Authentication is an important and challenging issue for the security of Fog computing since, services are offered to massive-scale end users (Fog users or Edge) by front Fog servers (or Fog nodes). In this paper, we propose a secure and efficient mutual authentication scheme for the Edge-Fog-Cloud network architecture, to mutually authenticate Fog users at the Edge of the network, with the Fog servers at the Fog layer. Our scheme requires a user - roaming randomly in the network { to hold only one long-lived master secret key (with long enough bit-length) allowing him to communicate with any of the Fog servers in the network, in a fully authenticated way. The Fog users are able to mutually authenticate with new Fog servers joining the network, without the need to re-register and without any extra overheads. Moreover, the servers in the Fog are required to store only one secret key for each Fog user. On the other hand, the Fog users are totally unrelated to any public-key infrastructure. The scheme requires the Fog user to perform very few hash invocations and symmetric encryptions/decryptions. Therefore, the scheme is suitable to be efficiently implemented on the Fog user's smart card/device.