Version History: Full 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).

Version History: Special 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].

Version History: Preliminary 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.

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.

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).

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.

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.

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.

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: Full version published in ECCC TR 06-026, February 2006. Updated full version published June 2006.

We consider the problem of random selection, where \(p\) players follow a protocol to jointly select a random element of a universe of size \(n\). However, some of the players may be adversarial and collude to force the output to lie in a small subset of the universe. We describe essentially the first protocols that solve this problem in the presence of a dishonest majority in the full-information model (where the adversary is computationally unbounded and all communication is via non-simultaneous broadcast). Our protocols are nearly optimal in several parameters, including the round complexity (as a function of \(n\)), the randomness complexity, the communication complexity, and the tradeoffs between the fraction of honest players, the probability that the output lies in a small subset of the universe, and the density of this subset.

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.

This report contains a discussion of the difficult problem of key management for multicast communication sessions.  It focuses on two main areas of concern with respect to key management, which are, initializing the multicast group with a common net key and rekeying the multicast group.  A rekey may be necessary upon the compromise of a user or for other reasons (e.g., periodic rekey).  In particular, this report identifies a technique which allows for secure compromise recovery, while also being robust against collusion of excluded users.  This is one important feature of multicast key management which has not been addressed in detail by most other multicast key management proposals [1,2,4].  The benefits of this proposed technique are that it minimizes the number of transmissions required to rekey the multicast group and it imposes minimal storage requirements on the multicast group.

Our transformation also applies to public-coin (aka Arthur-Merlin) computational zero-knowledge proofs: We transform any Arthur-Merlin proof system which is computational zero-knowledge with respect to the honest-verifier, into an Arthur-Merlin proof system which is computational zero-knowledge with respect to any probabilistic polynomial-time verifier.

A crucial ingredient in our analysis is a new lemma regarding 2-universal hashing functions.