Verifying and decoding in constant depth
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Algorithmic results in list decoding
Foundations and Trends® in Theoretical Computer Science
Uniform direct product theorems: simplified, optimized, and derandomized
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Hardness amplification proofs require majority
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The uniform hardcore lemma via approximate Bregman projections
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Proofs of Retrievability via Hardness Amplification
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
New direct-product testers and 2-query PCPs
Proceedings of the forty-first annual ACM symposium on Theory of computing
Chernoff-type direct product theorems
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Near-optimal extractors against quantum storage
Proceedings of the forty-second ACM symposium on Theory of computing
Hardness amplification within NP against deterministic algorithms
Journal of Computer and System Sciences
Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized
SIAM Journal on Computing
Hardness Amplification Proofs Require Majority
SIAM Journal on Computing
General hardness amplification of predicates and puzzles
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Studies in complexity and cryptography
Certifiable quantum dice: or, true random number generation secure against quantum adversaries
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Clustering in the boolean hypercube in a list decoding regime
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We consider the problem of approximately locally listdecoding direct product codes. For a parameter k, the kwise direct product encoding of an N-bit message msg is an N^{k}-length string over the alphabet {0, 1}^k indexed by ktuples (i_1, . . . , i_k) \in {1, . . . ,N}^k so that the symbol at position (i_1, . . . , i_k) of the codeword is msg(i_1) . . . msg(i_k). Such codes arise naturally in the context of hardness amplification of Boolean functions via the Direct Product Lemma (and the closely related Yao's XOR Lemma), where typically k \ll N (e.g., k = poly logN). We describe an efficient randomized algorithm for approximate local list-decoding of direct product codes. Given access to a word which agrees with the k-wise direct product encoding of some message msg in at least an \in fraction of positions, our algorithm outputs a list of poly(1/\in) Boolean circuits computing N-bit strings (viewed as truth tables of logN-variable Boolean functions) such that at least one of them agrees with msg in at least 1 - \deltafraction of positions, for \delta = O(k^{-0.1}), provided that \in =\Omega(poly(1/k)); the running time of the algorithm is polynomial in logN and 1/\in. When \in \ge e-^{k^{\alpha} } for a certain constant \alpha 0, we get a randomized approximate listdecoding algorithm that runs in time quasi-polynomial in 1/\in (i.e., (1/\in)^{poly log 1/\in}). By concatenating the k-wise direct product codes with Hadamard codes, we obtain locally list-decodable codes over the binary alphabet, which can be efficiently approximately list-decoded from fewer than 1/2 - \in fraction of corruptions as long as \in = \Omega(poly(1/k)). As an immediate application, we get uniform hardness amplification for P^{NP_\parallel} , the class of languages reducible to NP through one round of parallel oracle queries: If there is a language in P^{NP_\parallel} that cannot be decided by any BPP algorithm on more that 1 - 1/n^{\Omega(1)} fraction of inputs, then there is another language in P^{NP_\parallel} that cannot be decided by any BPP algorithm on more than 1/2 + 1/n^{\omega(1)} fraction of inputs.