DOCSLIB.ORG

Pseudorandom Generators Without the XOR Lemma

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Pseudorandom Generators Without the XOR Lemma Pseudorandom generators without the XOR Lemma [Extended Abstract] z x Madhu Sudany Luca Trevisan Salil Vadhan Abstract Intro duction Impagliazzo and Wigderson [IW97] have recently shown that if This paper continues the exploration of hardness versus random- O n there exists a decision problem solvable in time and hav- ness trade-offs, that is, results showing that randomized algorithms n can be efficiently simulated deterministically if certain complexity- n P ing circuit complexity (for all but finitely many ) then theoretic assumptions are true. We present two new approaches BPP . This result is a culmination of a series of works showing con- nections between the existence of hard predicates and the existence to proving the recent result of Impagliazzo and Wigderson [IW97] O n of good pseudorandom generators. that, if there is a decision problem computable in time and n n The construction of Impagliazzo and Wigderson goes through having circuit complexity for all but finitely many , then BPP three phases of “hardness amplification” (a multivariate polyno- P . Impagliazzo and Wigderson prove their result by pre- mial encoding, a first derandomized XOR Lemma, and a second senting a “randomness-efficient amplification of hardness” based derandomized XOR Lemma) that are composed with the Nisan– on a derandomized version of Yao’s XOR Lemma. The hardness- Wigderson [NW94] generator. In this paper we present two dif- amplification procedure is then composed with the Nisan–Wigderson ferent approaches to proving the main result of Impagliazzo and (NW) generator [NW94] and this gives the result. The hardness Wigderson. In developing each approach, we introduce new tech- amplification goes through three steps: an encoding using multi- niques and prove new results that could be useful in future improve- variate polynomials (from [BFNW93]), a first derandomized XOR ments and/or applications of hardness-randomness trade-offs. Lemma (from [Imp95]) and a second derandomized XOR Lemma Our first result is that when (a modified version of) the Nisan- (which is the technical contribution of [IW97]). Wigderson generator construction is applied with a “mildly” hard In our first result, we show how to construct a “pseudoentropy predicate, the result is a generator that produces a distribution indis- generator” starting from a predicate with “mild” hardness. Roughly tinguishable from having large min-entropy. An extractor can then speaking, a pseudoentropy generator takes a short random seed be used to produce a distribution computationally indistinguishable as input and outputs a distribution that is indistinguishable from from uniform. This is the first construction of a pseudorandom gen- having high min-entropy. Combining our pseudoentropy generator erator that works with a mildly hard predicate without doing hard- with an extractor, we obtain a pseudorandom generator. Interest- ness amplification. ingly, our pseudoentropy generator is (a modification of) the NW We then show that in the Impagliazzo–Wigderson construction generator itself. Along the way we prove that, when built out of only the first hardness-amplification phase (encoding with multi- a mildly hard predicate, the NW generator outputs a distribution variate polynomial) is necessary, since it already gives the required that is indistinguishable from having high Shannon entropy, a result average-case hardness. We prove this result by (i) establishing a that has not been observed before. The notion of a pseudoentropy connection between the hardness-amplification problem and a list- generator, and the idea that a pseudoentropy generator can be con- decoding problem for error-correcting codes based on multivariate verted into a pseudorandom generator using an extractor, are due polynomials; and (ii) presenting a list-decoding algorithm that im- to H˚astad et al. [HILL98]. Our construction is the first construc- proves and simplifies a previous one by Arora and Sudan [AS97]. tion of a pseudorandom generator that works using a mildly hard predicate and without hardness amplification. The full version of this paper appears as [STV98]. y We then revisit the hardness amplification problem, as consid- Laboratory for Computer Science, 545 Technology Square, MIT, Cambridge, MA 02141. E-mail: [email protected]. ered in [BFNW93, Imp95, IW97], and we show that the first step z Department of Computer Science, Columbia University, 500W 120th St., New alone (encoding with multivariate polynomials) is sufficient to am- York, NY 10027. Email: [email protected]. Work done at MIT. plify hardness to the desired level, so that the derandomized XOR x Laboratory for Computer Science, 545 Technology Square, MIT, Cam- Lemmas are not necessary in this context. Our proof is based on bridge, MA 02141. E-mail: [email protected]. URL: a list-decoding algorithm for multivariate polynomial codes and http://theory.lcs.mit.edu/˜salil. Supported by a DOD/NDSEG grad- uate fellowship and partially by DARPA grant DABT63-96-C-0018. exploits a connection between the list-decoding and the hardness- amplification problems. The list-decoding algorithm described in this paper is quantitatively better than a previous one by Arora and Sudan [AS97], and has a simpler analysis. 