In 2002, Mitsunari, Sakai and Kasahara formulated the Collusion Attack Algorithm with k traitors (known as k-CAA problem) and used it to develop the first traitor tracing scheme based on the bilinear pairings. Traitor tracing scheme is needed to discourage legitimate subscribers from sharing their secret keys to construct pirate decoders. In this paper, we propose a first signature scheme (IE-CBS-kCAA) based on k-CAA problem, which belongs to the fourth category of PKC schemes: the public key cryptography schemes based on an implicit and explicit certificates. The security analysis proves that our IE-CBS-kCAA scheme is secure against two game attacks in the random oracle model. The security is closely related to the difficulty of solving the modified k-CAA and discrete logarithm problems.