Paper 2011/068

Rational Secret Sharing with Honest Players over an Asynchronous Channel

William K. Moses Jr. and C. Pandu Rangan

Abstract

We consider the problem of rational secret sharing introduced by Halpern and Teague \cite{HT04}, where the players involved in secret sharing play only if it is to their advantage. This can be characterized in the form of preferences. Players would prefer to get the secret than to not get it and secondly with lesser preference, they would like as few other players to get the secret as possible. Several positive results have already been published to efficiently solve the problem of rational secret sharing. However, only a handful of papers have touched upon the use of an asynchronous broadcast channel, and in those papers, either the protocol involved cryptographic primitives \cite{FKN10} or else the protocol required the dealer to be interactively involved \cite{MSR08a}. However, \cite{OPRV09} did handle such a case through the use of an honest minority of players, but in their paper, they had placed a restriction on the number of honest players that could take part in relation to the total number number of players active in the protocol. In our paper, we propose an $m$-out-of-$n$ rational secret sharing scheme which can function over an asynchronous broadcast channel without the use of cryptographic primitives and with a non-interactive dealer. This is possible because our scheme uses a small number, $k+1$, of honest players. The protocol is resilient to coalitions of size up to $k$ and furthermore it is $\varepsilon$-resilient to coalitions of size up to $m-1$. The protocol will have a strict Nash equilibrium with probability $Pr(\frac{k+1}{n})$ and an $\varepsilon$-Nash equilibrium with probability $Pr(\frac{n-k-1}{n})$. Furthermore, our protocol is immune to backward induction. Later on in the paper, we extend our results to include malicious players as well. We also show that our protocol handles the possibility of a player deviating in order to force another player to get a wrong value. This type of deviation was discussed and handled by Asharov and Lindell \cite{AL09} by increasing the number of rounds. However, our protocol handles this in what we believe to be a more time efficient manner.