1 To be accurate, the term extractor comes fron [NZ96] and postdates the paper of H˚astad et al. [HILL98]. overview of previous results An The works of Blum and in increasingly sophisticated forms, in [Imp95, IW97]. Likewise, Micali [BM84] and Yao [Yao82] introduce the notion of a cryp- the NW generator and its original analysis have always been used tographically strong pseudorandom generator (csPRG) and show in conditional derandomization results since. Future progress in how to construct pseudorandom generators based on the existence the area will probably require a departure from this observance of of one-way permutations. A csPRG is a polynomial-time algorithm the NW methodology, or at least a certain amount of revisitation of that on input a randomly selected string of length n produces an its main parts. output of length n that is computationally indistinguishable from In this paper, we give two new ways to build pseudorandom n uniform by any adversary of p oly size, where is an arbitrar- generators with seeds of logarithmic length. Both approaches by- ily small constant. Yao also observes that a given polynomial-time pass the need for the XOR Lemma, and instead use tools (such as randomized algorithm can be simulated deterministically using a list decoding, extractors, and pseudoentropy generators) that did n p oly n csPRG in time by trying all the seeds and taking the not appear in the sequence of works from [NW94] to [IW97]. For majority answer. a diagram illustrating the steps leading up to the results of [IW97] In a seminal work, Nisan and Wigderson [NW94] explore the and how our techniques depart from that framework, see Figure 1. use of a weaker type of pseudorandom generator (PRG) in order to Both of our approaches are described in more detail below. derandomize randomized algorithm. They observe that, for the pur- pose of derandomization, one can consider generators computable t p oly t t in time p oly (instead of ) where is the length of the Worst−case hard seed, since the derandomization process cycles through all the seeds, polynomial t [BFNW93] and this induces an overhead factor anyway. They also observe encoding that one can restrict to generators that are good against adversaries whose running time is bounded by a fixed polynomial, instead of polynomial Mildly hard encoding Thm 10 XOR [Imp95] every polynomial. They then show how to construct a pseudo- + noisy Lemma random generator meeting this relaxed definition under weaker as- reconstruction (Thm 19) Constant hardness sumptions than those used to build cryptographically strong pseu- XOR Pseudoentropy Lemma [IW97] dorandom generators. Furthermore, they show that, under a suf- Generator ficiently strong assumption, one can build a PRG that uses seeds Extremely hard of logarithmic length (which would be impossible for a csPRG). [NW94] extractor Such a generator can be used to simulate randomized algorithms (Lem 13) BPP in polynomial time, and its existence implies P . The con- Pseudorandom Generator dition under which Nisan and Wigderson prove the existence of a PRG with seeds of logarithmic length is the existence of a decision n f g f g problem (i.e., a predicate P ) solvable in time n n O Figure 1: A comparison of our approach with previous ones. Dou- such that for some positive constant no circuit of size n ble arrows indicate our results. can solve the problem on more than a fraction of the inputs. This is a very strong hardness requirement, and it is of interest to obtain similar conclusions under weaker assumptions. Pseudo entropy Generator An example of a weaker assumption is the existence of a mildly A Nisan and Wigderson show hard predicate. We say that a predicate is mildly hard if for some that when their generator is constructed using a very hard-on-average n fixed no circuit of size can decide the predicate on more predicate, then the output of the generator is indistinguishable from p oly n than a fraction of the inputs. Nisan and Wigder- the uniform distribution. It is a natural question to ask what hap- son prove that mild hardness suffices to derive a pseudorandom pens if there are stronger or weaker conditions on the predicate. In log n generator with seed of O length, which in turn implies a this paper we consider the question of what happens if the predicate quasi-polynomial deterministic simulation of BPP. This result is is only mildly hard. Specifically we are interested in whether expo- proved by using Yao’s XOR Lemma [Yao82] (see, e.g., [GNW95] nential average-case hardness is really necessary for direct pseu- for a proof) to convert a mildly hard predicate over n inputs into dorandom generation.
