Mixing Transactions with Arbitrary Values on Blockchains

Wangze Ni, Peng Cheng, Lei Chen

Abstract: Due to the transparency of blockchain, adversaries can observe the details of a transaction, and then utilize the amount as a unique quasi-identifier to make deanonymization. Nowadays, to obscure the linkages between receivers and senders within a transaction on the blockchain, mixing services are widely applied in many real applications to enhance cryptocurrencies' anonymity. The basic idea of mixing services is to hide an output within several other outputs in a transaction such that adversaries cannot distinguish them by their amounts since they are purposely selected to have the same amount. For a set of original outputs with different amounts, mixing services need to decompose them into a set of decomposed outputs, where any decomposed output has some other decomposed outputs with the same amount. Since the transaction fee is related to the number of outputs, we are motivated to decompose original outputs into a minimal set of decomposed outputs, which is challenging to guarantee the privacy-preserving effect at the same time. In this paper, we formally define the anonymity-aware output decomposition (AA-OD) problem, which aims to find a c-decomposition with a minimum number of decomposed outputs for a given original output set. A c-decomposition guarantees that for any original output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$o$</tex> , there are at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$c$</tex> of all decomposed outputs with an amount of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x$</tex> coming from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$o$</tex> . We prove that the AA-OD problem is NP-hard. Thus, we propose an approximation algorithm, namely Boggart <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> Boggart is a magical creature in J. K. Rowling's Harry Potter series who can shift his shape and no one knows what it looks like., to solve the AA-OD problem with a (2/c + 3)-approximation bound on the number of decomposed outputs. We verify the efficiency and effectiveness of our approach through comprehensive experiments on both real and synthetic data sets.