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ABSTRACT: A q -ary error-correcting code C ⊆ {1,2,..., q } n is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C . We prove that in order for a q -ary code to be list-decodable up to radius (1-1/ q )(1- ε) n , we must have L = Ω(1/ ε<sup>2</sup>) . Specifically, we prove that there exists a constant cq > 0 and a function fq such that for small enough ε > 0, if C is list-decodable to radius (1-1/ q )(1- ε) n with list size cq / ε<sup>2</sup>, then C has at most fq ( ε) codewords, independent of n . This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n ) number of codewords are known for list size L = O (1/ ε<sup>2</sup>). A result similar to ours is implicit in Blinovsky ( Problems of Information Transmission, 1986) for the binary ( q =2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.
IEEE Transactions on Information Theory 12/2010; · 3.01 Impact Factor
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ABSTRACT: We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma, Umans, and Zuckerman (STOC "01) required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy (FOCS "05). Our expanders can be interpreted as near-optimal "randomness condensers," that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. (STOC "03) and improving upon it when the error parameter is small (e.g. 1/poly(n)).
Computational Complexity, 2007. CCC '07. Twenty-Second Annual IEEE Conference on; 07/2007
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ABSTRACT: We present a deterministic logspace algorithm for solving S-T CONNECTIVITY on directed graphs if (i) we are given a stationary distribution for random walk on the graph and (ii) the random walk which starts at the source vertex s has polynomial mixing time. This result generalizes the recent deterministic logspace algorithm for S-T CONNECTIVITY on undirected graphs [15]. It identifies knowledge of the stationary distribution as the gap between the S-T CONNECTIVITY problems we know how to solve in logspace (L) and those that capture all of randomized logspace (RL).
Computational Complexity, 2007. CCC '07. Twenty-Second Annual IEEE Conference on; 07/2007
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ABSTRACT: 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</sup>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.
Computational Complexity, 2005. Proceedings. Twentieth Annual IEEE Conference on; 07/2005
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ABSTRACT: 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, l}<sup>n</sup>). 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 P. 2) If H(X) ≤ k = n - O(log n), we show how to compress to length k + δ· (n - k) for any constant δ > 0; in quasi-polynomial time we show how to compress to length k + O(polylog log (n - k)) even if k = n -polylog(n). 3) If the support of X is the witness set for a self-reducible NP relation, then we show how to compress to expected length H(X) + 4.
Computational Complexity, 2004. Proceedings. 19th IEEE Annual Conference on; 07/2004
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ABSTRACT: Proc. of 30th STOC, pp. 644--652, 1998. 72. R. Impagliazzo, A. Wigderson, Randomness vs. Time: De-randomization under a uniform assumption, Proc. of the 39th FOCS, pp.734--743, 1998. 73. A. Ambainis, L. Schulman, A. Ta-Shma, U. Vazirani, A. Wigderson, The Quantum Communication Complexity of Sampling, Proc. of the 39th FOCS, pp. 342--351, 1998. 74. E. Ben-Sasson, A. Wigderson, Short Proofs are Narrow -- Resolution made Simple, Proc. of the 31th STOC, pp. 517--526, 1999. 75. A. Shpilka, A. Wigderson, Depth-3 Arithmetic Formulae over Fields of Characteristic Zero, Proc. of the 14th Conference on Computational Complexity, pp. 87--97, 1999. 76. Z. Bar-Yossef, O. Goldreich, A. Wigderson, Deterministic Amplification of Space-Bounded Probabilistic Algorithms Proc. of the 14th Conference on Computational Complexity, pp. 188--199, 1999. 77. O. Goldreich, A. Wigderson, Improved derandomization of BPP using a hitting set generator, Proc. of the RAMDOM 99 Conference, pp. 131--137, 1999. 78. R
12/2003;
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ABSTRACT: We show new lower bounds and impossibility results for general (possibly non-black-box) zero-knowledge proofs and arguments. Our main results are that, under reasonable complexity assumptions: 1. There does not exist a constant-round zero-knowledge strong proof (or argument) of knowledge (as defined by Goldreich, 2001) for a nontrivial language; 2. There does not exist a two-round zero-knowledge proof system with perfect completeness for an NP-complete language; 3. There does not exist a constant-round public-coin proof system for a nontrivial language that is resettable zero knowledge. This result also extends to bounded resettable zero knowledge. In contrast, we show that under reasonable assumptions, there does exist such a (computationally sound) argument system that is bounded-resettable zero knowledge.
