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Deterministic Extractors for Affine Sources over Large Fields [chapter]

Ariel Gabizon
2010 Monographs in Theoretical Computer Science. An EATCS Series  
We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n c (where c is a large enough constant).  ...  An (n, k)-affine source over a finite field F is a random variable X = (X 1 , ..., X n ) ∈ F n , which is uniformly distributed over an (unknown) k-dimensional affine subspace of F n .  ...  Finally, a big big thanks to Asaf Nussboim for introducing the first author to Weil's theorems.  ... 
doi:10.1007/978-3-642-14903-0_3 fatcat:vhndowpghrhq3inel3dquoif5u

Deterministic extractors for affine sources over large fields

Ariel Gabizon, Ran Raz
2008 Combinatorica  
We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n c (where c is a large enough constant).  ...  An (n, k)-affine source over a finite field F is a random variable X = (X 1 , ..., X n ) ∈ F n , which is uniformly distributed over an (unknown) k-dimensional affine subspace of F n .  ...  Finally, a big big thanks to Asaf Nussboim for introducing the first author to Weil's theorems.  ... 
doi:10.1007/s00493-008-2259-3 fatcat:dgm4unne35c4dbnyam53pad4cu

Deterministic Extractors for Affine Sources over Large Fields

A. Gabizon, R. Raz
46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)  
We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n c (where c is a large enough constant).  ...  An (n, k)-affine source over a finite field F is a random variable X = (X 1 , ..., X n ) ∈ F n , which is uniformly distributed over an (unknown) k-dimensional affine subspace of F n .  ...  Finally, a big big thanks to Asaf Nussboim for introducing the first author to Weil's theorems.  ... 
doi:10.1109/sfcs.2005.31 dblp:conf/focs/GabizonR05 fatcat:ffqm5xlns5b67bum5iwb2jivie

An Introduction to Randomness Extractors [chapter]

Ronen Shaltiel
2011 Lecture Notes in Computer Science  
Goal: Design extractors for "large" families of "interesting" sources.  ...  Existence of deterministic extractors: Let m ≤ n be integers, let ϵ > 0 and let C be a class with at most 2 poly(n/ϵ) distributions over {0, 1} n .  ...  Affine sources: Let F q be the finite field with q elements. Affine sources are distributions that are uniform over some affine subspace of the vector space F n q .  ... 
doi:10.1007/978-3-642-22012-8_2 fatcat:ldxe2dhplfdhvcdepfval6cxne

Extractors and Rank Extractors for Polynomial Sources [chapter]

Ariel Gabizon
2010 Monographs in Theoretical Computer Science. An EATCS Series  
A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields.  ...  It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields.  ...  We thank Andrej Bogdanov and Gil Alon for helpful conversations.  ... 
doi:10.1007/978-3-642-14903-0_4 fatcat:vzv5tremgnhfhcahi7brcmmr7q

Extractors and Rank Extractors for Polynomial Sources

Zeev Dvir, Ariel Gabizon, Avi Wigderson
2007 Foundations of Computer Science (FOCS), IEEE Annual Symposium on  
A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields.  ...  It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields.  ...  We thank Andrej Bogdanov and Gil Alon for helpful conversations.  ... 
doi:10.1109/focs.2007.4389479 fatcat:yqz4gbohmjcbzkzo3634tdwmua

Extractors and Rank Extractors for Polynomial Sources

Zeev Dvir, Ariel Gabizon, Avi Wigderson
2007 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)  
A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields.  ...  It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields.  ...  We thank Andrej Bogdanov and Gil Alon for helpful conversations.  ... 
doi:10.1109/focs.2007.9 dblp:conf/focs/EzraS07 fatcat:xofqsysjanb6xbgxsyegvetgca

Extractors And Rank Extractors For Polynomial Sources

Zeev Dvir, Ariel Gabizon, Avi Wigderson
2009 Computational Complexity  
A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields.  ...  It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields.  ...  We thank Andrej Bogdanov and Gil Alon for helpful conversations.  ... 
doi:10.1007/s00037-009-0258-4 fatcat:6id75dzsvfg5xkwrjwhegh6mfe

Improved Extractors for Affine Lines [article]

Ariel Gabizon
2013 arXiv   pre-print
We say that a function D:F^n0,1 is a 1 if for every affine line l⊆^n, D(X) is -close to uniform when X is uniformly distributed over l.  ...  Let F be the field of q elements.  ...  Acknowledgements I would like to thank Dana Moshkovitz and Ran Raz for helpful comments and discussions.  ... 
arXiv:1311.5622v2 fatcat:i24dtl7mrbbjjjw5ndzuvujfxe