Load more

Recommended publications
  • The Limits of Post-Selection Generalization
    The Limits of Post-Selection Generalization Kobbi Nissim∗ Adam Smithy Thomas Steinke Georgetown University Boston University IBM Research – Almaden [email protected] [email protected] [email protected] Uri Stemmerz Jonathan Ullmanx Ben-Gurion University Northeastern University [email protected] [email protected] Abstract While statistics and machine learning offers numerous methods for ensuring gener- alization, these methods often fail in the presence of post selection—the common practice in which the choice of analysis depends on previous interactions with the same dataset. A recent line of work has introduced powerful, general purpose algorithms that ensure a property called post hoc generalization (Cummings et al., COLT’16), which says that no person when given the output of the algorithm should be able to find any statistic for which the data differs significantly from the population it came from. In this work we show several limitations on the power of algorithms satisfying post hoc generalization. First, we show a tight lower bound on the error of any algorithm that satisfies post hoc generalization and answers adaptively chosen statistical queries, showing a strong barrier to progress in post selection data analysis. Second, we show that post hoc generalization is not closed under composition, despite many examples of such algorithms exhibiting strong composition properties. 1 Introduction Consider a dataset X consisting of n independent samples from some unknown population P. How can we ensure that the conclusions drawn from X generalize to the population P? Despite decades of research in statistics and machine learning on methods for ensuring generalization, there is an increased recognition that many scientific findings do not generalize, with some even declaring this to be a “statistical crisis in science” [14].
    [Show full text]
  • The Computational Complexity of Nash Equilibria in Concisely Represented Games∗
    The Computational Complexity of Nash Equilibria in Concisely Represented Games¤ Grant R. Schoenebeck y Salil P. Vadhanz August 26, 2009 Abstract Games may be represented in many di®erent ways, and di®erent representations of games a®ect the complexity of problems associated with games, such as ¯nding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payo®s, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payo®s are computed by a given boolean circuit, and graph games, where each agent's payo® is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, ¯nding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payo®s to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes. 1 Introduction In recent years, there has been a surge of interest at the interface between computer science and game theory. On one hand, game theory and its notions of equilibria provide a rich framework for modeling the behavior of sel¯sh agents in the kinds of distributed or networked environments that often arise in computer science and o®er mechanisms to achieve e±cient and desirable global outcomes in spite of the sel¯sh behavior.
    [Show full text]
  • Pseudodistributions That Beat All Pseudorandom Generators
    Electronic Colloquium on Computational Complexity, Report No. 19 (2021) Pseudodistributions That Beat All Pseudorandom Generators Edward Pyne∗ Salil Vadhany Harvard University Harvard University [email protected] [email protected] February 17, 2021 Abstract A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom gen- erator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ordered branching programs whose seed length has a better dependence on the error parameter " than the classic PRG construction of Nisan (STOC 1990 and Combinatorica 1992). In this work, we give an explicit construction of PRPGs that achieve parameters that are impossible to achieve by a PRG. In particular, we construct a PRPG for ordered permuta- tion branching programs of unbounded width with a single accept state that has seed length O~(log3=2 n) for error parameter " = 1= poly(n), where n is the input length. In contrast, re- cent work of Hoza et al. (ITCS 2021) shows that any PRG for this model requires seed length Ω(log2 n) to achieve error " = 1= poly(n). As a corollary, we obtain explicit PRPGs with seed length O~(log3=2 n) and error " = 1= poly(n) for ordered permutation branching programs of width w = poly(n) with an arbi- trary number of accept states. Previously, seed length o(log2 n) was only known when both the width and the reciprocal of the error are subpolynomial, i.e.
