Colluding attacks to a payment protocol and two signature exchange schemes

An untraceable fair network payment protocol is proposed by Wang in Asiacrypt'03, which employs the existent techniques of the off-line untraceable cash and a new technique called restrictive confirmation signature scheme (RCSS). It is claimed that the fair payment protocol has both the fairness such that the buyer obtains the digital goods if and only if the merchant gains the digital cash and the untraceability and unlinkability such that no one can tell who is the original owner of the money. In this paper we show that the fairness is breached under a simple colluding attack, by which a dishonest merchant can obtain the digital money without the buyer obtaining the goods. We also apply the attack to some of the schemes of fair exchange of digital signatures proposed by Ateniese in ACM CCS'99. Our study shows that two of them are subjected to the attack. A countermeasure against the attack is proposed for the fair exchange of digital signatures. However, we are unable to fix the fair payment protocol if the untraceability and unlinkability are the required features.