Identity Based Encryption is an approach to link the public key to an identity. It is an extremely useful asymmetric cryptography type in which public and private keys are computed from a known identifier such as an email address instead of being generated randomly. This allows more flexibility in managing ad-hoc public key encryption and ensuring secure communications. The aim of this work is to improve IBE scheme using the bilinear Tate pairing on genus two curves with ordinary Jacobian over large prime fields. We present a full description of functional IBE scheme using the optimization of the Tate pairing computations. The proposed application answers a question of Boneh and Franklin [2] about the possibility of using the Tate pairing in IBE schemes and represents the first IBE exploiting pairings in genus two. We provide a full description of a functional IBE scheme using the optimization of the Tate pairing computations.