TY - JOUR

T1 - Loss-less condensers, unbalanced expanders, and extractors

AU - Ta-Shma, A.

AU - Umans, C.

AU - Zuckerman, D.

PY - 2001

Y1 - 2001

N2 - An extractor is a procedure which extracts randomness from a defective random source using a few additional random bits. Explicit extractor constructions have numerous applications and obtaining such constructions is an important de-randomization goal. Trevisan recently introduced an elegant extractor construction, but the number of truly random bits required is suboptimal when the input source has low minentropy. Significant progress toward overcoming this bottleneck has been made, but so far has required complicated recursive techniques that lose the simplicity of Trevisan's construction. We give a clean method for overcoming this bottleneck by constructing loss-less condensers, which compress the n-bit input source without losing any min-entropy, using O(log n) additional random bits. Our condensers are built using a simple modification of Trevisan's construction, and yield the best extractor constructions to date. Loss-less condensers also produce unbalanced bipartite expander graphs with small (poly logarithmic) degree D and very strong expansion of (1 - ε)D. We give other applications of our construction, including dispersers with entropy loss O(log n), depth two super-concentrators whose size is within a polylog of optimal, and an improved hardness of approximation result.

AB - An extractor is a procedure which extracts randomness from a defective random source using a few additional random bits. Explicit extractor constructions have numerous applications and obtaining such constructions is an important de-randomization goal. Trevisan recently introduced an elegant extractor construction, but the number of truly random bits required is suboptimal when the input source has low minentropy. Significant progress toward overcoming this bottleneck has been made, but so far has required complicated recursive techniques that lose the simplicity of Trevisan's construction. We give a clean method for overcoming this bottleneck by constructing loss-less condensers, which compress the n-bit input source without losing any min-entropy, using O(log n) additional random bits. Our condensers are built using a simple modification of Trevisan's construction, and yield the best extractor constructions to date. Loss-less condensers also produce unbalanced bipartite expander graphs with small (poly logarithmic) degree D and very strong expansion of (1 - ε)D. We give other applications of our construction, including dispersers with entropy loss O(log n), depth two super-concentrators whose size is within a polylog of optimal, and an improved hardness of approximation result.

UR - http://www.scopus.com/inward/record.url?scp=0034832181&partnerID=8YFLogxK

U2 - 10.1145/380752.380790

DO - 10.1145/380752.380790

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.conferencearticle???

AN - SCOPUS:0034832181

SN - 0734-9025

SP - 143

EP - 152

JO - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

JF - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

Y2 - 6 July 2001 through 8 July 2001

ER -