# New Complexity-Theoretic Techniques in Cryptography (NSF CNS-0430336)

2009
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].

2008
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), 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.

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.

2007
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.

Canetti, Ran, Ron Rivest, Madhu Sudan, Luca Trevisan, Salil Vadhan, and Hoeteck Wee. “Amplifying collision-resistance: A complexity-theoretic treatment.” In A. Menezes, editor, Advances in Cryptology (CRYPTO '07), 4622:264-283. Lecture Notes in Computer Science, Springer-Verlag, 2007. Publisher's VersionAbstract

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.

2006
Vadhan, Salil. “An unconditional study of computational zero knowledge.” SIAM Journal on Computing: Special Issue on Randomness and Complexity 36, no. 4 (2006): 1160-1214. Publisher's VersionAbstract

Version History: Extended abstract in FOCS '04.

We prove a number of general theorems about $$\mathbf{ZK}$$, the class of problems possessing (computational) zero-knowledge proofs. Our results are unconditional, in contrast to most previous works on $$\mathbf{ZK}$$, which rely on the assumption that one-way functions exist. We establish several new characterizations of $$\mathbf{ZK}$$ and use these characterizations to prove results such as the following:

1. Honest-verifier $$\mathbf{ZK}$$ equals general $$\mathbf{ZK}$$.
2. Public-coin $$\mathbf{ZK}$$ equals private-coin $$\mathbf{ZK}$$.
3. $$\mathbf{ZK}$$ is closed under union.
4. $$\mathbf{ZK}$$ with imperfect completeness equals $$\mathbf{ZK}$$ with perfect completeness.
5. Any problem in $$\mathbf{ZK}$$ $$\cap$$ $$\mathbf{NP}$$ can be proven in computational zero knowledge by a $$\mathbf{BPP^{NP}}$$prover.
6. $$\mathbf{ZK}$$ with black-box simulators equals $$\mathbf{ZK}$$ with general, non–black-box simulators.

The above equalities refer to the resulting class of problems (and do not necessarily preserve other efficiency measures such as round complexity). Our approach is to combine the conditional techniques previously used in the study of $$\mathbf{ZK}$$ with the unconditional techniques developed in the study of $$\mathbf{SZK}$$, the class of problems possessing statistical zero-knowledge proofs. To enable this combination, we prove that every problem in $$\mathbf{ZK}$$ can be decomposed into a problem in $$\mathbf{SZK}$$ together with a set of instances from which a one-way function can be constructed.

Micciancio, Daniele, Shien Jin Ong, Amit Sahai, and Salil Vadhan. “Concurrent zero knowledge without complexity assumptions.” In S. Halevi and T. Rabin, eds., Proceedings of the Third Theory of Cryptography Conference (TCC '06), 3876:1-20. New York, NY, USA: Springer Verlag, Lecture Notes in Computer Science, 2006. Publisher's VersionAbstract

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).

Nguyen, Minh, and Salil Vadhan. “Zero knowledge with efficient provers.” In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC ‘06), 287-295. ACM, 2006. Publisher's VersionAbstract
We prove that every problem in NP that has a zero-knowledge proof also has a zero-knowledge proof where the prover can be implemented in probabilistic polynomial time given an NP witness. Moreover, if the original proof system is statistical zero knowledge, so is the resulting efficient-prover proof system. An equivalence of zero knowledge and efficient-prover zero knowledge was previously known only under the assumption that one-way functions exist (whereas our result is unconditional), and no such equivalence was known for statistical zero knowledge. Our results allow us to translate the many general results and characterizations known for zero knowledge with inefficient provers to zero knowledge with efficient provers.
Gradwohl, Ronen, Salil Vadhan, and David Zuckerman. “Random selection with an adversarial majority.” In Advances in Cryptology—CRYPTO ‘06, C. Dwork, ed. 4117:409–426. Springer Verlag, Lecture Notes in Computer Science, 2006. Publisher's VersionAbstract

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.

Nguyen, Minh-Huyen, Shien Jin Ong, and Salil Vadhan. “Statistical zero-knowledge arguments for NP from any one-way function.” In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS ‘06), 3-13. IEEE, 2006. Publisher's VersionAbstract

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)