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The Coin Problem and Pseudorandomness for Branching Programs

2010
*
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
*

*The*

*coin*

*problem*is open

*and*interesting in models that cannot compute

*the*majority function. In this paper we study

*the*

*coin*

*problem*in

*the*model of read-once width-w

*branching*

*programs*. ...

*The*

*Coin*

*Problem*is

*the*following

*problem*: a

*coin*is given, which lands on head with probability either 1/2 + β or 1/2 − β. ... We would like to thank Peter

*and*

*the*group

*for*their gracious hospitality. ...

##
###
Two Sides of the Coin Problem

2014
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

Brody

doi:10.4230/lipics.approx-random.2014.618
dblp:conf/approx/CohenGR14
fatcat:4u5sen5yszbkvnau6bjp3mccle
*and*Verbin [FOCS 2010] proved that a length n, width w read once*branching**program*cannot solve*the**coin**problem**for*β < O(1/(log n) 3w ). ... In other words,*the*strong independence assumption in*the**coin**problem*is completely redundant in*the*model of read once*branching**programs*. ... Motivated by*the*construction of*pseudorandom*generators*for*ROBP, Brody*and*Verbin [BV10] considered*the**coin**problem**for**the*model of ROBP with bounded width. ...##
###
Pseudorandomness for Regular Branching Programs via Fourier Analysis
[chapter]

2013
*
Lecture Notes in Computer Science
*

Our

doi:10.1007/978-3-642-40328-6_45
fatcat:2h62one4rvgwdogc2lztfhfq5m
*pseudorandom*generator is similar to*the*one used by Gopalan et al. (FOCS 2012)*for*read-once CNFs, but*the*analysis is quite different; ours is based on Fourier analysis of*branching**programs*. ... 2+o(1)*for*arbitrary*branching**programs*of size s). ... Relation to*Coin*Theorem*The**Coin*Theorem of Brody*and*Verbin [6] shows that general (non-regular) oblivious, read-once*branching**programs*of width w cannot distinguish n independent*and*unbiased*coin*...##
###
Pseudorandomness for Regular Branching Programs via Fourier Analysis
[article]

2013
*
arXiv
*
pre-print

+o(1)

arXiv:1306.3004v2
fatcat:z6n6fbio2facld4szkvi6knmh4
*for*arbitrary*branching**programs*of size s). ... We present an explicit*pseudorandom*generator*for*oblivious, read-once, permutation*branching**programs*of constant width that can read their input bits in any order. ... Relation to*Coin*Theorem*The**Coin*Theorem of Brody*and*Verbin [6] shows that general (non-regular) oblivious, read-once*branching**programs*of width w cannot distinguish n independent*and*unbiased*coin*...##
###
Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs

2014
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

We present an explicit

doi:10.4230/lipics.approx-random.2014.885
dblp:conf/approx/SteinkeVW14
fatcat:p4hfp6do5be6bimdu3ikxbquvq
*pseudorandom*generator*for*oblivious, read-once, width-3*branching**programs*, which can read their input bits in any order.*The*generator has seed lengthÕ(log 3 n). ... Our work generalizes a recent result of Reingold, Steinke,*and*Vadhan (RANDOM '13)*for*permutation*branching**programs*. ... Algorithm*for*G a,b,n,ε : {0, 1} s a,b,n,ε → {0, 1} n . Parameters: n ∈ N, ε > 0. Input: A random seed of length s a,b,n,ε . ...##
###
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs

1992
*
Journal of computer and system sciences (Print)
*

We also give a new length-width trade-off

doi:10.1016/0022-0000(92)90047-m
fatcat:e4mxtxumwjhh3btjtgng2itrdy
*for*oblivious*branching**programs*; in particular, our bound implies new lower bounds on*the*size of arbitrary*branching**programs*, or on*the*size of Boolean formulas ... We give several applications of*the*lower bounds.*The*tirst application is a*pseudorandom*generator*for*Logspace. ... bounds*for**branching**programs*. ...##
###
Pseudorandomness and Fourier Growth Bounds for Width 3 Branching Programs
[article]

2014
*
arXiv
*
pre-print

We present an explicit

arXiv:1405.7028v1
fatcat:kdfrrdtjfnh73m6qtgcwvantaa
*pseudorandom*generator*for*oblivious, read-once, width-3*branching**programs*, which can read their input bits in any order.*The*generator has seed length Õ( ^3 n ). ... Our work generalizes a recent result of Reingold, Steinke,*and*Vadhan (RANDOM '13)*for*permutation*branching**programs*. ... Introduction*Pseudorandom*Generators*for*Space-Bounded Computation A major open*problem*in*the*theory of*pseudorandomness*is to construct an "optimal"*pseudorandom*generator*for*space-bounded computation ...##
###
Using nondeterminism to amplify hardness

