High Entropy Random Selection Protocols

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.