In this paper, we propose a new variant of indistinguishability-based security model for the RFID authentication protocol, which allows an adversary to obtain an authentication result and secret key of a target tag. Ng et al. showed that symmetric-key based RFID authentication protocols cannot be resilient to the above information leakage simultaneously in the Paise-Vaudenay security model. We review the existing result and extend the Juels-Weis security model to satisfy these properties by using a suitable restriction. Moreover, we give two example protocols that satisfy the modified security model.