Improvement of one escrow-free identity-based signature

In the common identity-based cryptosystem, a trusted party called the private key generator (PKG), is responsible for generating the secret key for each user´s identity. As the PKG knows the secret key for all users, the PKG must be absolutely trustworthy. In a real world scenario, this is not desirable because a malicious PKG can impersonate any user. This is known as the key escrow problem. In EuroPKI´2009, Yuen et al proposed an escrow-free identity-based signature scheme. The scheme has to introduce six generators in the underlying group. This leads to an impressive cost. In this paper, we reduce the quantity of the involved generators as less as possible and improve it. The improved scheme only requires three generators. It saves about  cost.