We consider the classical secret sharing problem in the case where all agents are selfish but rational. In recent work, Kol and Naor show that, when there are two players, in the non-simultaneous communication model, i.e. when rushing is possible, there is no Nash equilibrium that ensures both players learn the secret. However, they describe a mechanism for this problem, for any number of players, that is an epsilon-Nash equilibrium, in that no player can gain more than epsilon utility by deviating from it. Unfortunately, the Kol and Naor mechanism, and, to the best of our knowledge, all previous mechanisms for this problem require each agent to send O(n) messages in expectation, where n is the number of agents. This may be problematic for some applications of rational secret sharing such as secure multi-party computation and simulation of a mediator. We address this issue by describing mechanisms for rational secret sharing that are designed for large n. Both of our results hold for n > 2, and are Nash equilbria, rather than just epsilon-Nash equilbria. Our first result is a mechanism for n-out-of-n rational secret sharing that is scalable in the sense that it requires each agent to send only an expected O(log n) bits. Moreover, the latency of this mechanism is O(log n) in expectation, compared to O(n) expected latency for the Kol and Naor result. Our second result is a mechanism for a relaxed variant of rational m-out-of-n secret sharing where m = Theta(n). It requires each processor to send O(log n) bits and has O(log n) latency. Both of our mechanisms are non-cryptographic, and are not susceptible to backwards induction.
In @cite_19 we overcame this difficulty, providing a algorithm for rational secret sharing, in which each player only sends @math bits per round and the expected number of rounds is constant (although each round takes @math time). Moreover, following the protocol is an @math -Nash equilibrium. Unfortunately, a certain bad event with small but constant probability caused some players, when they recognized it, to deviate from the protocol so that the equilibrium is not everlasting. This paper is the full version of @cite_19 . However, we improve on the work in @cite_19 in two ways. First, we remove all probability of error for @math -out-of- @math secret sharing, and improve the probability of error for @math -out-of- @math from a constant to an inverse polynomial. Second, we show that our new protocol is a Nash equilibrium, not just an @math -Nash equilibrium, as long as @math .
@cite_19
2001578254
We consider the classical secret sharing problem in the case where all agents are selfish but rational. In recent work, Kol and Naor show that in the non-simultaneous communciation model (i.e. when rushing is possible), there is no Nash equilibrium that ensures all agents learn the secret. However, they describe a mechanism for this problem that is an e-Nash equilibrium, i.e. it is close to an equilibrium in the sense that no player can gain more than e utility by deviating from it. Unfortunately, the Kol and Naor mechanism, and, to the best of our knowledge, all previous mechanisms for this problem require each agent to send O(n) messages in expectation, where n is the number of agents. This may be problematic for some applications of rational secret sharing such as secure multiparty computation and simulation of a mediator. We address this issue by describing a mechanism for rational n-out-of-n secret sharing that is an e-Nash equilibrium, and is scalable in the sense that it requires each agent to send only an expected O(1) bits. Moreover, the latency of our mechanism is O(log n) in expectation, compared to O(n) expected latency for the Kol and Naor result. We also design mechanisms for a relaxed variant of rational m-out-of-n secret sharing where m = Θ(n) that require each processor to send O(log n) bits and have O( n) latency. Our mechanisms are non-cryptographic, and are not susceptible to backwards induction.