Abstract Secure computation with penalties aims to achieve fairness in secure computation protocols by imposing monetary penalties on adversarial parties. A fundamental problem in constructing protocols that involves monetary entities is how to formalize them in a computational model. Bentov and Kumaresan (CRYPTO 2014) introduced a new computational model with special atomic entities called coins that capture currency and showed a protocol for secure computation with penalties in the model. Their model, secure computation with coins, assumes coins have several properties that are natural in the sense of expressing currency. However, on the other hand, it also requires the unnatural assumption “all coins are indistinguishable from each other" for a technical reason to accomplish the security proof. The motivation of this work is to remove this assumption to make the model a more general. We propose a new model, secure computation with color coins, such that coins have identifiable colors and do not hold the indistinguishability property. Furthermore, our model allows us to set different prices for each color. We show that secure computation with penalties can be realized in our model. To do this, we make some modifications to the ideal functionality of secure computation with penalties to adapt it to our model without losing the essence, i.e., fairness with penalties.

It is known that Bitcoin enables achieving fairness in secure computation by imposing monetary penalties on adversarial parties. This functionality is called secure computation with penalties. Bentov and Kumaresan (2014) [9] introduced the claim-or-refund functionality that can be implemented via Bitcoin. They achieved secure computation with penalties with O(n) rounds and O(n) broadcasts for any function, where n is the number of parties. After that, Kumaresan and Bentov (2014) [8] showed a constant-round protocol. Unfortunately, this protocol requires O(n2) broadcasts. As far as we know, no protocol achieves O(1) rounds and O(n) broadcasts based on Bitcoin. This work accomplishes such efficiency in secure computation with penalties. We first show a protocol in a slightly relaxed setting called secure computation with non-equivalent penalties. This setting is the same as secure computation with penalties except that every honest party receives more than a predetermined amount of compensation, while the previous one requires that every honest party receives the same amount of compensation. Namely, our setting allows the compensations for honest parties to be non-equivalent. Moreover, we present a technique to remove the non-equivalence of our protocol without sacrificing efficiency. We then propose a new ideal functionality called claim-refund-or-give that can be implemented via Bitcoin.