The standard digital signature scheme can be easily subject to key exposure problem In order to overcome this problem; a feasible and effective approach is employed by key-evolving signature scheme. In this paper, we study key-evolving signature within the UC framework and propose an appropriate ideal functionality that captures the basic security requirements of key-evolving signature. Then, we present a generic way to transform a key-evolving signature scheme into a real-life protocol. Finally, we show that UC definition of security is equivalent to previous definition of security which is termed as EUF-ACMA security.