On invertible sampling and adaptive security

Secure multiparty computation (MPC) is one of the most general and well studied problems in cryptography. We focus on MPC protocols that are required to be secure even when the adversary can adaptively corrupt parties during the protocol, and under the assumption that honest parties cannot reliably erase their secrets prior to corruption. Previous feasibility results for adaptively secure MPC in this setting applied either to deterministic functionalities or to randomized functionalities which satisfy a certain technical requirement. The question whether adaptive security is possible for all functionalities was left open. We provide the first convincing evidence that the answer to this question is negative, namely that some (randomized) functionalities cannot be realized with adaptive security. We obtain this result by studying the following related invertible sampling problem: given an efficient sampling algorithm A, obtain another sampling algorithm B such that the output of B is computationally indistinguishable from the output of A, but B can be efficiently inverted (even if A cannot). This invertible sampling problem is independently motivated by other cryptographic applications. We show, under strong but well studied assumptions, that there exist efficient sampling algorithms A for which invertible sampling as above is impossible. At the same time, we show that a general feasibility result for adaptively secure MPC implies that invertible sampling is possible for every A, thereby reaching a contradiction and establishing our main negative result.