A forward secure identity based signature scheme (FSIBS) provides forward secrecy of secret keys. In order to mitigate the damage when keys are leaked, it is desirable to evolve all the secret keys, i.e., both the user keys and the master key. In this paper, we propose a new RSA-based FSIBS scheme which requires constant size keys and generates constant size signatures. The experimental results show that it takes 3 ms to generate a signature in the proposed scheme while it takes 75 ms in the existing pairing based approach. The proposed scheme is provably secure under the factoring assumption in the random oracle model.