Weakness of Kurosawa and Heng's Witness Indistinguishable Protocol Based on DH-tuple

The concept of witness indistinguishability and witness hiding was introduced by Feige and Shamir in 1990. Recently, B. Kurosawa and S.H. Heng proposed 3-move undeniable signature scheme at Eurocrypt 2005, which has the confirmation and disvowal protocols for proving the message-signature pair is or is not valid respectively. They also proposed an new DH-tuple Witness indistinguishable (WI) protocol, which is the foundation of the comfirmation protocol in this new undeniable signature scheme. In this paper, we will first show a weakness in Kurosawa and Heng's witness indistinguishable protocol. In general, there are two or more witnesses in a WT protocol. The weakness in Kurosawa and Heng's WI protocol is that a prover with a witness, but does not konw another witness, can forge to produce a cheating proof, which is considered to be from another witness. So, from the concept of witness hiding, there will be an new witness, which is difference from the two original witnesses. We also investigate the reason cuased the weakness in Kurosawa et al.'s protocol.