How to define the security of undeniable signature schemes is a challenging task. This paper presents two security definitions of undeniable signature schemes which are more useful or natural than the existing definition. It then proves their equivalence. We first define the UC-security, where UC means universal compos-ability. We next show that there exists a UC-secure undeniable signature scheme which does not satisfy the standard definition of security that has been believed to be adequate so far. More precisely, it does not satisfy the invisibility defined by [10]. We then show a more adequate definition of invisibility which captures a wider class of (naturally secure) undeniable signature schemes. We finally prove that the UC-security against non-adaptive adversaries is equivalent to this definition of invisibility and the strong unforge-ability in F{sub}(ZK)-hybrid model, where F{sub}(ZK) is the ideal ZK functionality. Our result of equivalence implies that all the known proven secure undeniable signature schemes (including Chaum's scheme) are UC-secure if the confirmation/disavowal protocols are both UC zero-knowledge.
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