A behavioral authentication (BA) system uses the behavioral characteristics of users to verify their identity claims. A BA verification algorithm can be constructed by training a neural network (NN) classifier on users' profiles. The trained NN model classifies the presented verification data, and if the classification matches the claimed identity, the verification algorithm accepts the claim. This classification-based approach removes the need to maintain a profile database. However, similar to other NN architectures, the NN classifier of the BA system is vulnerable to privacy attacks. To protect the privacy of training and test data used in an NN different techniques are widely used. In this paper, our focus is on a non-crypto-based approach, and we used random projection (RP) to ensure data privacy in an NN model. RP is a distance-preserving transformation based on a random matrix. Before sharing the profiles with the verifier, users will transform their profiles by RP and keep their matrices secret. To reduce the computation load in RP, we use sparse random projection, which is very effective for low-compute devices. Along with correctness and security properties, our system can ensure the changeability property of the BA system. We also introduce an ML-based privacy attack, and our proposed system is robust against this and other privacy and security attacks. We implemented our approach on three existing behavioral BA systems and achieved a below 2.0% FRR and a below 1.0% FAR rate. Moreover, the machine learning-based privacy attacker can only recover below 3.0% to 12.0% of features from a portion of the projected profiles. However, these recovered features are not sufficient to know details about the users' behavioral pattern or to be used in a subsequent attack. Our approach is general and can be used in other NN-based BA systems as well as in traditional biometric systems.