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
Verifying and decoding in constant depth
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
Chernoff-Type Direct Product Theorems
Journal of Cryptology
New direct-product testers and 2-query PCPs
Proceedings of the forty-first annual ACM symposium on Theory of computing
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The classical direct product theorem for circuits says that if a Boolean function $f:\{0,1\}^n\to\{0,1\}$ is somewhat hard to compute on average by small circuits, then the corresponding $k$-wise direct product function $f^k(x_1,\dots,x_k)=(f(x_1),\dots,f(x_k))$ (where each $x_i\in\{0,1\}^n$) 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 $\epsilon$, there is an efficient randomized algorithm $A$ with the following property. Given a circuit $C$ that computes $f^k$ on at least $\epsilon$ fraction of inputs, the algorithm $A$ outputs with probability at least $3/4$ a list of $O(1/\epsilon)$ circuits such that at least one of the circuits on the list computes $f$ on more than $1-\delta$ fraction of inputs, for $\delta=O((\log1/\epsilon)/k)$; moreover, each output circuit is an $\mathsf{AC}^0$ circuit (of size $\mathrm{poly}(n,k,\log1/\delta,1/\epsilon)$), with oracle access to the circuit $C$. Using the Goldreich-Levin decoding algorithm [O. Goldreich and L. A. Levin, A hard-core predicate for all one-way functions, in Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing, Seattle, 1989, pp. 25-32], we also get a fully uniform version of Yao's XOR lemma [A. C. Yao, Theory and applications of trapdoor functions, in Proceedings of the Twenty-Third Annual IEEE Symposium on Foundations of Computer Science, Chicago, 1982, pp. 80-91] with optimal parameters, up to constant factors. Our results simplify and improve those in [R. Impagliazzo, R. Jaiswal, and V. Kabanets, Approximately list-decoding direct product codes and uniform hardness amplification, in Proceedings of the Forty-Seventh Annual IEEE Symposium on Foundations of Computer Science, Berkeley, CA, 2006, pp. 187-196]. 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.