{"title":"关于“极高端”prg和快速非随机化所需的硬度假设","authors":"Ronen Shaltiel, Emanuele Viola","doi":"10.4230/LIPIcs.ITCS.2022.116","DOIUrl":null,"url":null,"abstract":"The hardness vs.~randomness paradigm aims to explicitly construct pseudorandom generators $G:\\{0,1\\}^r \\rightarrow \\{0,1\\}^m$ that fool circuits of size $m$, assuming the existence of explicit hard functions. A ``high-end PRG'' with seed length $r=O(\\log m)$ (implying BPP=P) was achieved in a seminal work of Impagliazzo and Wigderson (STOC 1997), assuming the high-end hardness assumption: there exist constants $0<\\beta<1<B$, and functions computable in time $2^{B \\cdot n}$ that cannot be computed by circuits of size $2^{\\beta \\cdot n}$. Recently, motivated by fast derandomization of randomized algorithms, Doron et al.~(FOCS 2020) and Chen and Tell (STOC 2021), construct ``extreme high-end PRGs'' with seed length $r=(1+o(1))\\cdot \\log m$, under qualitatively stronger assumptions. We study whether extreme high-end PRGs can be constructed from the following scaled version of the assumption which we call ``the extreme high-end hardness assumption'', and in which $\\beta=1-o(1)$ and $B=1+o(1)$. We give a partial negative answer, showing that certain approaches cannot yield a black-box proof. (A longer abstract with more details appears in the PDF file)","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"126 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"On Hardness Assumptions Needed for \\\"Extreme High-End\\\" PRGs and Fast Derandomization\",\"authors\":\"Ronen Shaltiel, Emanuele Viola\",\"doi\":\"10.4230/LIPIcs.ITCS.2022.116\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The hardness vs.~randomness paradigm aims to explicitly construct pseudorandom generators $G:\\\\{0,1\\\\}^r \\\\rightarrow \\\\{0,1\\\\}^m$ that fool circuits of size $m$, assuming the existence of explicit hard functions. A ``high-end PRG'' with seed length $r=O(\\\\log m)$ (implying BPP=P) was achieved in a seminal work of Impagliazzo and Wigderson (STOC 1997), assuming the high-end hardness assumption: there exist constants $0<\\\\beta<1<B$, and functions computable in time $2^{B \\\\cdot n}$ that cannot be computed by circuits of size $2^{\\\\beta \\\\cdot n}$. Recently, motivated by fast derandomization of randomized algorithms, Doron et al.~(FOCS 2020) and Chen and Tell (STOC 2021), construct ``extreme high-end PRGs'' with seed length $r=(1+o(1))\\\\cdot \\\\log m$, under qualitatively stronger assumptions. We study whether extreme high-end PRGs can be constructed from the following scaled version of the assumption which we call ``the extreme high-end hardness assumption'', and in which $\\\\beta=1-o(1)$ and $B=1+o(1)$. We give a partial negative answer, showing that certain approaches cannot yield a black-box proof. (A longer abstract with more details appears in the PDF file)\",\"PeriodicalId\":11639,\"journal\":{\"name\":\"Electron. Colloquium Comput. Complex.\",\"volume\":\"126 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-11-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electron. Colloquium Comput. Complex.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.ITCS.2022.116\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electron. Colloquium Comput. Complex.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.ITCS.2022.116","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5
摘要
硬度vs随机性范式旨在显式地构建伪随机生成器G: {0,1} r→{0,1}m，假设存在显式硬函数，它愚弄大小为m的电路。Impagliazzo和Wigderson (STOC 1997)在一项开创性的工作中实现了种子长度r = O (log m)(意味着BPP=P)的“高端PRG”，假设高端硬度假设:存在常数0 < β < 1 < B，并且在2b·n时间内可计算的函数不能被2 β·n大小的电路计算。最近，Doron等人(FOCS 2020)和Chen和Tell (STOC 2021)在随机算法的快速非随机化的激励下，在更强的定性假设下，构建了种子长度为r = (1 + o(1))·log m的“极端高端prg”。研究了β = 1−0(1)和B = 1+ 0(1)的硬度假设能否构造出极值高端prg，我们称之为极值高端硬度假设。为了证明这一点，我们从硬函数中建立了(一般)黑箱PRG构造的一个新性质:在固定硬函数的几个比特的同时，可以固定构造的许多输出比特。这一特性将PRG结构与典型的提取器结构区分开来，这可能解释了为什么PRG结构很难设计。m→，1 r m Ω(1) g2: {0,1 r 2→{，1 m。第一个PRG是从高端假设到(1)的PRG是单向函数。我们表明，在高端情况下，放大的黑盒证明必须有m个查询。已知从是和使用放大，不能用于构建prgg2从极端高端硬度假设
On Hardness Assumptions Needed for "Extreme High-End" PRGs and Fast Derandomization
The hardness vs.~randomness paradigm aims to explicitly construct pseudorandom generators $G:\{0,1\}^r \rightarrow \{0,1\}^m$ that fool circuits of size $m$, assuming the existence of explicit hard functions. A ``high-end PRG'' with seed length $r=O(\log m)$ (implying BPP=P) was achieved in a seminal work of Impagliazzo and Wigderson (STOC 1997), assuming the high-end hardness assumption: there exist constants $0<\beta<1
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