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Nearly Optimal Pseudorandomness From Hardness

Dean Doron, Dana Moshkovitz, Justin Oh, David Zuckerman
2022 Journal of the ACM  
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  ...  Such a slowdown is nearly optimal for t close to n , since under standard complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown.  ...  thankful to Ronen Shaltiel for pointing out an error in a previous version, for suggesting an elegant way of signiicantly simplifying the contents of Section 6, and for very useful discussions about hardness  ... 
doi:10.1145/3555307 fatcat:xxuzk5vsqrdsnp2i7q7uow3ixy

Optimal Cryptographic Hardness of Learning Monotone Functions [chapter]

Dana Dachman-Soled, Homin K. Lee, Tal Malkin, Rocco A. Servedio, Andrew Wan, Hoeteck Wee
2008 Lecture Notes in Computer Science  
Some of our results show cryptographic hardness of learning polynomial-size monotone circuits to accuracy only slightly greater than 1/2 + 1/ √ n; this accuracy bound is close to optimal by known positive  ...  Over the years a range of positive algorithmic results have been obtained for learning various classes of monotone Boolean functions from uniformly distributed random examples.  ...  Theorem 5 is thus nearly optimal in terms of the size of the constant-depth circuits for which it establishes hardness of learning.  ... 
doi:10.1007/978-3-540-70575-8_4 fatcat:v6zgngwvpzb27jhxalzfytr6d4

Randomness extractors -- applications and constructions

Avi Wigderson, Marc Herbstritt
2009 Foundations of Software Technology and Theoretical Computer Science  
Randomness extractors are efficient algorithms which convert weak random sources into nearly perfect ones.  ...  We will highlight some of the applications, as well as recent constructions achieving near-optimal extraction.  ...  It is not hard to see that a seed is essential for an extractor to work in this general setting.  ... 
doi:10.4230/lipics.fsttcs.2009.2340 dblp:conf/fsttcs/Wigderson09 fatcat:bhsprjlp5zeblpzlsepybhojyy

Hardness vs randomness

Noam Nisan, Avi Wigderson
1994 Journal of computer and system sciences (Print)  
It stretches a short string of truly random bits into a long string that looks random to any algorithm from a complexity class C (e.g., P, NC, PSPACE, ...) using an arbitrary function that is hard for  ...  We present a simple new construction of a pseudorandom bit generator.  ...  The Main Lemma Given a "hard" function, it is intuitively easy to generate one pseudorandom bit from it since the value of the function must look random to any small circuit.  ... 
doi:10.1016/s0022-0000(05)80043-1 fatcat:vqgigqdbhzehzohlyvvhqga7cu

Hardness vs. randomness

N. Nisan, A. Wigderson
1988 [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science  
It stretches a short string of truly random bits into a long string that looks random to any algorithm from a complexity class C (e.g., P, NC, PSPACE, ...) using an arbitrary function that is hard for  ...  We present a simple new construction of a pseudorandom bit generator.  ...  The Main Lemma Given a "hard" function, it is intuitively easy to generate one pseudorandom bit from it since the value of the function must look random to any small circuit.  ... 
doi:10.1109/sfcs.1988.21916 dblp:conf/focs/NisanW88 fatcat:qtvqglf5yzbnbmjrapdz6rzc7a

Pseudorandom Generators from Regular One-Way Functions: New Constructions with Improved Parameters [chapter]

Yu Yu, Xiangxue Li, Jian Weng
2013 Lecture Notes in Computer Science  
be ε-hard to invert, we give a neat (and tighter) proof for the folklore construction of pseudorandom generator of seed length Θ(n) by making a single call to the underlying one-way function.  ...  Here the number of calls is also optimal by matching the lower bounds of Holenstein and Sinha (FOCS 2012).  ...  -Pseudorandomness extraction from X.  ... 
doi:10.1007/978-3-642-42045-0_14 fatcat:lvnvsb5zynfqjosjvkh5fwijuy

Pseudorandom generators from regular one-way functions: New constructions with improved parameters

Yu Yu, Xiangxue Li, Jian Weng
2015 Theoretical Computer Science  
to be ε-hard to invert, we give a neat (and tighter) proof for the folklore construction of pseudorandom generator of seed length Θ(n) by making a single call to the underlying one-way function. • For  ...  Here the number of calls is also optimal by matching the lower bounds of Holenstein and Sinha (FOCS 2012).  ...  . • Pseudorandomness extraction from X.  ... 
doi:10.1016/j.tcs.2014.12.013 fatcat:aafkuq6inrflzoc3quqtoxxtpa

Pseudorandomness and Combinatorial Constructions [article]

