TY - GEN
T1 - Uniform direct product theorems
T2 - 40th Annual ACM Symposium on Theory of Computing, STOC 2008
AU - Impagliazzo, Russell
AU - Jaiswal, Ragesh
AU - Kabanets, Valentine
AU - Wigderson, Avi
PY - 2008
Y1 - 2008
N2 - 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 fc-wise direct product function fk (x1,..., xk) = (f(x 1),..., f(xk)) (where each xi ε {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 G that computes fk on at least e 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 / on more than 1 - δ fraction of inputs, for δ = O((log 1/ε)/k). Moreover, each output circuit is an ACo 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 fc-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 fe-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 fc) increase in the input length. Finally, this general setting naturally allows the decoding of concatenated codes, which further yields nearly optimal derandomized amplification.
AB - 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 fc-wise direct product function fk (x1,..., xk) = (f(x 1),..., f(xk)) (where each xi ε {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 G that computes fk on at least e 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 / on more than 1 - δ fraction of inputs, for δ = O((log 1/ε)/k). Moreover, each output circuit is an ACo 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 fc-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 fe-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 fc) increase in the input length. Finally, this general setting naturally allows the decoding of concatenated codes, which further yields nearly optimal derandomized amplification.
UR - http://www.scopus.com/inward/record.url?scp=51849156726&partnerID=8YFLogxK
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AN - SCOPUS:51849156726
SN - 9781605580470
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 579
EP - 588
BT - STOC'08
Y2 - 17 May 2008 through 20 May 2008
ER -