Foundations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on; 11/2003
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Computational Complexity, 2002. Proceedings. 17th IEEE Annual Conference on; 02/2002
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ABSTRACT: Impagliazzo and Wigderson (1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP = BPP ). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. We obtain an optimal worst-case to average-case connection for EXP : if EXP BPTIME (( )), EXP has problems that are cannot be solved on a fraction 1/2 1/'( ) of the inputs by BPTIME ('( )) algorithms, for ' = <sup>1</sup>. We exhibit a PSPACE -complete downward self-reducible and random self-reducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson (1998), which used a a P -complete problem with these properties. We argue that the results in Impagliazzo and Wigderson (1998) and in this paper cannot be proved via "black-box" uniform reductions
Computational Complexity, 2002. Proceedings. 17th IEEE Annual Conference on; 02/2002
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Proceedings of the $34$th STOC; 01/2002
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ABSTRACT: The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. We consider the problem of deterministically converting a weak source of randomness into an almost uniform distribution. Previously, deterministic extraction procedures were known only for sources satisfying strong independence requirements. We look at sources which are samplable, i.e. can be generated by an efficient sampling algorithm. We seek an efficient deterministic procedure that, given a sample from any samplable distribution of sufficiently large min-entropy, gives an almost uniformly distributed output. We explore the conditions under which such deterministic extractors exist. We observe that no deterministic extractor exists if the sampler is allowed to use more computational resources than the extractor. On the other hand, if the extractor is allowed (polynomially) more resources than the sampler, we show that deterministic extraction becomes possible. This is true unconditionally in the nonuniform setting (i.e., when the extractor can be computed by a small circuit), and (necessarily) relies on complexity assumptions in the uniform setting
Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on; 02/2000
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ABSTRACT: The main contribution is a new type of graph product, which we
call the zig-zag product. Taking a product of a large graph with a small
graph, the resulting graph inherits (roughly) its size from the large
one, its degree from the small one, and its expansion properties from
both. Iteration yields simple explicit constructions of constant-degree
expanders of every size, starting from one constant-size expander.
Crucial to our intuition (and simple analysis) of the properties of this
graph product is the view of expanders as functions which act as
“entropy wave” propagators-they transform probability
distributions in which entropy is concentrated in one area to
distributions where that concentration is dissipated. In these terms,
the graph product affords the constructive interference of two such
waves. A variant of this product can be applied to extractors, giving
the first explicit extractors whose seed length depends
(poly)logarithmically on only the entropy deficiency of the source
(rather than its length) and that extract almost all the entropy of high
min-entropy sources. These high min-entropy extractors have several
interesting applications, including the first constant-degree explicit
expanders which beat the “eigenvalue bound”
Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on; 02/2000
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ABSTRACT: Summary form only given. R. Impagliazzo and A. Wigderson (1997) have recently shown that if there exists a decision problem solvable in time 2<sup>O(n)</sup> and having circuit complexity 2<sup>Ω(n) </sup> (for all but finitely many n) then P=BPP. This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of “hardness amplification” (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed with the Nisan-Wigderson (1994) generator. In this paper we present two different approaches to proving the main result of Impagliazzo and Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and/or applications of hardness-randomness trade-offs
Computational Complexity, 1999. Proceedings. Fourteenth Annual IEEE Conference on; 02/1999
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ABSTRACT: We consider the following (promise) problem, denoted ED (for
Entropy Difference): The input is a pair of circuits, and YES instances
(resp., NO instances) are such pairs in which the first (resp., second)
circuit generates a distribution with noticeably higher entropy. On one
hand we show that any language having a (honest-verifier) statistical
zero-knowledge proof is Karp-reducible to ED. On the other hand, we
present a public-coin (honest-verifier) statistical zero-knowledge proof
for ED. Thus, we obtain an alternative proof of Okamoto's result by
which HVSZK: (i.e., honest-verifier statistical zero knowledge) equals
public-coin HVSZK. The new proof is much simpler than the original one.
The above also yields a trivial proof that HVSZK: is closed under
complementation (since ED easily reduces to its complement). Among the
new results obtained is an equivalence of a weak notion of statistical
zero knowledge to the standard one
Computational Complexity, 1999. Proceedings. Fourteenth Annual IEEE Conference on; 02/1999
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ABSTRACT: We efficiently combine unpredictability and verifiability by
extending the Goldreich-Goldwasser-Micali (1986) construction of
pseudorandom functions f<sub>s</sub> from a secret seed s, so that
knowledge of s not only enables one to evaluate f<sub>s</sub> at any
point x, but also to provide an NP-proof that the value f<sub>s</sub>(x)
is indeed correct without compromising the unpredictability of f<sub>s
</sub> at any other point for which no such a proof was provided
Foundations of Computer Science, 1999. 40th Annual Symposium on; 02/1999
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ABSTRACT: An extractor is a function which extracts (almost) truly random bits from a weak random source, using a small number of additional random bits as a catalyst. We present a general method to reduce the error of any extractor. Our method works particularly well in the case that the original extractor extracts up to a constant function of the source min-entropy and achieves a polynomially small error. In that case, we are able to reduce the error to (almost) any ε, using only O(log(1/ε)) additional truly random bits (while keeping the other parameters of the original extractor more or less the same). In other cases (e.g. when the original extractor extracts all the min-entropy or achieves only a constant error), our method is not optimal but it is still quite efficient and leads to improved constructions of extractors. Using our method, we are able to improve almost all known extractors in the case where the error required is relatively small (e.g. less than a polynomially small error). In particular, we apply our method to the new extractors of L. Trevisan (1999) and R. Raz et al. (1999) to obtain improved constructions in almost all cases. Specifically, we obtain extractors that work for sources of any min-entropy on strings of length n which (a) extract any 1/n<sup>γ</sup> fraction of the min-entropy using O[log n+log(1/ε)] truly random bits (for any γ>0), (b) extract any constant fraction of the min-entropy using O[log<sup>2</sup>n+log(1/ε)] truly random bits, and (c) extract all the min-entropy using O[log<sup>3</sup>n+log n·log(1/ε)] truly random bits
Foundations of Computer Science, 1999. 40th Annual Symposium on; 02/1999
