High Entropy Random Selection Protocols
- PDF / 2,221,108 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 67 Downloads / 212 Views
High Entropy Random Selection Protocols Harry Buhrman, et al. [full author details at the end of the article] Received: 18 January 2018 / Accepted: 17 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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(n2 ) 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.
Matthias Christandl work done while visiting CWI. Michal Koucký work done while visiting CWI. Zvi Lotker work done while visiting CWI. Nikolay Vereshchagin work was partially done while visiting CWI.
13
Vol.:(0123456789)
Algorithmica
1 Introduction We study the following communication problem. Alice and Bob want to select a common random string. They are not at the same location so they do not see what the other player does. They communicate messages according to some protocol, and in the end they output a string of n bits which is a function of the messages communicated. This string should be as random as possible, and in our case we measure the amount of randomness by Shannon entropy or min entropy of the probability distribution that is generated by this protocol. The messages they communicate may depend on random experiments the players pe
Data Loading...
