '08), 4948:501-534. Springer-Verlag, Lecture Notes in Computer Science, 2008. Publisher's VersionAbstract

Version History: 

  • Preliminary versions of this work previously appeared on the Cryptology ePrint Archive and in the second author’s undergraduate thesis.
  • Chailloux, A., Kerenidis, I.: The role of help in classical and quantum zero-knowledge. Cryptology ePrint Archive, Report 2007/421 (2007),
  • Ciocan, D.F., Vadhan, S.: Interactive and noninteractive zero knowledge coincide in the help model. Cryptology ePrint Archive, Report 2007/389 (2007),
  • Ciocan, D.: Constructions and characterizations of non-interactive zero-knowledge. Undergradute thesis, Harvard University (2007) 

We show that interactive and noninteractive zero-knowledge are equivalent in the ‘help model’ of Ben-Or and Gutfreund (J. Cryptology, 2003). In this model, the shared reference string is generated by a probabilistic polynomial-time dealer who is given access to the statement to be proven. Our results do not rely on any unproven complexity assumptions and hold for statistical zero knowledge, for computational zero knowledge restricted to AM, and for quantum zero knowledge when the help is a pure quantum state.

Ong, Shien Jin, and Salil Vadhan. “An equivalence between zero knowledge and commitments.” In R. Canetti, editor, Proceedings of the Third Theory of Cryptography Conference (TCC ‘08), 4948:482-500. Springer Verlag, Lecture Notes in Computer Science, 2008. Publisher's VersionAbstract

We show that a language in NP has a zero-knowledge protocol if and only if the language has an “instance-dependent” commitment scheme. An instance-dependent commitment schemes for a given language is a commitment scheme that can depend on an instance of the language, and where the hiding and binding properties are required to hold only on the yes and no instances of the language, respectively.

The novel direction is the only if direction. Thus, we confirm the widely held belief that commitments are not only sufficient for zero knowledge protocols, but necessary as well. Previous results of this type either held only for restricted types of protocols or languages, or used nonstandard relaxations of (instance-dependent) commitment schemes.

Gutfreund, Dan, and Salil Vadhan. “Limitations on hardness vs. randomness under uniform reductions.” In Proceedings of the 12th International Workshop on Randomization and Computation (RANDOM ‘08), Lecture Notes in Computer Science, 5171:469-482. Springer-Verlag, 2008. Publisher's VersionAbstract

We consider (uniform) reductions from computing a function \({f}\) to the task of distinguishing the output of some pseudorandom generator \({G}\) from uniform. Impagliazzo and Wigderson [10] and Trevisan and Vadhan [24] exhibited such reductions for every function \({f}\) in PSPACE. Moreover, their reductions are “black box,” showing how to use any distinguisher \({T}\), given as oracle, in order to compute \({f}\) (regardless of the complexity of \({T}\) ). The reductions are also adaptive, but with the restriction that queries of the same length do not occur in different levels of adaptivity. Impagliazzo and Wigderson [10] also exhibited such reductions for every function \({f}\) in EXP, but those reductions are not black-box, because they only work when the oracle \({T}\) is computable by small circuits.

Our main results are that:

– Nonadaptive black-box reductions as above can only exist for functions \({f}\) in BPPNP (and thus are unlikely to exist for all of PSPACE).

– Adaptive black-box reductions, with the same restriction on the adaptivity as above, can only exist for functions \({f}\) in PSPACE (and thus are unlikely to exist for all of EXP).

Beyond shedding light on proof techniques in the area of hardness vs. randomness, our results (together with [10,24]) can be viewed in a more general context as identifying techniques that overcome limitations of black-box reductions, which may be useful elsewhere in complexity theory (and the foundations of cryptography).

Bogdanov, Andrej, Elchanan Mossel, and Salil Vadhan. “The complexity of distinguishing Markov random fields.” In Proceedings of the 12th International Workshop on Randomization and Computation (RANDOM ‘08), Lecture Notes in Computer Science, 5171:331-342. Springer-Verlag, 2008. Publisher's VersionAbstract

Markov random fields are often used to model high dimensional distributions in a number of applied areas. A number of recent papers have studied the problem of reconstructing a dependency graph of bounded degree from independent samples from the Markov random field. These results require observing samples of the distribution at all nodes of the graph. It was heuristically recognized that the problem of reconstructing the model where there are hidden variables (some of the variables are not observed) is much harder.

