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A user authentication scheme based upon the properties of Euclidean geometry properties is proposed. The center authority (CA) constructs the shared information by using both its secret and the userpsilas password in the user registration phase. Then the request message is computed by using both the userpsilas password and the registered information based on properties of Euclidean geometry and one-way hash functions. In the system verification phase, the CA verifies the validity of a user based on its request message and the CApsilas secrets. Only the CA can do the verification because only it knows its secret. The verification is fast because it does not need to access the database and its computation is simple. The timestamp allows the proposed scheme to withstand replay attacks. The security analysis shows that our scheme is also secure against eavesdropping attacks, and dictionary attacks. A formal approach is adopted to prove the proposed scheme is secure. The experimental results show that our scheme is more applicable for large-scale network environments than a password-based system or a PKI certificate-based system.