## Abstract

We study the two party problem of randomly selecting a common string among all the strings of length n. We want the protocol to have the property that the output distribution has high Shannon entropy or high min entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, Shannon entropy and simultaneously min entropy close to n/2. In the literature the randomness guarantee is usually expressed in terms of “resilience”. The notion of Shannon entropy is not directly comparable to that of resilience, 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 Shannon entropy n- O(1) and has O(log ^{∗}n) rounds, improving over the protocol of Goldreich et al. (SIAM J Comput 27: 506–544, 1998) that also achieves this entropy but needs O(n) rounds. Both these protocols need O(n^{2}) bits of communication. Next we reduce the number of rounds and the length of communication in our protocols. We show the existence, non-explicitly, of a protocol that has 6 rounds, O(n) bits of communication and yields Shannon entropy n- O(log n) and min entropy n/ 2 - O(log n). Our protocol achieves the same Shannon entropy bound as, also non-explicit, protocol of Gradwohl et al. (in: Dwork (ed) Advances in Cryptology—CRYPTO ‘06, 409–426, Technical Report , 2006), however achieves much higher min entropy: n/ 2 - O(log n) versus O(log n). Finally we exhibit a very simple 3-round explicit “geometric” protocol with communication length O(n). We connect the security parameter of this protocol with the well studied Kakeya problem motivated by Harmonic Analysis and Analytic Number Theory. We prove that this protocol has Shannon entropy n- o(n). 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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Pages (from-to) | 667-694 |

Number of pages | 28 |

Journal | Algorithmica |

Volume | 83 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2021 |

### Bibliographical note

Funding Information: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 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 partially supported by the Russian Academic Excellence Project ‘5-100’ and by the RFBR Grant 19-01-00563.

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