# Proof Systems

Chen, Yi-Hsiu, Mika Goos, Salil P. Vadhan, and Jiapeng Zhang. “A tight lower bound for entropy flattening.” In 33rd Computational Complexity Conference (CCC 2018), 102:23:21-23:28. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik: Leibniz International Proceedings in Informatics (LIPIcs), 2018. Publisher's VersionAbstract

Version History: Preliminary version posted as ECCC TR18-119.

We study entropy flattening: Given a circuit $$C_X$$ implicitly describing an n-bit source $$X$$ (namely, $$X$$ is the output of $$C_X$$  on a uniform random input), construct another circuit $$C_Y$$ describing a source $$Y$$ such that (1) source $$Y$$ is nearly flat (uniform on its support), and (2) the Shannon entropy of $$Y$$ is monotonically related to that of $$X$$. The standard solution is to have $$C_Y$$ evaluate $$C_X$$ altogether $$\Theta(n^2)$$ times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit $$C_Y$$ for entropy flattening that repeatedly queries $$C_X$$ as an oracle requires $$\Omega(n^2)$$queries.

Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [12, 22, 13, 6, 11, 10, 7, 24]. It is also used in reductions between problems complete for statistical zero-knowledge [19, 23, 4, 25]. The $$\Theta(n^2)$$ query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.

Mahmoody, Mohammad, Tal Moran, and Salil Vadhan. “Publicly verifiable proofs of sequential work.” In Innovations in Theoretical Computer Science (ITCS ‘13), 373-388. ACM, 2013. Publisher's VersionAbstract

Version HistoryPreliminary version posted as Cryptology ePrint Archive Report 2011/553, under title “Non-Interactive Time-Stamping and Proofs of Work in the Random Oracle Model”.

We 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” $$\mathcal{P} \overset{}\gets \mathbf{D}_n$$, where $$n$$ is the security parameter and $$\mathbf{D}_n$$ 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 $$\Omega(N)$$ sequential evaluations of the hash function after receiving $$\mathcal{P}$$. Thus, valid solutions constitute a “proof” that $$\Omega(N)$$ parallel time elapsed since $$\mathcal{P}$$ was received. Solutions can be publicly and efficiently verified in time $$\mathrm{poly}(n) \cdot \mathrm{polylog}(N)$$. Applications of these “time-lock puzzles” include noninteractive timestamping of documents (when the distribution over the possible documents corresponds to the puzzle distribution $$\mathbf{D}_n$$) 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 negative result 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.

Rothblum, Guy N., Salil Vadhan, and Avi Wigderson. “Interactive proofs of proximity: delegating computation in sublinear time.” In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC ‘13), 793-802. New York, NY: ACM, 2013. Publisher's VersionAbstract

We 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 $$\mathcal{NC}$$ have Interactive Proofs of Proximity with roughly $$\sqrt{n}$$ query and communication and complexities, and $$\mathrm{polylog} (n)$$ 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 $$\mathcal{NC}^1$$, 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 $$\tilde{\Omega}(\sqrt{n})$$.

• 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.

Dodis, Yevgeniy, Thomas Ristenpart, and Salil Vadhan. “Randomness condensers for efficiently samplable, seed-dependent sources.” In Ronald Cramer, editor, Proceedings of the 9th IACR Theory of Cryptography Conference (TCC ‘12), Lecture Notes on Computer Science, 7194:618-635. Springer-Verlag, 2012. Publisher's VersionAbstract

We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the condenser. That is, we seek functions $$\mathsf{Cond} : \{0,1\}^n \times \{0,1\}^d \to \{0,1\}^m$$such that if we choose a random seed $$S \gets \{0,1\}^d$$, and a source $$X = \mathcal{A}(S)$$ is generated by a randomized circuit $$\mathcal{A}$$ of size $$t$$ such that $$X$$ has min- entropy at least $$k$$ given $$S$$, then $$\mathsf{Cond}(X ; S)$$ should have min-entropy at least some $$k'$$ given $$S$$. The distinction from the standard notion of randomness 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 $$\mathcal{A}$$. Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy $$k' = m – \mathcal{O}(\log t)$$, i.e. logarithmic entropy deficiency.

• Randomness condensers suffice for key derivation in many cryptographic applications: when an adversary has negligible success probability (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.

Dvir, Zeev, Dan Gutfreund, Guy Rothblum, and Salil Vadhan. “On approximating the entropy of polynomial mappings.” In Proceedings of the Second Symposium on Innovations in Computer Science (ICS 2011), 460-475. Tsinghua University Press, 2011. Publisher's VersionAbstract

Version HistoryFull version posted as ECCC TR10-60.