2004
*
Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04
*

We revisit

doi:10.1145/1007352.1007389
dblp:conf/stoc/HealyVV04
fatcat:qtcq6cji2ffozoxfag2heaumfm
*the**problem*of hardness amplification in N P, as recently studied by O'Donnell (STOC '02). ... We also prove impossibility results demonstrating that both our use of nondeterminism*and**the*hypothesis that f is balanced are necessary*for*"black-box" hardness amplification procedures (such as ours ... A generator G : {0, 1} l → ({0, 1} n ) k is -*pseudorandom*against*branching**programs*of size s*and*block-size n if*for*every*branching**program*B of size s*and*block-size n: Pr[B(G(U l )) = 1] − Pr[B(U ...##
###
Using Nondeterminism to Amplify Hardness

2006
*
SIAM journal on computing (Print)
*

We revisit

doi:10.1137/s0097539705447281
fatcat:ulfaf5cazzeppah3svqctlb3im
*the**problem*of hardness amplification in N P, as recently studied by O'Donnell (STOC '02). ... We also prove impossibility results demonstrating that both our use of nondeterminism*and**the*hypothesis that f is balanced are necessary*for*"black-box" hardness amplification procedures (such as ours ... A generator G : {0, 1} l → ({0, 1} n ) k is -*pseudorandom*against*branching**programs*of size s*and*block-size n if*for*every*branching**program*B of size s*and*block-size n: Pr[B(G(U l )) = 1] − Pr[B(U ...##
###
Pseudorandom Bit Generators That Fool Modular Sums
[chapter]

2009
*
Lecture Notes in Computer Science
*

*The*

*problem*we study is a generalization of

*the*

*problem*of constructing small bias distributions [NN], which are solutions to

*the*M = 2 case. ... (A similar construction was independently discovered by Meka

*and*Zuckerman [MZ]).

*The*second construction works

*for*every M, n,

*and*has seed length O(log n + log(M/ ) log(M log(1/ ))). ... Acknowledgments We thank Emanuele Viola

*for*drawing our attention to this

*problem*. We thank Andrej Bogdanov

*for*helpful discussions. ...

##
###
The Distinguishability of Product Distributions by Read-Once Branching Programs

2013
*
2013 IEEE Conference on Computational Complexity
*

However,

doi:10.1109/ccc.2013.33
dblp:conf/coco/Steinberger13
fatcat:rmcf7jxpkjcevibk6cet7rrafy
*the**problem*seems both natural*and*interesting*for*space-bounded distinguishers,*and*in particular*for*distinguishers having only a constant amount of space. ... As their main result, Brody*and*Verbin [7] give bounds on*the*ability of constant width read-once*branching**programs*(ROBPs) to distinguish biased*coins*. ... I would like to thank Kevin Matulef, Joshua Brody*and*Elad Verbin*for*helpful conversations at all stages of this work. ...##
###
Pseudorandomness for Read-Once, Constant-Depth Circuits
[article]

2015
*
arXiv
*
pre-print

Our work makes use of Fourier analytic techniques

arXiv:1504.04675v2
fatcat:6dctoleq3vcwviagefkltfltom
*for**pseudorandomness*introduced by Reingold, Steinke,*and*Vadhan (RANDOM '13) to show that*the*generator of Gopalan et al. ...*The*previous best seed length known*for*this model was Õ(^D+4 n), obtained by Trevisan*and*Xue (CCC '13)*for*all of AC^0 (not just read-once). ... In Section 4 we verify that*the*analysis in [26] of their*pseudorandom*restriction generator*for**branching**programs*applies to our setting of readonce AC 0*and*use*the*results of*the*preceding sections ...##
###
My favorite ten complexity theorems of the past decade
[chapter]

1994
*
Lecture Notes in Computer Science
*

We review

doi:10.1007/3-540-58715-2_130
fatcat:oqj4nco6ozaxpclwtipoor2egi
*the*past ten years in computational complexity theory by focusing on ten theorems that*the*author enjoyed*the*most. ... We use each of*the*theorems as a springboard to discuss work done in various areas of complexity theory. ... Lokam,Dieter Van Melkebeek*and*Sophie Laplante*for*their comments*and*help on this paper. ...##
###
One Way Function Candidate based on the Collatz Problem
[article]

2018
*
arXiv
*
pre-print

*The*analysis shows why

*the*

*problem*is mathematically so inaccessible

*and*how

*the*algorithm conditional

*branching*structure can be used to construct one way functions. ...

*The*one way function based on

*the*Collatz

*problem*is proposed. ...

*For*example:

*pseudorandom*number generators,

*pseudorandom*functions

*and*various cryptographic protocols. Informally OWF is easy to compute given input x. ...

##
###
Coin Theorems and the Fourier Expansion
[article]

2019
*
arXiv
*
pre-print

shows that known level 1

arXiv:1906.03743v1
fatcat:aiayfo2mtjchjkpxhz32klkehi
*and*total influence bounds*for*some classes of interest (such as constant-width read-once*branching**programs*) in fact follow as a black-box from*the*corresponding*coin*theorems ... In this note we compare two measures of*the*complexity of a class F of Boolean functions studied in (unconditional)*pseudorandomness*: F's ability to distinguish between biased*and*uniform*coins*(*the**coin*... its proof in this note;*and**the*anonymous reviewers*for*their helpful comments*and*suggestions. ...
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