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 keyevolving 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 EU-CMA security.