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Göös et al. (ITCS, ) have recently introduced the notion of  (). 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  and , respectively, and Merlin, the prover, attempts to convince them that some public function  evaluates to 1 on (, ). In addition to standard completeness and soundness, Göös 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 (, ) to an external observer. In this paper, we relate this new notion to the well-studied model of  () that was originally suggested by Feige et al. (STOC, ). Roughly speaking, we show that the randomness complexity of  corresponds to the communication complexity of  and that the communication complexity of  corresponds to the randomness complexity of . This relation works in both directions where different variants of  are being used. As a secondary contribution, we reveal new connections between different variants of  protocols which we believe to be of independent interest. Our results give rise to better  protocols based on existing  protocols, and to better protocols for conditional disclosure of secrets (a variant of ) from existing s.

From Private Simultaneous Messages to Zero-Information Arthur–Merlin Protocols and Back