This paper proposes a novel secure biometric authentication scheme. The scheme deals with fingerprint minutiae as the biometric feature and the matching is checked by a widely used technique. To discuss security, we formalize the model of secure biometric authentication scheme by abstracting the related and proposed schemes. The schemes which satisfy all the proposed security requirements are guaranteed to prevent leakage of biometric information and impersonation. In particular, the definition captures well-known and practical attacks including replay attacks and hill-climbing attacks. We prove that the proposed scheme achieves all the requirements if the additive homomorphic encryption scheme used in the scheme satisfies some additional properties. As far as we know, the proposed scheme is the first one that satisfies all the requirements. Also, we show that modified Elgamal cryptosystem satisfies all the properties under the decisional Diffie-Hellman assumption.