    [Show full text]
  • Hardness of Non-Interactive Differential Privacy from One-Way
    Hardness of Non-Interactive Differential Privacy from One-Way Functions Lucas Kowalczyk* Tal Malkin† Jonathan Ullman‡ Daniel Wichs§ May 30, 2018 Abstract A central challenge in differential privacy is to design computationally efficient non-interactive algorithms that can answer large numbers of statistical queries on a sensitive dataset. That is, we would like to design a differentially private algorithm that takes a dataset D Xn consisting of 2 some small number of elements n from some large data universe X, and efficiently outputs a summary that allows a user to efficiently obtain an answer to any query in some large family Q. Ignoring computational constraints, this problem can be solved even when X and Q are exponentially large and n is just a small polynomial; however, all algorithms with remotely similar guarantees run in exponential time. There have been several results showing that, under the strong assumption of indistinguishability obfuscation (iO), no efficient differentially private algorithm exists when X and Q can be exponentially large. However, there are no strong separations between information-theoretic and computationally efficient differentially private algorithms under any standard complexity assumption. In this work we show that, if one-way functions exist, there is no general purpose differen- tially private algorithm that works when X and Q are exponentially large, and n is an arbitrary polynomial. In fact, we show that this result holds even if X is just subexponentially large (assuming only polynomially-hard one-way functions). This result solves an open problem posed by Vadhan in his recent survey [Vad16]. *Columbia University Department of Computer Science.
    [Show full text]
  • Zero Knowledge and Soundness Are Symmetric∗
    Zero Knowledge and Soundness are Symmetric∗ Shien Jin Ong Salil Vadhan Division of Engineering and Applied Sciences Harvard University Cambridge, Massachusetts, USA. E-mail: {shienjin,salil}@eecs.harvard.edu November 14, 2006 Abstract We give a complexity-theoretic characterization of the class of problems in NP having zero- knowledge argument systems that is symmetric in its treatment of the zero knowledge and the soundness conditions. From this, we deduce that the class of problems in NP ∩ 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. Keywords: zero-knowledge argument systems, statistical zero knowledge, complexity classes, clo- sure under complement, distributional one-way functions. ∗Both the authors were supported by NSF grant CNS-0430336 and ONR grant N00014-04-1-0478.
    [Show full text]
  • Pseudorandomness and Combinatorial Constructions
    Pseudorandomness and Combinatorial Constructions Luca Trevisan∗ September 20, 2018 Abstract In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or unknown. In computer science, probabilistic al- gorithms are sometimes simpler and more efficient than the best known deterministic algorithms for the same problem. Despite this evidence for the power of random choices, the computational theory of pseu- dorandomness shows that, under certain complexity-theoretic assumptions, every probabilistic algorithm has an efficient deterministic simulation and a large class of applications of the the probabilistic method can be converted into explicit constructions. In this survey paper we describe connections between the conditional “derandomization” re- sults of the computational theory of pseudorandomness and unconditional explicit constructions of certain combinatorial objects such as error-correcting codes and “randomness extractors.” 1 Introduction 1.1 The Probabilistic Method in Combinatorics In extremal combinatorics, the probabilistic method is the following approach to proving existence of objects with certain properties: prove that a random object has the property with positive probability. This simple idea has beem amazingly successful, and it gives the best known bounds for arXiv:cs/0601100v1 [cs.CC] 23 Jan 2006 most problems in extremal combinatorics. The idea was introduced (and, later, greatly developed) by Paul Erd˝os [18], who originally applied it to the following question: define R(k, k) to be the minimum value n such that every graph on n vertices has either an independent set of size at least k or a clique of size at least k;1 It was known that R(k, k) is finite and that it is at most 4k, and the question was to prove a lower bound.
    [Show full text]
  • Computational Complexity
    Computational Complexity The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Vadhan, Salil P. 2011. Computational complexity. In Encyclopedia of Cryptography and Security, second edition, ed. Henk C.A. van Tilborg and Sushil Jajodia. New York: Springer. Published Version http://refworks.springer.com/mrw/index.php?id=2703 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33907951 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Computational Complexity Salil Vadhan School of Engineering & Applied Sciences Harvard University Synonyms Complexity theory Related concepts and keywords Exponential time; O-notation; One-way function; Polynomial time; Security (Computational, Unconditional); Sub-exponential time; Definition Computational complexity theory is the study of the minimal resources needed to solve computational problems. In particular, it aims to distinguish be- tween those problems that possess efficient algorithms (the \easy" problems) and those that are inherently intractable (the \hard" problems). Thus com- putational complexity provides a foundation for most of modern cryptogra- phy, where the aim is to design cryptosystems that are \easy to use" but \hard to break". (See security (computational, unconditional).) Theory Running Time. The most basic resource studied in computational com- plexity is running time | the number of basic \steps" taken by an algorithm. (Other resources, such as space (i.e., memory usage), are also studied, but they will not be discussed them here.) To make this precise, one needs to fix a model of computation (such as the Turing machine), but here it suffices to informally think of it as the number of \bit operations" when the input is given as a string of 0's and 1's.
