Secure identification and authentication are essential processes in sensitive areas of application such as e-Government or e-Health. In Austria, the official eID is the so called the Austrian citizen card and a means of choice for secure citizen identification and authentication. To facilitate the adoption of citizen card authentication at service providers within the Austrian e-Government strategy, the open source module MOA-ID has been developed. It acts as identity provider for different service providers and manages identification and authentication based on the Austrian citizen card. Currently, MOA-ID is deployed locally in every service provider's domain and is assumed to be fully trusted. With the increasing use of eIDs, however, a move into a public cloud might be advantageous due to benefits provided cloud computing, e.g., cost savings or scalability. Nevertheless, the move of a trusted service into the public cloud brings up new obstacles, in particular with respect to security and privacy. Therefore, in this paper we introduce and evaluate three different approaches on how the Austrian eID system based on MOA-ID could be securely moved into the cloud without violating any privacy or data protection aspects. To achieve this, we rely on various cryptographic methods and focus on minimum changes of the current identification and authentication process flow. Based on an evaluation of these three different approaches, we propose a model which can be generically used for eID identification and authentication in privacy-invasive environments such as the public cloud.