Here we prove that the problem of reconstructing bounded-degree models with hidden nodes is hard. Specifically, we show that unless NP = RP,

  • It is impossible to decide in randomized polynomial time if two mod- els generate distributions whose statistical distance is at most 1/3 or at least 2/3.
  • Given two generating models whose statistical distance is promised to be at least 1/3, and oracle access to independent samples from one of the models, it is impossible to decide in randomized polynomial time which of the two samples is consistent with the model.

The second problem remains hard even if the samples are generated efficiently, albeit under a stronger assumption.

Reingold, Omer, Luca Trevisan, Madhur Tulsiani, and Salil Vadhan. “Dense subsets of pseudorandom sets.” In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS ‘08), 76-85. IEEE, 2008. Publisher's VersionAbstract
A theorem of Green, Tao, and Ziegler can be stated (roughly) as follows: if R is a pseudorandom set, and D is a dense subset of R, then D may be modeled by a set M that is dense in the entire domain such that D and M are indistinguishable. (The precise statement refers to "measures" or distributions rather than sets.) The proof of this theorem is very general, and it applies to notions of pseudo-randomness and indistinguishability defined in terms of any family of distinguishers with some mild closure properties. The proof proceeds via iterative partitioning and an energy increment argument, in the spirit of the proof of the weak Szemeredi regularity lemma. The "reduction" involved in the proof has exponential complexity in the distinguishing probability. We present a new proof inspired by Nisan's proof of Impagliazzo's hardcore set theorem. The reduction in our proof has polynomial complexity in the distinguishing probability and provides a new characterization of the notion of "pseudoentropy" of a distribution. A proof similar to ours has also been independently discovered by Gowers [2]. We also follow the connection between the two theorems and obtain a new proof of Impagliazzo's hardcore set theorem via iterative partitioning and energy increment. While our reduction has exponential complexity in some parameters, it has the advantage that the hardcore set is efficiently recognizable.
Ong, Shien Jin, and Salil Vadhan. “Zero knowledge and soundness are symmetric.” In Advances in Cryptology–EUROCRYPT '07, 4515:187-209. Barcelona, Spain: Springer Verlag, Lecture Notes in Computer Science, M. Naor, ed. 2007. Publisher's VersionAbstract

Version History: Recipient of Best Paper Award. Preliminary version posted on ECCC as TR06-139, November 2006.

We give a complexity-theoretic characterization of the class of problems in NP having zero-knowledge argument systems. This characterization is symmetric in its treatment of the zero knowledge and the soundness conditions, and thus we deduce that the class of problems in NP \(\bigcap\) coNP having zero-knowledge arguments is closed under complement. Furthermore, we show that a problem in NP has a statistical zero-knowledge argument \(\)system if and only if its complement has a computational zero-knowledge proof system. What is novel about these results is that they are unconditional, i.e., do not rely on unproven complexity assumptions such as the existence of one-way functions.

Our characterization of zero-knowledge arguments also enables us to prove a variety of other unconditional results about the class of problems in NP having zero-knowledge arguments, such as equivalences between honest-verifier and malicious-verifier zero knowledge, private coins and public coins, inefficient provers and efficient provers, and non-black-box simulation and black-box simulation. Previously, such results were only known unconditionally for zero-knowledge proof systems, or under the assumption that one-way functions exist for zero-knowledge argument systems.

EUROCRYPT 2007.pdf ECCC 2006.pdf
Ron, Dana, Amir Rosenfeld, and Salil Vadhan. “The hardness of the expected decision depth problem.” Information Processing Letters 101, no. 3 (2007): 112-118. Publisher's VersionAbstract

Given a function \(f\) over \(n\) binary variables, and an ordering of the \(n\) variables, we consider the Expected Decision Depth problem. Namely, what is the expected number of bits that need to be observed until the value of the function is determined, when bits of the input are observed according to the given order. Our main finding is that this problem is (essentially) #P-complete. Moreover, the hardness holds even when the function f is represented as a decision tree.