Leakage-Resilient Secret Sharing in Non-Compartmentalized Models

Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, Huaxiong Wang, Daniel Wichs, Yael Tauman Kalai, Adam D. Smith
2020 Conference on Information-Theoretic Cryptography  
For affine leakage functions, we obtain schemes with threshold access structure that are leakage-resilient as long as there is a substantial difference between the total amount of information obtained  ...  Specifically, we have a threshold scheme with parameters similar to Shamir's scheme and is leakage-resilient against affine leakage.  ...  For structured entropy source, for example, the source is a flat distribution over an affine subspace of the universal space {0, 1} n , there exists a single function that turns any such source distribution  ... 
doi:10.4230/lipics.itc.2020.7 dblp:conf/icits/LinCGSW20 fatcat:mv7zwyq6xbd6lcgekvhone7wem

Extractors for Low-Weight Affine Sources

Anup Rao
2009 2009 24th Annual IEEE Conference on Computational Complexity  
This gives new extractors for low entropy bit-fixing sources with exponentially small error, a parameter that is important for the application of these extractors to cryptography.  ...  We give polynomial time computable extractors for low-weight affince sources. A distribution is affine if it samples a random points from some unknown low dimensional subspace of F n 2 .  ...  sources over GF (2) (0.5 + α)n, for positive constant α 2 −Ω(n) Ω(n) [KZ] Extractor for affine sources over a large field, |F| > n 20 Extractor for low-weight affine sources over GF (2) k > log c (n)  ... 
doi:10.1109/ccc.2009.36 dblp:conf/coco/Rao09 fatcat:e6ml6i7uivdbhfln7fnsyushqq

Two-sources Randomness Extractors for Elliptic Curves [article]

Abdoul Aziz Ciss
2014 arXiv   pre-print
This paper studies the task of two-sources randomness extractors for elliptic curves defined over finite fields K, where K can be a prime or a binary field.  ...  In other words, for a ginven elliptic curve E defined over a finite field F_q and two random points P ∈P and Q∈Q, where P and Q are two subgroups of E(F_q), our function extracts the least significant  ...  Section 3 introduces our first contribution, ie. a new construction of a two-source deterministic randomness extractor for elliptic curves defined over prime fields.  ... 
arXiv:1404.2226v2 fatcat:t5a5bi3hhrh6pkulzj6zu2qfne

Simulating independence

B. Barak, G. Kindler, R. Shaltiel, B. Sudakov, A. Wigderson
2010 Journal of the ACM  
We give the following new explicit constructions (namely, poly(n)-time computable functions) of deterministic extractors, dispersers and related objects. All work for any fixed rate δ > 0.  ...  A distribution X over binary strings of length n has minentropy k if every string has probability at most 2 −k in X. 1 We say that X is a δ-source if its rate k/n is at least δ.  ...  Temporary unrealistic assumption: We'll assume a deterministic 1-source extractor (called ext * ) for any linear entropy sources.  ... 
doi:10.1145/1734213.1734214 fatcat:odaxswxlvrbyjlusmoynqadyci

Simulating independence

Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, Avi Wigderson
2005 Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05  
We give the following new explicit constructions (namely, poly(n)-time computable functions) of deterministic extractors, dispersers and related objects. All work for any fixed rate δ > 0.  ...  A distribution X over binary strings of length n has minentropy k if every string has probability at most 2 −k in X. 1 We say that X is a δ-source if its rate k/n is at least δ.  ...  Temporary unrealistic assumption: We'll assume a deterministic 1-source extractor (called ext * ) for any linear entropy sources.  ... 
doi:10.1145/1060590.1060592 dblp:conf/stoc/BarakKSSW05 fatcat:6lziswiaf5fpbd33eydzexvtwi

How to get more mileage from randomness extractors

Ronen Shaltiel
2008 Random structures & algorithms (Print)  
A deterministic randomness extractor for C is a function E : {0, 1} n → {0, 1} m such that for any X in C the distribution E(X) is statistically close to the uniform distribution.  ...  Using our transformation we give an explicit construction of a two-source extractor E : {0, 1} n × {0, 1} n → {0, 1} m such that for every two independent distributions X 1 and X 2 over {0, 1} n with min-entropy  ...  many discussions on various aspects of randomness extractors.  ... 
doi:10.1002/rsa.20207 fatcat:ekydvlvjzfbuthkgi6kzvoqe6a
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