# Publications

2009
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]).

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.

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.
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.

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.

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
Ben-Sasson, Eli, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil Vadhan. “Robust PCPs of proximity, shorter PCPs, and applications to coding.” SIAM Journal on Computing: Special Issue on Randomness and Complexity 36, no. 4 (2006): 889-974. Publisher's VersionAbstract

Version History. Extended abstract in STOC '04.

We continue the study of the trade-off between the length of probabilistically checkable proofs (PCPs) and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size $$n$$):

1. We present PCPs of length $$\exp(o(\log \log n)^2) ·n$$ that can be verified by making $$o(\log \log n)$$ Boolean queries.
2. For every $$ε > 0$$, we present PCPs of length $$\exp(\log^ε n) · n$$ that can be verified by making a constant number of Boolean queries.

In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi polylogarithmic in the first case (of query complexity $$o(\log \log n)$$), and is $$2^{(\log n)^ε}$$, for any $$ε > 0$$, in the second case (of constant query complexity). Our techniques include the introduction of a new variant of PCPs that we call “robust PCPs of proximity.” These new PCPs facilitate proof composition, which is a central ingredient in the construction of PCP systems. (A related notion and its composition properties were discovered independently by Dinur and Reingold.) Our main technical contribution is a construction of a “length-efficient” robust PCP of proximity. While the new construction uses many of the standard techniques used in PCP constructions, it does differ from previous constructions in fundamental ways, and in particular does not use the “parallelization” step of Arora et al. [J. ACM, 45 (1998), pp. 501–555]. The alternative approach may be of independent interest. We also obtain analogous quantitative results for locally testable codes. In addition, we introduce a relaxed notion of locally decodable codes and present such codes mapping $$k$$ information bits to codewords of length $$k^{1+ε}$$ for any $$ε > 0$$.

Healy, Alex, Salil Vadhan, and Emanuele Viola. “Using nondeterminism to amplify hardness.” SIAM Journal on Computing: Special Issue on STOC '04 35, no. 4 (2006): 903-931. Publisher's VersionAbstract

We revisit the problem of hardness amplification in $$\mathcal{NP}$$ as recently studied by O’Donnell [J. Comput. System Sci., 69 (2004), pp. 68–94]. We prove that if $$\mathcal{NP}$$ has a balanced function $$f$$ such that any circuit of size $$s(n)$$ fails to compute $$f$$ on a $$1/\mathrm{poly}(n)$$ fraction of inputs, then $$\mathcal{NP}$$ has a function $$f'$$ such that any circuit of size $$s'(n) = s(\sqrt{n})^{\Omega(1)}$$ fails to compute $$f$$ on a $$1/2 − 1/s' (n)$$ fraction of inputs. In particular,

1. if $$s(n) = n^{\omega(1)}$$, we amplify to hardness $$1/2 - 1/n^{\omega(1)}$$;
2. if $$s(n) = 2^{n^{\Omega(1)}}$$, we amplify to hardness $$1/2 - 1/2^{n^{\Omega(1)}}$$;
3. if $$s(n) = 2^{\Omega(n)}$$, we amplify to hardness $$1/2 - 1/2^{\Omega(\sqrt{n})}$$.

Our results improve those of of O’Donnell, which amplify to$$1/2 - 1/ \sqrt{n}$$. O’Donnell also proved that no construction of a certain general form could amplify beyond $$1/2 - 1/n$$. We bypass this barrier by using both derandomization and nondeterminism in the construction of $$f'$$. We also prove impossibility results demonstrating that both our use of nondeterminism and the hypothesis that $$f$$ is balanced are necessary for “black-box” hardness amplification procedures (such as ours).

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.

Reingold, Omer, Luca Trevisan, and Salil Vadhan. “Pseudorandom walks in regular digraphs and the RL vs. L problem.” In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC ‘06), 457-466, 2006, 457-466. Publisher's VersionAbstract

Version History: Preliminary version as ECCC TR05-22, February 2005 (https://eccc.weizmann.ac.il/report/2005/022/; attached as ECCC2005.pdf).

We revisit the general $$\mathbf{RL}$$ vs. $$\mathbf{L}$$ question, obtaining the following results.

1. Generalizing Reingold’s techniques to directed graphs, we present a deterministic, log-space algorithm that given a regular directed graph G (or, more generally, a digraph with Eulerian connected components) and two vertices s and t, finds a path between s and t if one exists.
2. If we restrict ourselves to directed graphs that are regular and consistently labelled, then we are able to produce pseudorandom walks for such graphs in logarithmic space (this result already found an independent application).
3. We prove that if (2) could be generalized to all regular directed graphs (including ones that are not consistently labelled) then $$\mathbf{L=RL}$$. We do so by exhibiting a new complete promise problem for $$\mathbf{RL}$$, and showing that such a problem can be solved in deterministic logarithmic space given a log-space pseudorandom walk generator for regular directed graphs.
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)

2005
Trevisan, Luca, Salil Vadhan, and David Zuckerman. “Compression of samplable sources.” Computational Complexity: Special Issue on CCC'04 14, no. 3 (2005): 186-227. Publisher's VersionAbstract

We study the compression of polynomially samplable sources. In particular, we give efficient prefix-free compression and decompression algorithms for three classes of such sources (whose support is a subset of $$\{0, 1\}^n$$).

1. We show how to compress sources $$X$$ samplable by logspace machines to expected length $$H(X) + O(1)$$. Our next results concern flat sources whose support is in $$\mathbf{P}$$.
2. If $$H(X) ≤ k = n−O(\log n)$$, we show how to compress to expected length $$k + \mathrm{polylog}(n − k)$$.
3. If the support of $$X$$ is the witness set for a self-reducible $$\mathbf{NP}$$ relation, then we show how to compress to expected length $$H(X)+ 5$$.
Ben-Sasson, Eli, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil Vadhan. “Short PCPs verifiable in polylogarithmic time.” In Proceedings of the 20th Annual IEEE Conference on Computational Complexity (CCC '05), 120-134, 2005, 120-134. Publisher's VersionAbstract
We show that every language in NP has a probabilistically checkable proof of proximity (i.e., proofs asserting that an instance is "close" to a member of the language), where the verifier's running time is polylogarithmic in the input size and the length of the probabilistically checkable proof is only polylogarithmically larger that the length of the classical proof. (Such a verifier can only query polylogarithmically many bits of the input instance and the proof. Thus it needs oracle access to the input as well as the proof, and cannot guarantee that the input is in the language - only that it is close to some string in the language.) If the verifier is restricted further in its query complexity and only allowed q queries, then the proof size blows up by a factor of 2/sup (log n)c/q/ where the constant c depends only on the language (and is independent of q). Our results thus give efficient (in the sense of running time) versions of the shortest known PCPs, due to Ben-Sasson et al. (STOC '04) and Ben-Sasson and Sudan (STOC '05), respectively. The time complexity of the verifier and the size of the proof were the original emphases in the definition of holographic proofs, due to Babai et al. (STOC '91), and our work is the first to return to these emphases since their work. Of technical interest in our proof is a new complete problem for NEXP based on constraint satisfaction problems with very low complexity constraints, and techniques to arithmetize such constraints over fields of small characteristic.