TY - JOUR
T1 - Extracting randomness using few independent sources
AU - Barak, Boaz
AU - Impagliazzo, Russell
AU - Wigderson, Avi
PY - 2006
Y1 - 2006
N2 - In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ > 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0,1}n each having min-entropy δn. These extractors output n bits that are 2-n close to uniform. This construction uses several results from additive number theory, and in particular a recent result of Bourgain et al. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with δ-entropy ω(log n), and outputs every possible m-bit string with positive probability. The main tool we use is a variant of the "stepping-up lemma" of Erdos and Hajnal used in establishing a lower bound on the Ramsey number for hypergraphs.
AB - In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ > 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0,1}n each having min-entropy δn. These extractors output n bits that are 2-n close to uniform. This construction uses several results from additive number theory, and in particular a recent result of Bourgain et al. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with δ-entropy ω(log n), and outputs every possible m-bit string with positive probability. The main tool we use is a variant of the "stepping-up lemma" of Erdos and Hajnal used in establishing a lower bound on the Ramsey number for hypergraphs.
KW - Ramsey graphs
KW - Randomness extractors
KW - Sum-product theorem
UR - http://www.scopus.com/inward/record.url?scp=34247566037&partnerID=8YFLogxK
U2 - 10.1137/S0097539705447141
DO - 10.1137/S0097539705447141
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AN - SCOPUS:34247566037
SN - 0097-5397
VL - 36
SP - 1095
EP - 1118
JO - SIAM Journal on Computing
JF - SIAM Journal on Computing
IS - 4
ER -