Relations among security notions for undeniable signature schemes

In this paper, we conduct a thorough study among various notions of security of undeniable signature schemes and establish some relationships among them. We focus on two adversarial goals which are unforgeability and invisibility and two attacks which are chosen message attack and full attack. In particular,we show that unforgeability against chosen message attack is equivalent to unforgeability against full attack, and invisibility against chosen message attack is equivalent to invisibility against full attack. We also present an undeniable signature scheme whose unforgeability is based on the factoring assumption and whose invisibility is based on the composite decision Diffie-Hellman assumption.