TY - GEN

T1 - Dispersers for affine sources with sub-polynomial entropy

AU - Shaltiel, Ronen

PY - 2011

Y1 - 2011

N2 - We construct an explicit disperser for affine sources over double-struck F 2 n with entropy k=2 log0.9n=n o(1). This is a polynomial time computable function D:double-struck F 2 n → {0, 1} such that for every affine space V of double-struck F 2 n that has dimension at least k, D(V)= {0,1}. This improves the best previous construction of Ben-Sasson and Kop party (STOC 2009) that achieved k = Ω(n 4/5).Our technique follows a high level approach that was developed in Barak, Kindler, Shaltiel, Sudakov and Wigderson (J. ACM 2010) and Barak, Rao, Shaltiel and Wigderson (STOC 2006) in the context of dispersers for two independent general sources. The main steps are: • Adjust the high level approach to make it suitable for affine sources. • Implement a "challenge-response game" for affine sources (in the spirit of the two aforementioned papers that introduced such games for two independent general sources). • In order to implement the game, we construct extractors for affine block-wise sources. For this we use ideas and components by Rao (CCC 2009). • Combining the three items above, we obtain dispersers for affine sources with entropy larger than √n. We use a recursive win-win analysis in the spirit of Rein gold, Shaltiel and Wigderson (SICOMP 2006) and Barak, Rao, Shaltiel and Wigderson (STOC 2006) to get affine dispersers with entropy less than √n.

AB - We construct an explicit disperser for affine sources over double-struck F 2 n with entropy k=2 log0.9n=n o(1). This is a polynomial time computable function D:double-struck F 2 n → {0, 1} such that for every affine space V of double-struck F 2 n that has dimension at least k, D(V)= {0,1}. This improves the best previous construction of Ben-Sasson and Kop party (STOC 2009) that achieved k = Ω(n 4/5).Our technique follows a high level approach that was developed in Barak, Kindler, Shaltiel, Sudakov and Wigderson (J. ACM 2010) and Barak, Rao, Shaltiel and Wigderson (STOC 2006) in the context of dispersers for two independent general sources. The main steps are: • Adjust the high level approach to make it suitable for affine sources. • Implement a "challenge-response game" for affine sources (in the spirit of the two aforementioned papers that introduced such games for two independent general sources). • In order to implement the game, we construct extractors for affine block-wise sources. For this we use ideas and components by Rao (CCC 2009). • Combining the three items above, we obtain dispersers for affine sources with entropy larger than √n. We use a recursive win-win analysis in the spirit of Rein gold, Shaltiel and Wigderson (SICOMP 2006) and Barak, Rao, Shaltiel and Wigderson (STOC 2006) to get affine dispersers with entropy less than √n.

KW - Dispersers

KW - Explicit construction

KW - Pseudorandomness

KW - Randomness extractors

UR - http://www.scopus.com/inward/record.url?scp=84863333239&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2011.37

DO - 10.1109/FOCS.2011.37

M3 - Conference contribution

AN - SCOPUS:84863333239

SN - 9780769545714

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 247

EP - 256

BT - Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011

T2 - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011

Y2 - 22 October 2011 through 25 October 2011

ER -