To profit of the high bandwidth and low cost of the IEEE 802.11 wireless local area network (WLAN) and to benefit of the large zone coverage of the 3G network, the third Generation Partnership Project (3GPP) has specified the interworking architecture for the 3G-WLAN and develops System Architecture Evolution (SAE)/ Long Term Evolution (LTE) architecture. This inter-working introduces a new challenge to ensure the security of both networks and to manage a secure and fast vertical handover (VH). The Extensible Authentication Protocol and Authentication Key Agreement (EAP-AKA) is the authentication mechanism adopted by the 3GPP for VH between the 3G and the WLAN. However, the EAP-AKA method suffers from several weaknesses, such as user identity showing, sequence number synchronization and additional bandwidth consumption. These expose legitimate user to risk and increase the authentication delay. In this paper we propose a new fast authentication method (EAP-FAKA) which delegates the user authentication to the WLAN on behalf of the 3G network. The new method is based on the Elliptic Curve Diffie-Hellman (ECDH) and symmetric cryptosystem. The proposed protocol achieves fast and mutual authentication with definition of a new key framework. The security properties of the new method are checked by using a formal verification (HLPSL) which has proved a high talent in finding potential attacks automatically in security protocols.