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.

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]).

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), http://eprint.iacr.org/
• Ciocan, D.F., Vadhan, S.: Interactive and noninteractive zero knowledge coincide in the help model. Cryptology ePrint Archive, Report 2007/389 (2007), http://eprint.iacr.org/
• 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.

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.

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).

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.

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.

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.

We initiate a complexity-theoretic treatment of hardness amplification for collision-resistant hash functions, namely the transformation of weakly collision-resistant hash functions into strongly collision-resistant ones in the standard model of computation. We measure the level of collision resistance by the maximum probability, over the choice of the key, for which an efficient adversary can find a collision. The goal is to obtain constructions with short output, short keys, small loss in adversarial complexity tolerated, and a good trade-off between compression ratio and computational complexity. We provide an analysis of several simple constructions, and show that many of the parameters achieved by our constructions are almost optimal in some sense.

Version History. Full version available at https://eccc.weizmann.ac.il//eccc-reports/2005/TR05-093/ (Attached as ECCC2005).

We provide unconditional constructions of concurrent statistical zero-knowledge proofs for a variety of non-trivial problems (not known to have probabilistic polynomial-time algorithms). The problems include Graph Isomorphism, Graph Nonisomorphism, Quadratic Residuosity, Quadratic Nonresiduosity, a restricted version of Statistical Difference, and approximate versions of the (\(\mathsf{coNP}\) forms of the) Shortest Vector Problem and Closest Vector Problem in lattices. For some of the problems, such as Graph Isomorphism and Quadratic Residuosity, the proof systems have provers that can be implemented in polynomial time (given an \(\mathsf{NP}\) witness) and have \(\tilde{O}(\log n)\) rounds, which is known to be essentially optimal for black-box simulation. To the best of our knowledge, these are the first constructions of concurrent zero-knowledge proofs in the plain, asynchronous model (i.e., without setup or timing assumptions) that do not require complexity assumptions (such as the existence of one-way functions).

Version History: Merged with STOC '07 paper of Haitner and Reingold. Also available as a journal version. Full version invited to SIAM J. Computing Special Issue on FOCS ‘06

We show that every language in NP has a statistical zero-knowledge argument system under the (minimal) complexity assumption that one-way functions exist. In such protocols, even a computationally unbounded verifier cannot learn anything other than the fact that the assertion being proven is true, whereas a polynomial-time prover cannot convince the verifier to accept a false assertion except with negligible probability. This resolves an open question posed by Naor et al. (1998). Departing from previous works on this problem, we do not construct standard statistically hiding commitments from any one-way function. Instead, we construct a relaxed variant of commitment schemes called "1-out-of-2-binding commitments," recently introduced by Nguyen et al. (2006)

We propose the use of complete problems to unify and extend the study of statistical zero knowledge. To this end, we examine several consequences of our Completeness Theorem and its proof, such as:

• A way to make every (honest-verifier) statistical zero-knowledge proof very communication efficient, with the prover sending only one bit to the verifier (to achieve soundness error 1/2).
• Simpler proofs of many of the previously known results about statistical zero knowledge, such as the Fortnow and Aiello--Håstad upper bounds on the complexity of SZK and Okamoto's result that SZK is closed under complement.
• Strong closure properties of SZK which amount to constructing statistical zero-knowledge proofs for complex assertions built out of simpler assertions already shown to be in SZK.
• New results about the various measures of "knowledge complexity," including a collapse in the hierarchy corresponding to knowledge complexity in the "hint" sense.
• Algorithms for manipulating the statistical difference between efficiently samplable distributions, including transformations which "polarize" and "reverse" the statistical relationship between a pair of distributions.

We give several efficient transformations for manipulating the statistical difference (variation distance) between a pair of probability distributions. The effects achieved include increasing the statistical difference, decreasing the statistical difference, "polarizing" the statistical relationship, and "reversing" the statistical relationship. We also show that a boolean formula whose atoms are statements about statistical difference can be transformed into a single statement about statistical difference. All of these transformations can be performed in polynomial time, in the sense that, given circuits which sample from the input distributions, it only takes polynomial time to compute circuits which sample from the output distributions.

By our prior work (see FOCS 97), such transformations for manipulating statistical difference are closely connected to results about SZK, the class of languages possessing statistical zero-knowledge proofs. In particular, some of the transformations given in this paper are equivalent to the closure of SZK under complement and under certain types of Turing reductions. This connection is also discussed briefly in this paper.