In a profile-based authentication system, a user profile is stored at the verifier and later used to verify their authentication claim. A profile includes user-specific information that is privacy sensitive. In this paper we propose a non-cryptographic approach to providing privacy for user profile data in profile-based authentication systems, using an efficient construction of random projection: a linear dimension reducing transform that projects the profile and the verification data to a lower dimension space, while preserving relative distances of the vectors and so correctness of authentication. We define privacy measures for two types of profiles: a single vector profile and a multivector profile, derive theoretical bounds on the privacy and correctness of privacy enhanced systems, and verify the results experimentally on two profile-based authentication systems: a face-biometric system  and a behavioural based authentication system. We discuss our results and propose directions for future research.