Using nondeterminism to amplify hardness
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
On uniform amplification of hardness in NP
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Key agreement from weak bit agreement
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Foundations and Trends® in Theoretical Computer Science
Algorithmic results in list decoding
Foundations and Trends® in Theoretical Computer Science
If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances
Computational Complexity
Pseudorandomness and Average-Case Complexity Via Uniform Reductions
Computational Complexity
List-decoding reed-muller codes over small fields
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 Complexity of Local List Decoding
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
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
Hardness amplification within NP against deterministic algorithms
Journal of Computer and System Sciences
Hardness Amplification Proofs Require Majority
SIAM Journal on Computing
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Computational randomness from generalized hardcore sets
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
List Decoding Tensor Products and Interleaved Codes
SIAM Journal on Computing
Hardness amplification via space-efficient direct products
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
On the complexity of hard-core set constructions
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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We show that Yao's XOR Lemma, and its essentially equivalent rephrasing as a Direct Product Lemma, can be re-interpreted as a way of obtaining error-correcting codes with good list-decoding algorithms from error-correcting codes having weak unique-decoding algorithms. To get codes with good rate and efficient list decoding algorithms one needs a proof of the Direct Product Lemma that, respectively, is strongly derandomized, and uses very small advice.We show how to reduce advice in Impagliazzo's proof of the Direct Product Lemma for pairwise independent inputs, which leads to error-correcting codes with O(n2) encoding length, \bar 0(n^2 ) encoding time, and probabilistic \bar 0(n) list-decoding time. (Note that the decoding time is sub-linear in the length of the encoding.)Back to complexity theory, our advice-efficient proof of Impagliazzo's "hard-core set" results yields a (weak) uniform version of O'Donnell results on amplification of hardness in NP. We show that if there is a problem in NP that cannot be solved by BPP algorithms on more than a 1 - 1/(log n)c fraction of inputs, then there is a problem in NP that cannot be solved by BPP algorithms on more than a 3/4+1/(logn)c fraction of inputs, where c 0 is an absolute constant.