Nearly Optimal Pseudorandomness From Hardness
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 ag...
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| Veröffentlicht in: | Proceedings / annual Symposium on Foundations of Computer Science S. 1057 - 1068 |
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01.11.2020
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Doron, Dean Oh, Justin Moshkovitz, Dana Zuckerman, David |
| Author_xml | – sequence: 1 givenname: Dean surname: Doron fullname: Doron, Dean email: ddoron@stanford.edu organization: Stanford University,Department of Computer Science,Stanford,CA,USA – sequence: 2 givenname: Dana surname: Moshkovitz fullname: Moshkovitz, Dana email: danama@cs.utexas.edu organization: University of Texas at Austin,Department of Computer Science,Austin,TX,USA – sequence: 3 givenname: Justin surname: Oh fullname: Oh, Justin email: sjo@cs.utexas.edu organization: University of Texas at Austin,Department of Computer Science,Austin,TX,USA – sequence: 4 givenname: David surname: Zuckerman fullname: Zuckerman, David email: diz@cs.utexas.edu organization: University of Texas at Austin,Department of Computer Science,Austin,TX,USA |
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| Snippet | Existing proofs that deduce \text{BPP} =\mathrm{P} from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large... |
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| SubjectTerms | Computer science derandomization Generators Hybrid power systems Integrated circuit modeling list-recoverable codes Probabilistic logic pseudo-entropy pseudorandom generators Runtime Testing |
| Title | Nearly Optimal Pseudorandomness From Hardness |
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