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Recently, there has been a lot of interest in studying the transfer of assets across different blockchains in the form of cross-chain atomic swaps. Unfortunately, the current candidates of atomic swaps (hash-lock time contracts) offer no privacy; the identities as well as the exact trade that happened between any two parties is publicly visible. In this work, we explore the different notions of privacy that we can hope for in an atomic swap protocol. Concretely, we define an atomic swap as a two-party protocol and formalize the different notions of privacy in the form of anonymity, confidentiality and indistinguishability of swap transactions. As a building block, we abstract out the primitive of Atomic Release of Secrets ( ARS ) which captures atomic exchange of a secret for a pre-decided transaction. We then show how ARS can be used to build privacy-preserving cross-chain swaps. We also show that the recently introduced notion of adapter signatures [Poe18, War17] is a concrete instantiation of ARS under the framework of Schnorr signatures [Sch91] and thus, construct a private cross-chain swap using Schnorr signatures.