TY - JOUR
T1 - Extractors from Reed-Muller codes
AU - Ta-Shma, Amnon
AU - Zuckerman, David
AU - Safra, Shmuel
N1 - Funding Information:
* Corresponding author. E-mail addresses: [email protected] (A. Ta-Shma), [email protected] (D. Zuckerman), [email protected] (S. Safra). URLs: http://www.cs.tau.ac.il/~amnon (A. Ta-Shma), http://www.cs.utexas.edu/~diz (D. Zuckerman), http://www.cs.tau.ac.il/~safra (S. Safra). 1 Some of this work was done while the author was at the University of California at Berkeley, and supported in part by a David and Lucile Packard Fellowship for Science and Engineering and NSF NYI Grant CCR-9457799. 2 Some of this work was done while the author was on leave at the University of California at Berkeley. Supported in part by a David and Lucile Packard Fellowship for Science and Engineering, NSF Grants CCR-0310960, CCR-9912428, and CCR-9457799, and an Alfred P. Sloan Research Fellowship.
PY - 2006/8
Y1 - 2006/8
N2 - Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. Previous research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, researchers had failed to build extractors directly from a good code without using other tools from pseudorandomness. We succeed in constructing an extractor directly from a Reed-Muller code. To do this, we develop a novel proof technique. Furthermore, our construction is the first to achieve degree close to linear. In contrast, the best previous constructions brought the log of the degree within a constant of optimal, which gives polynomial degree. This improvement is important for certain applications. For example, it was used [E. Mossel, C. Umans, On the complexity of approximating the VC dimension, J. Comput. System Sci. 65 (2002) 660-671] to show that approximating VC dimension to within a factor of N1 - δ is AM-hard for any positive δ.
AB - Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. Previous research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, researchers had failed to build extractors directly from a good code without using other tools from pseudorandomness. We succeed in constructing an extractor directly from a Reed-Muller code. To do this, we develop a novel proof technique. Furthermore, our construction is the first to achieve degree close to linear. In contrast, the best previous constructions brought the log of the degree within a constant of optimal, which gives polynomial degree. This improvement is important for certain applications. For example, it was used [E. Mossel, C. Umans, On the complexity of approximating the VC dimension, J. Comput. System Sci. 65 (2002) 660-671] to show that approximating VC dimension to within a factor of N1 - δ is AM-hard for any positive δ.
KW - Derandomization
KW - Expanders
KW - Extractors
KW - Inapproximability
KW - Pseudorandomness
KW - Reed-Muller codes
UR - http://www.scopus.com/inward/record.url?scp=33646520833&partnerID=8YFLogxK
U2 - 10.1016/j.jcss.2005.05.010
DO - 10.1016/j.jcss.2005.05.010
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AN - SCOPUS:33646520833
SN - 0022-0000
VL - 72
SP - 786
EP - 812
JO - Journal of Computer and System Sciences
JF - Journal of Computer and System Sciences
IS - 5
ER -