    [Show full text]
  • Pseudorandomness for Regular Branching Programs Via Fourier Analysis
    Electronic Colloquium on Computational Complexity, Report No. 86 (2013) Pseudorandomness for Regular Branching Programs via Fourier Analysis Omer Reingold∗ Thomas Steinkey Salil Vadhanz [email protected] [email protected] [email protected] June 2013 Abstract We present an explicit pseudorandom generator for oblivious, read-once, permutation branching pro- grams of constant width that can read their input bits in any order. The seed length is O(log2 n), where n is the length of the branching program. The previous best seed length known for this model was n1=2+o(1), which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of s1=2+o(1) for arbitrary branching programs of size s). Our techniques also give seed length n1=2+o(1) for general oblivious, read-once branching programs of width o(1) 2n , which is incomparable to the results of Impagliazzo et al. Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width w has Fourier mass at most (2w)2k at level k, independent of the length of the program. ∗Microsoft Research Silicon Valley, 1065 La Avenida, Mountain View, CA. ySchool of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge MA. Work done in part while at Stanford University. Supported by NSF grant CCF-1116616 and the Lord Rutherford Memorial Research Fellowship.
    [Show full text]
  • A Note on Extracting Randomness from Santha-Vazirani Sources
    A Note on Extracting Randomness from Santha-Vazirani Sources Omer Reingold∗ Salil Vadhany Incumbent of the Walter and Elise Haas Division of Engineering & Applied Sciences Career Development Chair Harvard University Department of Computer Science Cambridge, MA, USA Weizmann Institute of Science [email protected] Rehovot, Israel [email protected] Avi Wigdersonz School of Mathematics Institute for Advanced Study Princeton, NJ, USA [email protected] October 11, 2004 Abstract Santha and Vazirani [SV86] proposed the notion of a semi-random source of bits (also known as Santha-Vazirani sources), and proved that it is impossible to (deterministically) extract an almost-uniform random bit from such a source (while it is possible given several independent Santha-Vazirani sources). We provide a simpler and more transparent proof of their impossibility result. Our proof has the advantage of applying to a natural strengthening of Santha-Vazirani sources. Moreover, our proof technique has been used in [DOPS04] to obtain impossibility results on doing cryptography with semi-random sources. A source of length n is a random variable X taking values in f0; 1gn. We will denote the individual bits of X by X = X1 ··· Xn. If Y is a random variable taking values in f0; 1g, the bias of Y is j Pr[Y = 0] − Pr[Y = 1]j, i.e. the smallest δ such that (1 − δ)=2 ≤ Pr[Y = 0] ≤ (1 + δ)=2. Definition 1 For δ 2 [0; 1], a source X of length n is a Santha-Vazirani (SV) source (or semi- random source) with bias δ if for every i 2 [n] and every x1; : : : ; xi 2 f0; 1g, the bias of Xi conditioned on X1 = x1;:::;Xi−1 = xi−1 is at most δ.
    [Show full text]
  • Verifiable Random Functions
    Verifiable Random Functions y z Silvio Micali Michael Rabin Salil Vadhan Abstract random string of the proper length. The possibility thus ex- ists that, if it so suits him, the party knowing the seed s may We efficiently combine unpredictability and verifiability by declare that the value of his pseudorandom oracle at some x f x extending the Goldreich–Goldwasser–Micali construction point is other than s without fear of being detected. It f s of pseudorandom functions s from a secret seed , so that is for this reason that we refer to these objects as “pseudo- s f knowledge of not only enables one to evaluate s at any random oracles” rather than using the standard terminology f x x NP point , but also to provide an -proof that the value “pseudorandom functions” — the values s come “out f x s is indeed correct without compromising the unpre- of the blue,” as if from an oracle, and the receiver must sim- s f dictability of s at any other point for which no such a proof ply trust that they are computed correctly from the seed . was provided. Therefore, though quite large, the applicability of pseu- dorandom oracles is limited: for instance, to settings in which (1) the “seed owner”, and thus the one evaluating 1Introduction the pseudorandom oracle, is totally trusted; or (2) it is to the seed-owner’s advantage to evaluate his pseudorandom oracle correctly; or (3) there is absolutely nothing for the PSEUDORANDOM ORACLES. Goldreich, Goldwasser, and seed-owner to gain from being dishonest. Micali [GGM86] show how to simulate a random ora- f x One efficient way of enabling anyone to verify that s b cle from a-bit strings to -bit strings by means of a con- f x really is the value of pseudorandom oracle s at point struction using a seed, that is, a secret and short random clearly consists of publicizing the seed s.However,this string.
