Goos, Pitassi and Watson (ITCS, 2015) have recently introduced the notion of Zero-Information Arthur-Merlin Protocols (ZAM). In this model, which can be viewed as a private version of the standard Arthur-Merlin communication complexity game, Alice and Bob are holding a pair of inputs $x$ and $y$ respectively, and Merlin, the prover, attempts to convince them that some public function $f$ evaluates to 1 on $(x,y)$. In addition to standard completeness and soundness, Goos et al., require a ``zero-knowledge'' property which asserts that on each yes-input, the distribution of Merlin's proof leaks no information about the inputs $(x,y)$ to an external observer.

In this paper, we relate this new notion to the well-studied model of Private Simultaneous Messages (PSM) that was originally suggested by Feige, Naor and Kilian (STOC, 1994). Roughly speaking, we show that the randomness complexity of ZAM corresponds to the communication complexity of PSM, and that the communication complexity of ZAM corresponds to the randomness complexity of PSM. This relation works in both directions where different variants of PSM are being used. As a secondary contribution, we reveal new connections between different variants of PSM protocols which we believe to be of independent interest.

Our results give rise to better ZAM protocols based on existing PSM protocols, and to better protocols for conditional disclosure of secrets (a variant of PSM) from existing ZAM's.