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Most existing secure biometric authentication schemes are server-centric, and users must fully trust the server to store, process, and manage their biometric data. As a result, users’ biometric data could be leaked by outside attackers or the service provider itself. This paper first constructs the EDZKP protocol based on the inner product, which proves whether the secret value is the Euclidean distance of the secret vectors. Then, combined with the Cuproof protocol, we propose a novel user-centric biometric authentication scheme called BAZKP. In this scheme, all the biometric data remain encrypted during authentication phase, so the server will never see them directly. Meanwhile, the server can determine whether the Euclidean distance of two secret vectors is within a pre-defined threshold by calculation. Security analysis shows BAZKP satisfies completeness, soundness, and zero-knowledge. Based on BAZKP, we propose a privacy-preserving biometric authentication system, and its evaluation demonstrates that it provides reliable and secure authentication.