We define in the context of this work a new EAP method, that we call EAP-CGA. This method performs a mutual authentication of the hosts based on their CGAs. The verification of the authenticity of the provided CGAs decides the failure or the success of the authentication process. We provide also a new way on how to integrate this new authentication method in the cryptographic protocol of HIP, known as the HIP Base Exchange. Because HIP is a recent protocol proposal dedicated mainly to solve the mobility problem in IP networks, such an idea paves the way to authentication extensions related to this solution and has other benefits related to hosts identification, mainly coupling two identifiers of the same identity. Our protocols proposals are validated using an automatic validation tool and the results we obtained show their robustness mainly against Man-in-the-Middle and replay attacks