## 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 minentropy. 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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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings |

Publisher | Springer Verlag |

Pages | 366-379 |

Number of pages | 14 |

ISBN (Print) | 9783540742074 |

State | Published - 2007 |

Externally published | Yes |

Event | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 - Princeton, NJ, United States Duration: 20 Aug 2007 → 22 Aug 2007 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4627 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 |
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Country/Territory | United States |

City | Princeton, NJ |

Period | 20/08/07 → 22/08/07 |