### Abstract

A merger is a probabilistic procedure which extracts the randomness out of any (arbitrarily correlated) set of random variables, as long as one of them is uniform. Our main result is an efficient, simple, optimal (to constant factors) merger, which, for k random vairables on n bits each, uses a O(log(nk)) seed, and whose error is 1/nk. Our merger can be viewed as a derandomized version of the merger of Lu, Reingold, Vadhan and Wigderson (2003). Its analysis generalizes the recent resolution of the Kakeya problem in finite fields of Dvir (2008). Following the plan set forth by Ta-Shma (1996), who defined mergers as part of this plan, our merger provides the last "missing link" to a simple and modular construction of extractors for all entropies, which is optimal to constant factors in all parameters. This complements the elegant construction of optimal extractor by Guruswami, Vadhan and Umans (2007). We also give simple extensions of our merger in two directions. First, we generalize it to handle the case where no source is uniform - in that case the merger will extract the entropy present in the most random of the given sources. Second, we observe that the merger works just as well in the computational setting, when the sources are efficiently samplable, and computational notions of entropy replace the information theoretic ones.

Original language | English (US) |
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Title of host publication | Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |

Pages | 625-633 |

Number of pages | 9 |

DOIs | |

State | Published - Dec 30 2008 |

Externally published | Yes |

Event | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States Duration: Oct 25 2008 → Oct 28 2008 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |
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Country | United States |

City | Philadelphia, PA |

Period | 10/25/08 → 10/28/08 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

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## Cite this

*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008*(pp. 625-633). [4690995] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2008.23