The aim of this paper is to present a signature scheme in which the ability to sign messages of a signer is limited to a fixed number k of signatures. It is an identity-based signature scheme in which each signature can be used only once. We called such schemes "bounded life span". It is based on mental games and it uses zero-knowledge tools. A validation center is needed to initialize this identity-based scheme. A credential center is used to insure the unicity and the bounded life-span aspects. It allows delegation and numerous practical applications.