## Abstract

We study the two party problem of randomly selecting a string among all the strings of length n. We want the protocol to have the property that the output distribution has high entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, entropy. In the literature the randomness guarantee is usually expressed as being close to the uniform distribution or in terms of resiliency. The notion of entropy is not directly comparable to that of resiliency, but we establish a connection between the two that allows us to compare our protocols with the existing ones. We construct an explicit protocol that yields entropy n − O(1) and has 4 log^{∗} n rounds, improving over the protocol of Goldreich et al. [3] that also achieves this entropy but needs O(n) rounds. Both these protocols need O(n^{2}) bits of communication. Next we reduce the communication in our protocols. We show the existence, non-explicitly, of a protocol that has 6 rounds, 2n + 8 log n bits of communication and yields entropy n − O(log n) and min-entropy n/2 − O(log n). Our protocol achieves the same entropy bound as the recent, also non-explicit, protocol of Gradwohl et al. [4], however achieves much higher min-entropy: n/2−O(log n) versus O(log n). Finally we exhibit very simple explicit protocols. We connect the security parameter of these geometric protocols with the well studied Kakeya problem motivated by harmonic analysis and analytical number theory. We are only able to prove that these protocols have entropy 3n/4 but still n/2 − O(log n) min-entropy. Therefore they do not perform as well with respect to the explicit constructions of Gradwohl et al. [4] entropy-wise, but still have much better min-entropy. We conjecture that these simple protocols achieve n − o(n) entropy. Our geometric construction and its relation to the Kakeya problem follows a new and different approach to the random selection problem than any of the previously known protocols.

Original language | English |
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Journal | Dagstuhl Seminar Proceedings |

Volume | 7411 |

State | Published - 2008 |

Externally published | Yes |

Event | Algebraic Methods in Computational Complexity 2007 - Warden, Germany Duration: 7 Oct 2007 → 12 Oct 2007 |

### Bibliographical note

Publisher Copyright:© 2008 Dagstuhl Seminar Proceedings.

### Funding

We would like to thank to Troy Lee and John Tromp for useful discussions and Navin Goyal for pointing us to the problem of Kakeya. We also thank anonymous referees for valuable comments on the preliminary version of this paper. Part of the work was done while the second, third, fourth, and sixth author were visiting CWI, Amsterdam. H. Buhrman was supported by EU project QAP and BRICKS project AFM1. H. Buhrman and M. Koucky´ were supported in part by an NWO VICI grant (639.023.302). M. Koucky´ was supported in part by grant GA Cˇ R 201/07/P276, 201/05/0124, project No. 1M0021620808 of MSˇ MT Cˇ R and Institutional Research Plan No. AV0Z10190503. The work of N. Vereshchagin was supported in part by RFBR grant 06-01-00122. We would like to thank to Troy Lee and John Tromp for useful discussions and Navin Goyal for pointing us to the problem of Kakeya. We also thank anonymous referees for valuable comments on the preliminary version of this paper. Part of the work was done while the second, third, fourth, and sixth author were visiting CWI, Amsterdam. H. Buhrman was supported by EU project QAP and BRICKS project AFM1. H. Buhrman and M. Koucký were supported in part by an NWO VICI grant (639.023.302). M. Koucký was supported in part by grant GA ČR 201/07/P276, 201/05/0124, project No. 1M0021620808 of MŠMT ČR and Institutional Research Plan No. AV0Z10190503. The work of N. Vereshchagin was supported in part by RFBR grant 06-01-00122.

Funders | Funder number |
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MSˇ MT Cˇ R | AV0Z10190503 |

European Commission | |

Ministerstvo Školství, Mládeže a Tělovýchovy | |

Russian Foundation for Basic Research | 06-01-00122 |

Nederlandse Organisatie voor Wetenschappelijk Onderzoek | 639.023.302, 201/05/0124, GA Cˇ R 201/07/P276, 1M0021620808 |