We investigate the complexity of the following computational problem:

Polynomial Entropy Approximation (PEA): Given a low-degree polynomial mapping $$p : \mathbb{F}^n → \mathbb{F}^m$$, where F is a finite field, approximate the output entropy $$H(p(U_n))$$, where $$U_n$$ is the uniform distribution on $$\mathbb{F}^n$$ and $$H$$ may be any of several entropy measures.

We show:

• Approximating the Shannon entropy of degree 3 polynomials $$p : \mathbb{F}_2^n \to \mathbb{F}^m_2$$ over $$\mathbb{F}_2$$ to within an additive constant (or even $$n^.9$$) is complete for $$\mathbf{SZKP_L}$$, the class of problems having statistical zero-knowledge proofs where the honest verifier and its simulator are computable in logarithmic space. ($$\mathbf{SZKP_L}$$contains most of the natural problems known to be in the full class $$\mathbf{SZKP}$$.)

• For prime fields $$\mathbb{F} \neq \mathbb{F}_2$$ and homogeneous quadratic polynomials $$p : \mathbb{F}^n \to \mathbb{F}^m$$, there is a probabilistic polynomial-time algorithm that distinguishes the case that $$p(U_n)$$) has entropy smaller than k from the case that $$p(U_n))$$ has min-entropy (or even Renyi entropy) greater than $$(2 + o(1))k$$.

• For degree d polynomials $$p : \mathbb{F}^n_2 \to \mathbb{F}^m_2$$ , there is a polynomial-time algorithm that distinguishes the case that $$p(U_n)$$ has max-entropy smaller than $$k$$ (where the max-entropy of a random variable is the logarithm of its support size) from the case that $$p(U_n)$$ has max-entropy at least $$(1 + o(1)) \cdot k^d$$ (for fixed $$d$$ and large $$k$$).

Rothblum, Guy, and Salil Vadhan. “Are PCPs inherent in efficient arguments?Computational Complexity 19, no. 2 (2010): 265-304. Publisher's VersionAbstract

Version HistorySpecial Issue on CCC '09.

Starting with Kilian (STOC ‘92), several works have shown how to use probabilistically checkable proofs (PCPs) and cryptographic primitives such as collision-resistant hashing to construct very efficient argument systems (a.k.a. computationally sound proofs), for example with polylogarithmic communication complexity. Ishai et al. (CCC ‘07) raised the question of whether PCPs are inherent in efficient arguments, and if so, to what extent. We give evidence that they are, by showing how to convert any argument system whose soundness is reducible to the security of some cryptographic primitive into a PCP system whose efficiency is related to that of the argument system and the reduction (under certain complexity assumptions).

Birrell, Eleanor, and Salil Vadhan. “Composition of zero-knowledge proofs with efficient provers.” In Daniele Micciancio, editor, Proceedings of the 7th IACR Theory of Cryptography Conference (TCC ‘10), Lecture Notes on Computer Science, 5978:572-587. Springer-Verlag, 2010. Publisher's VersionAbstract

We revisit the composability of different forms of zero- knowledge proofs when the honest prover strategy is restricted to be polynomial time (given an appropriate auxiliary input). Our results are:

1. When restricted to efficient provers, the original Goldwasser–Micali–Rackoff (GMR) definition of zero knowledge (STOC ‘85), here called plain zero knowledge, is closed under a constant number of sequential compositions (on the same input). This contrasts with the case of unbounded provers, where Goldreich and Krawczyk (ICALP ‘90, SICOMP ‘96) exhibited a protocol that is zero knowledge under the GMR definition, but for which the sequential composition of 2 copies is not zero knowledge.

2. If we relax the GMR definition to only require that the simulation is indistinguishable from the verifier’s view by uniform polynomial-time distinguishers, with no auxiliary input beyond the statement being proven, then again zero knowledge is not closed under sequential composition of 2 copies.

3. We show that auxiliary-input zero knowledge with efficient provers is not closed under parallel composition of 2 copies under the assumption that there is a secure key agreement protocol (in which it is easy to recognize valid transcripts). Feige and Shamir (STOC ‘90) gave similar results under the seemingly incomparable assumptions that (a) the discrete logarithm problem is hard, or (b) $$\mathcal{UP}\nsubseteq\mathcal{BPP}$$ and one-way functions exist.
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).

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].

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.

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.

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.

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-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)