High Entropy Random Selection Protocols

In this paper, we construct protocols for two parties that do not trust each other, to generate random variables with high Shannon entropy. We improve known bounds for the trade off between the number of rounds, length of communication and the entropy of the outcome

Inverting Onto Functions and Polynomial Hierarchy

In this paper we construct an oracle under which the polynomial hierarchy is infinite but there are non-invertible polynomial time computable multivalued onto functions

Catalytic space: non-determinism and hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation

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(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 Kakey

Randomised Individual Communication Complexity

In this paper we study the individual communication complexity of the following problem. Alice receives an input string x and Bob an input string y, and Alice has to output y. For deterministic protocols it has been shown in Buhrman et al. (2004), that C(y) many bits need to be exchanged even if the actual amount of information C(y|x) is much smaller than C(y). It turns out that for randomised protocols the situation is very diffe

Inverting Onto Functions and Polynomial Hierarchy

In this paper we construct an oracle under which the polynomial hierarchy is infinite but there are non-invertible polynomial time computable multivalued onto functions

High Entropy Random Selection Protocols

In this paper, we construct protocols for two parties that do not trust each other, to generate random variables with high Shannon entropy. We improve known bounds for the trade off between the number of rounds, length of communication and the entropy of the outcome