One of the main challenges faced by Biometric-based authentication systems is the need to offer secure authentication while maintaining the privacy of the biometric data. Previous solutions, such as Secure Sketch and Fuzzy Extractors, rely on assumptions that cannot be guaranteed in practice, and often affect the authentication accuracy. In this paper, we introduce HoneyFaces: the concept of adding a large set of synthetic faces (indistinguishable from real) into the biometric "password file". This password inflation protects the privacy of users and increases the security of the system without affecting the accuracy of the authentication. In particular, privacy for the real users is provided by "hiding" them among a large number of fake users (as the distributions of synthetic and real faces are equal). In addition to maintaining the authentication accuracy, and thus not affecting the security of the authentication process, HoneyFaces offer several security improvements: increased exfiltration hardness, improved leakage detection, and the ability to use a Two-server setting like in HoneyWords. Finally, HoneyFaces can be combined with other security and privacy mechanisms for biometric data. We implemented the HoneyFaces system and tested it with a password file composed of 270 real users. The "password file" was then inflated to accommodate up to $2^{36.5}$ users (resulting in a 56.6 TB "password file"). At the same time, the inclusion of additional faces does not affect the true acceptance rate or false acceptance rate which were 93.33\% and 0.01\%, respectively.