    [Show full text]
  • Arxiv:1912.04524V1 [Cs.CC] 10 Dec 2019
    High-precision Estimation of Random Walks in Small Space AmirMahdi Ahmadinejad Jonathan Kelner Stanford University Massachusetts Institute of Technology [email protected] [email protected] Jack Murtagh∗ John Peebles† Aaron Sidford‡ Harvard University Yale University Stanford University [email protected] [email protected] [email protected] Salil Vadhan§ Harvard University [email protected] December 11, 2019 Abstract In this paper, we provide a deterministic O˜(log N)-space algorithm for estimating the ran- dom walk probabilities on Eulerian directed graphs (and thus also undirected graphs) to within inverse polynomial additive error (ε = 1/poly(N)) where N is the length of the input. Previ- ously, this problem was known to be solvable by a randomized algorithm using space O(log N) (Aleliunas et al., FOCS ‘79) and by a deterministic algorithm using space O(log3/2 N) (Saks and Zhou, FOCS ‘95 and JCSS ‘99), both of which held for arbitrary directed graphs but had not been improved even for undirected graphs. We also give improvements on the space com- plexity of both of these previous algorithms for non-Eulerian directed graphs when the error is negligible (ε = 1/N ω(1)), generalizing what Hoza and Zuckerman (FOCS ‘18) recently showed for the special case of distinguishing whether a random walk probability is 0 or greater than ε. We achieve these results by giving new reductions between powering Eulerian random-walk matrices and inverting Eulerian Laplacian matrices, providing a new notion of spectral approx- imation for Eulerian graphs that is preserved under powering, and giving the first deterministic O˜(log N)-space algorithm for inverting Eulerian Laplacian matrices.
    [Show full text]
  • A Study of Statistical Zero-Knowledge Proofs Salil Pravin
    A Study of Statistical Zero-Knowledge Proofs by Salil Pravin Vadhan A.B., Mathematics and Computer Science, Harvard University (1995) Certificate of Advanced Study in Mathematics, Cambridge University (1996) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1999 @ Salil Pravin Vadhan, 1999. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. A uthor ........... .'v --~-- g -. - --- ---. - .---------------------------- Dtopartment of Mathematics August 6, 1999 ,/1 I; Certified by ....... .. .. - .. -. .... - ...- .. .-- fi Gold asser RSA Professor of Computer Science Thesis Supervisor .. V... ...... V J .. ................ Accepted by ......... c alS p. e Michael Sipser Chairman, Applied Mathematics Committee A A ccepted by ................. ...--.. - -- -- - Tomasz Mrowka Chairman, Departmental Committee on Graduate Students MASSACHUSETTS INSTITUTE LIBRARIES A Study of Statistical Zero-Knowledge Proofs by Sail Pravin Vadhan Submitted to the Department of Mathematics on August 6, 1999, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Zero-knowledge interactive proofs, introduced by Goldwasser, Micali, and Rackoff, are fas- cinating constructs which enable one party (the "prover") to convince another party (the "verifier") of an assertion, with the property that the verifier learns nothing other than the fact that the assertion being proven is true. In addition to being powerful tools for construct- ing secure cryptographic protocols, zero-knowledge proofs yield rich classes of computational problems that are of both complexity-theoretic and cryptographic interest. This thesis is a detailed investigation of statisticalzero-knowledge proofs, which are zero- knowledge proofs in which the condition that the verifier "learns nothing" is interpreted in a strong statistical sense.
    [Show full text]