TY - GEN

T1 - Nearly optimal pseudorandomness from hardness

AU - Doron, Dean

AU - Moshkovitz, Dana

AU - Oh, Justin

AU - Zuckerman, David

N1 - Funding Information:
Dean Doron is supported by NSF Grant CCF-1763311, and most of this work was done while at UT Austin, supported by NSF Grant CCF-1705028. Dana Moshkovitz is supported in part by NSF Grant CCF-1705028 and CCF-1648712. Justin Oh is supported by NSF Grant CCF-1705028. David Zuckerman is supported in part by NSF Grant CCF-1705028 and a Simons Investigator Award (#409864).
Publisher Copyright:
© 2020 IEEE.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - Existing proofs that deduce text{BPP} = mathrm{P} from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against randomized single-valued nondeterministic (SVN) circuits, we convert any randomized algorithm over inputs of length n running in time t geq n to a deterministic one running in time t{2+ alpha} for an arbitrarily small constant alpha > 0. Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits. Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1 + α)log s, under the assumption that there exists a function f E that requires randomized SVN circuits of size at least 2(1-α')n, where. α=O(α'). The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.

AB - Existing proofs that deduce text{BPP} = mathrm{P} from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against randomized single-valued nondeterministic (SVN) circuits, we convert any randomized algorithm over inputs of length n running in time t geq n to a deterministic one running in time t{2+ alpha} for an arbitrarily small constant alpha > 0. Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits. Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1 + α)log s, under the assumption that there exists a function f E that requires randomized SVN circuits of size at least 2(1-α')n, where. α=O(α'). The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.

KW - derandomization

KW - list-recoverable codes

KW - pseudo-entropy

KW - pseudorandom generators

UR - http://www.scopus.com/inward/record.url?scp=85100334036&partnerID=8YFLogxK

U2 - 10.1109/FOCS46700.2020.00102

DO - 10.1109/FOCS46700.2020.00102

M3 - Conference contribution

AN - SCOPUS:85100334036

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 1057

EP - 1068

BT - Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020

PB - IEEE Computer Society

T2 - 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020

Y2 - 16 November 2020 through 19 November 2020

ER -