A fast and simple randomized parallel algorithm for the maximal independent set problem
Journal of Algorithms
Unbiased bits from sources of weak randomness and probabilistic communication complexity
SIAM Journal on Computing - Special issue on cryptography
Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Journal of Computer and System Sciences
Randomness-optimal oblivious sampling
Proceedings of the workshop on Randomized algorithms and computation
Extracting all the randomness and reducing the error in Trevisan's extractors
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Extracting randomness: a survey and new constructions
Journal of Computer and System Sciences
Extractors and pseudo-random generators with optimal seed length
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators
SIAM Journal on Discrete Mathematics
Loss-less condensers, unbalanced expanders, and extractors
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Extractors and pseudorandom generators
Journal of the ACM (JACM)
Extractors: optimal up to constant factors
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Extracting randomness via repeated condensing
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Simple Extractors for All Min-Entropies and a New Pseudo-Random Generator
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Encryption against Storage-Bounded Adversaries from On-Line Strong Extractors
Journal of Cryptology
Extracting Randomness Using Few Independent Sources
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Deterministic Extractors for Bit-Fixing Sources by Obtaining an Independent Seed
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Extractors with weak random seeds
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Deterministic extractors for small-space sources
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The bit extraction problem or t-resilient functions
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Simple construction of almost k-wise independent random variables
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Generalized strong extractors and deterministic privacy amplification
IMA'05 Proceedings of the 10th international conference on Cryptography and Coding
Extracting randomness from multiple independent sources
IEEE Transactions on Information Theory
Exposure-Resilient Extractors and the Derandomization of Probabilistic Sublinear Time
Computational Complexity
Simple Direct Reduction of String (1,2)-OT to Rabin's OT without Privacy Amplification
ICITS '08 Proceedings of the 3rd international conference on Information Theoretic Security
Extracting Computational Entropy and Learning Noisy Linear Functions
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Deterministic extractors for independent-symbol sources
IEEE Transactions on Information Theory
Computational randomness from generalized hardcore sets
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
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In this paper, we consider the task of deterministically extracting randomness from sources consisting of a sequence of n independent symbols from {0,1}d. The only randomness guarantee on such a source is that the whole source has min-entropy k. We give an explicit deterministic extractor which can extract Ω(logk – logd – loglog(1/ε)) bits with error ε, for any n,d,k∈ℕ and ε∈(0,1). For sources with a larger min-entropy, we can extract even more randomness. When k≥n1/2+γ, for any constant γ∈(0,1/2), we can extract m=k–O(d log(1/ε)) bits with any error $\varepsilon \ge 2^{-\Omega(n^{\gamma})}$. When k≥logcn, for some constant c0, we can extract m=k–d (1/ε)O(1) bits with any error ε≥k−−Ω(1). Our results generalize those of Kamp & Zuckerman and Gabizon et al. which only work for bit-fixing sources (with d=1 and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a non-explicit deterministic extractor which can extract m=k–O(log(1/ε)) bits whenever k=ω(d+log(n/ε)). Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss k–m = Ω(log(1/ε)). This generalizes a lower bound of Radhakrishnan & Ta-Shma with respect to general sources.