## Extractors with weak random seeds (2005)

Venue: | In Proceedings of the 37th Annual ACM Symposium on Theory of Computing |

Citations: | 62 - 6 self |

### BibTeX

@INPROCEEDINGS{Raz05extractorswith,

author = {Ran Raz},

title = {Extractors with weak random seeds},

booktitle = {In Proceedings of the 37th Annual ACM Symposium on Theory of Computing},

year = {2005},

pages = {11--20}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show how to extract random bits from two or more independent weak random sources in cases where only one source is of linear min-entropy and all other sources are of logarithmic min-entropy. Our main results are as follows: 1. A long line of research, starting by Nisan and Zuckerman [15], gives explicit constructions of seeded-extractors, that is, extractors that use a short seed of truly random bits to extract randomness from a weak random source. For every such extractor E, with seed of length d, we construct an extractor E ′ , with seed of length d ′ = O(d), that achieves the same parameters as E but only requires the seed to be of min-entropy larger than (1/2 + δ) · d ′ (rather than fully random), where δ is an arbitrary small constant. 2. Fundamental results of Chor and Goldreich and Vazirani [6, 22] show how to extract Ω(n) random bits from two (independent) sources of length n and min-entropy larger than (1/2 + δ) · n, where δ is an arbitrary small constant. We show how to extract Ω(n) random bits (with optimal probability of error) when only one source is of min-entropy (1/2 + δ) · n and the other source is of logarithmic min-entropy. 1 3. A recent breakthrough of Barak, Impagliazzo and Wigderson [4] shows how to extract Ω(n) random bits from a constant number of (independent) sources of length n and min-entropy larger than δn, where δ is an arbitrary small constant. We show how to extract Ω(n) random bits (with optimal probability of error) when only one source is of min-entropy δn and all other (constant number of) sources are of logarithmic min-entropy. 4. A very recent result of Barak, Kindler, Shaltiel, Sudakov and Wigderson [5] shows how to extract a constant number of random bits from three (independent) sources of length n and min-entropy larger than δn, where δ is an arbitrary small constant. We show how to extract Ω(n) random bits, with sub-constant probability of error, from one source of min-entropy δn and two sources of logarithmic min-entropy.