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
Randomness in interactive proofs
Computational Complexity
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
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Randomness-optimal oblivious sampling
Proceedings of the workshop on Randomized algorithms and computation
A Combinatorial Consistency Lemma with Application to Proving the PCP Theorem
SIAM Journal on Computing
Pseudorandom generators without the XOR lemma
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Hard-core distributions for somewhat hard problems
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Pseudorandomness and Average-Case Complexity via Uniform Reductions
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
On uniform amplification of hardness in NP
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Assignment Testers: Towards a Combinatorial Proof of the PCP Theorem
SIAM Journal on Computing
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Chernoff-type direct product theorems
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
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
Security Amplification for Interactive Cryptographic Primitives
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
On the complexity of differentially private data release: efficient algorithms and hardness results
Proceedings of the forty-first annual ACM symposium on Theory of computing
Extractors Using Hardness Amplification
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
A strong direct product theorem for disjointness
Proceedings of the forty-second ACM symposium on Theory of computing
Local list-decoding and testing of random linear codes from high error
Proceedings of the forty-second ACM symposium on Theory of computing
Hardness amplification within NP against deterministic algorithms
Journal of Computer and System Sciences
Some recent results on local testing of sparse linear codes
Property testing
Some recent results on local testing of sparse linear codes
Property testing
General hardness amplification of predicates and puzzles
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Parallel repetition of entangled games
Proceedings of the forty-third annual ACM symposium on Theory of computing
Almost optimal bounds for direct product threshold theorem
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
Derandomized Parallel Repetition Theorems for Free Games
Computational Complexity
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The classical Direct-Product Theorem for circuits says that if a Boolean function f: {0,1}n - {0,1} is somewhat hard to compute on average by small circuits, then the corresponding k-wise direct product function fk(x1,...,xk)=(f(x1),...,f(xk)) (where each xi - {0,1}n) is significantly harder to compute on average by slightly smaller circuits. We prove a fully uniform version of the Direct-Product Theorem with information-theoretically optimal parameters, up to constant factors. Namely, we show that for given k and ε, there is an efficient randomized algorithm A with the following property. Given a circuit C that computes fk on at least ε fraction of inputs, the algorithm A outputs with probability at least 3/4 a list of O(1/ε) circuits such that at least one of the circuits on the list computes f on more than 1-δ fraction of inputs, for δ = O((log 1/ε)/k). Moreover, each output circuit is an AC0 circuit (of size poly(n,k,log 1/δ,1/ε)), with oracle access to the circuit C. Using the Goldreich-Levin decoding algorithm [5], we also get a fully uniform version of Yao's XOR Lemma [18] with optimal parameters, up to constant factors. Our results simplify and improve those in [10]. Our main result may be viewed as an efficient approximate, local, list-decoding algorithm for direct-product codes (encoding a function by its values on all k-tuples) with optimal parameters. We generalize it to a family of "derandomized" direct-product codes, which we call intersection codes, where the encoding provides values of the function only on a subfamily of k-tuples. The quality of the decoding algorithm is then determined by sampling properties of the sets in this family and their intersections. As a direct consequence of this generalization we obtain the first derandomized direct product result in the uniform setting, allowing hardness amplification with only constant (as opposed to a factor of k) increase in the input length. Finally, this general setting naturally allows the decoding of concatenated codes, which further yields nearly optimal derandomized amplification.