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The Coin Problem and Pseudorandomness for Branching Programs

Joshua Brody, Elad Verbin
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.  ... 
doi:10.1109/focs.2010.10 dblp:conf/focs/BrodyV10 fatcat:ea4qck6ffvfndcd4cro5yuuloy

Two Sides of the Coin Problem

Gil Cohen, Anat Ganor, Ran Raz, Marc Herbstritt
2014 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Brody 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.  ... 
doi:10.4230/lipics.approx-random.2014.618 dblp:conf/approx/CohenGR14 fatcat:4u5sen5yszbkvnau6bjp3mccle

Pseudorandomness for Regular Branching Programs via Fourier Analysis [chapter]

Omer Reingold, Thomas Steinke, Salil Vadhan
2013 Lecture Notes in Computer Science  
Our 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  ... 
doi:10.1007/978-3-642-40328-6_45 fatcat:2h62one4rvgwdogc2lztfhfq5m

Pseudorandomness for Regular Branching Programs via Fourier Analysis [article]

Omer Reingold, Thomas Steinke, Salil Vadhan
2013 arXiv   pre-print
+o(1) 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  ... 
arXiv:1306.3004v2 fatcat:z6n6fbio2facld4szkvi6knmh4

Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs

Thomas Steinke, Salil Vadhan, Andrew Wan, Marc Herbstritt
2014 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We present an explicit 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 .  ... 
doi:10.4230/lipics.approx-random.2014.885 dblp:conf/approx/SteinkeVW14 fatcat:p4hfp6do5be6bimdu3ikxbquvq

Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs

László Babai, Noam Nisant, Márió Szegedy
1992 Journal of computer and system sciences (Print)  
We also give a new length-width trade-off 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.  ... 
doi:10.1016/0022-0000(92)90047-m fatcat:e4mxtxumwjhh3btjtgng2itrdy

Pseudorandomness and Fourier Growth Bounds for Width 3 Branching Programs [article]

Thomas Steinke, Salil Vadhan, Andrew Wan
2014 arXiv   pre-print
We present an explicit 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  ... 
arXiv:1405.7028v1 fatcat:kdfrrdtjfnh73m6qtgcwvantaa

Using nondeterminism to amplify hardness

Alexander Healy, Salil Vadhan, Emanuele Viola
2004 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04  
We revisit 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  ... 
doi:10.1145/1007352.1007389 dblp:conf/stoc/HealyVV04 fatcat:qtcq6cji2ffozoxfag2heaumfm

Using Nondeterminism to Amplify Hardness

Alexander Healy, Salil Vadhan, Emanuele Viola
2006 SIAM journal on computing (Print)  
We revisit 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  ... 
doi:10.1137/s0097539705447281 fatcat:ulfaf5cazzeppah3svqctlb3im

Pseudorandom Bit Generators That Fool Modular Sums [chapter]

Shachar Lovett, Omer Reingold, Luca Trevisan, Salil Vadhan
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.  ... 
doi:10.1007/978-3-642-03685-9_46 fatcat:xafdsn5p2fcadib7fd7ayrd2ey

The Distinguishability of Product Distributions by Read-Once Branching Programs

John Steinberger
2013 2013 IEEE Conference on Computational Complexity  
However, 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.  ... 
doi:10.1109/ccc.2013.33 dblp:conf/coco/Steinberger13 fatcat:rmcf7jxpkjcevibk6cet7rrafy

Pseudorandomness for Read-Once, Constant-Depth Circuits [article]

Sitan Chen, Thomas Steinke, Salil Vadhan
2015 arXiv   pre-print
Our work makes use of Fourier analytic techniques 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  ... 
arXiv:1504.04675v2 fatcat:6dctoleq3vcwviagefkltfltom

My favorite ten complexity theorems of the past decade [chapter]

Lance Fortnow
1994 Lecture Notes in Computer Science  
We review 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.  ... 
doi:10.1007/3-540-58715-2_130 fatcat:oqj4nco6ozaxpclwtipoor2egi

One Way Function Candidate based on the Collatz Problem [article]

Rade Vuckovac
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.  ... 
arXiv:1801.05079v3 fatcat:bnta3zhaanc65f3jdphdxdwzda

Coin Theorems and the Fourier Expansion [article]

Rohit Agrawal
2019 arXiv   pre-print
shows that known level 1 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.  ... 
arXiv:1906.03743v1 fatcat:aiayfo2mtjchjkpxhz32klkehi
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