Luca Trevisan
2006 arXiv   pre-print
Despite this evidence for the power of random choices, the computational theory of pseudorandomness shows that, under certain complexity-theoretic assumptions, every probabilistic algorithm has an efficient  ...  In this survey paper we describe connections between the conditional "derandomization" results of the computational theory of pseudorandomness and unconditional explicit constructions of certain combinatorial  ...  Shannon [59] independently applied the same idea to prove the existence of encoding schemes that can optimally correct from errors in a noisy channel and optimally compress data.  ... 
arXiv:cs/0601100v1 fatcat:jcidgz6cdzdfjdrbac6hkuwxlu

Page 1587 of Mathematical Reviews Vol. , Issue 93c [page]

1993 Mathematical Reviews  
ISBN 0-262-14051-9 This book considers “pseudorandom generators”, that is, processes which generate sequences of bits nearly indistinguishable from sequences of truly random bits (given certain resource  ...  Marius Zimand (1-RCT-C) 93c:68032 68Q15 03D15 65C10 Nisan, Noam (IL-HEBR-C) * Using hard problems to create pseudorandom generators. ACM Distinguished Dissertations.  ... 

Almost-natural proofs

Timothy Y. Chow
2011 Journal of computer and system sciences (Print)  
In this paper, we show that under the same pseudorandomness hypothesis, there do exist nearly-linear-time-computable Boolean function properties with only slightly lower density (namely, 2 −q(n) for a  ...  quasi-polynomial function q) that not only exclude P /poly, but even separate NP from P /poly.  ...  Specifically, under the same 2 n -hard pseudorandomness assumption made in the original Razborov-Rudich paper, we can explicitly exhibit a nearly-linear-timecomputable property that separates NP from P  ... 
doi:10.1016/j.jcss.2010.06.017 fatcat:zwwsd3zhpfbkfpiubbucoybe2a

Almost-Natural Proofs

Timothy Y. Chow
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
In this paper, we show that under the same pseudorandomness hypothesis, there do exist nearly-linear-time-computable Boolean function properties with only slightly lower density (namely, 2 −q(n) for a  ...  quasi-polynomial function q) that not only exclude P /poly, but even separate NP from P /poly.  ...  Specifically, under the same 2 n -hard pseudorandomness assumption made in the original Razborov-Rudich paper, we can explicitly exhibit a nearly-linear-timecomputable property that separates NP from P  ... 
doi:10.1109/focs.2008.16 dblp:conf/focs/Chow08 fatcat:bl2s4tzucbhp3p3f4ubjtfhweu

Almost-natural proofs [article]

Timothy Y. Chow
2009 arXiv   pre-print
Razborov and Rudich have shown that so-called "natural proofs" are not useful for separating P from NP unless hard pseudorandom number generators do not exist.  ...  Specifically, under the same pseudorandomness assumption that Razborov and Rudich make, a simple, explicit property that we call "discrimination" suffices to separate P/poly from NP; discrimination is  ...  NP unless 2 n ǫ -hard pseudorandom number generators do not exist.  ... 
arXiv:0805.1385v3 fatcat:gv3pxtayo5gi5lkhpkciizaohm

Page 611 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
So the choice of a suitable algorithm for random combinations is as hard as be- fore. P.  ...  The other two versions run nearly as fast and progressively remove a mea- sure zero locus of failures present in the first version.  ... 

Pseudorandomness and the Minimum Circuit Size Problem

Rahul Santhanam, Michael Wagner
2020 Innovations in Theoretical Computer Science  
(Pseudorandomness from Zero-Error Average-Case Hardness) We show that for a given size function s, the following are equivalent: Pseudorandom distributions supported on strings describable by s(O(n))-size  ...  We also show that for a certain natural variant of MCSP, there is a polynomial-time reduction from approximating the problem well in the worst case to solving it on average.  ...  Theorem 16 gives a connection from zero-error average-case hardness of MCSP to succinct pseudorandom distributions.  ... 
doi:10.4230/lipics.itcs.2020.68 dblp:conf/innovations/Santhanam20 fatcat:pgatorc5k5hlrknytijhfgjufy

On derandomization and average-case complexity of monotone functions

George Karakostas, Jeff Kinne, Dieter van Melkebeek
2012 Theoretical Computer Science  
We prove similar results in the settings of pseudorandom generators and average-case hard functions -that a pseudorandom generator secure against monotone circuits is also secure with somewhat weaker parameters  ...  against general circuits, and that an average-case hard function for monotone circuits is also hard with somewhat weaker parameters for general circuits.  ...  Proof of Theorem 5 We follow the standard proof from the general setting and keep track of monotonicity to verify the final circuit is monotone or anti-monotone.  ... 
doi:10.1016/j.tcs.2012.02.017 fatcat:6mk5ovihtzaxvhrexnd2vt5tci
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