One-way functions and Pseudorandom generators
Combinatorica - Theory of Computing
A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Towards the parallel repetition conjecture
Theoretical Computer Science - Special issue on complexity theory and the theory of algorithms as developed in the CIS
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
A Combinatorial Consistency Lemma with Application to Proving the PCP Theorem
SIAM Journal on Computing
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Two-Prover Protocols---Low Error at Affordable Rates
SIAM Journal on Computing
List-Decoding Using The XOR Lemma
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Error Reduction by Parallel Repetition—A Negative Result
Combinatorica
Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Parallel repetition: simplifications and the no-signaling case
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Assignment Testers: Towards a Combinatorial Proof of the PCP Theorem
SIAM Journal on Computing
Parallel repetition in projection games and a concentration bound
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Uniform direct product theorems: simplified, optimized, and derandomized
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
PCPs with small soundness error
ACM SIGACT News
Locally Testing Direct Product in the Low Error Range
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Two Query PCP with Sub-Constant Error
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A Counterexample to Strong Parallel Repetition
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Local list-decoding and testing of random linear codes from high error
Proceedings of the forty-second ACM symposium on Theory of computing
The structure of winning strategies in parallel repetition games
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized
SIAM Journal on Computing
Composition of low-error 2-query PCPs using decodable PCPs
Property testing
Some recent results on local testing of sparse linear codes
Property testing
Composition of low-error 2-query PCPs using decodable PCPs
Property testing
Some recent results on local testing of sparse linear codes
Property testing
Parallel repetition of entangled games
Proceedings of the forty-third annual ACM symposium on Theory of computing
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The "direct product code" of a function f gives its values on all k-tuples (f(x1),...,f(xk)). This basic construct underlies "hardness amplification" in cryptography, circuit complexity and PCPs. Goldreich and Safra [12] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [5] enabled for the first time testing proximity to this important code in the "list-decoding" regime. In particular, they give a 2-query test which works for polynomially small success probability 1/kα, and show that no such test works below success probability 1/k. Our main result is a 3-query test which works for exponentially small success probability exp(-kα). Our techniques (based on recent simplified decoding algorithms for the same code [15]) also allow us to considerably simplify the analysis of the 2-query test of [5]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of k, and success probability 1/kα. Finally we show the applicability of the new tests to PCPs. Starting with a 2-query PCP over an alphabet Σ and with soundness error 1-δ, Rao [19] (building on Raz's (k-fold) parallel repetition theorem [20] and Holenstein's proof [13]) obtains a new 2-query PCP over the alphabet Σk with soundness error exp(-δ2 k). Our techniques yield a 2-query PCP with soundness error exp(-δ √k). Our PCP construction turns out to be essentially the same as the miss-match proof system defined and analyzed by Feige and Kilian [8], but with simpler analysis and exponentially better soundness error.