Foundations of Coin Mixing Services

Abstract

Coin mixing services allow users to mix their cryptocurrency coins and thus enable unlinkable payments in a way that prevents tracking of honest users' coins by both the service provider and the users themselves. The easy bootstrapping of new users and backwards compatibility with cryptocurrencies (such as Bitcoin) with limited support for scripts are attractive features of this architecture, which has recently gained considerable attention in both academia and industry. A recent work of Tairi et al. [IEEE S&P 2021] formalizes the notion of a coin mixing service and proposes A2L, a new cryptographic protocol that simultaneously achieves high efficiency and interoperability. In this work, we identify a gap in their formal model and substantiate the issue by showing two concrete counterexamples: we show how to construct two encryption schemes that satisfy their definitions but lead to a completely insecure system. To amend this situation, we investigate secure constructions of coin mixing services. First, we develop the notion of blind conditional signatures (BCS), which acts as the cryptographic core for coin mixing services. We propose game-based security definitions for BCS and propose A2L+, a modified version of the protocol by Tairi et al. that satisfies our security definitions. Our analysis is in an idealized model (akin to the algebraic group model) and assumes the hardness of the one-more discrete logarithm problem. Finally, we propose A2L-UC, another construction of BCS that achieves the stronger notion of UC-security (in the standard model), albeit with a significant increase in computation cost. This suggests that constructing a coin mixing service protocol secure under composition requires more complex cryptographic machinery than initially thought.