Individual authentication technologies are essential for electronic systems as social infrastructures. Especially, biometrics-based authentication has been receiving increasing attention and is expected to be implemented on systems with portable devices such as mobile phones for realizing more useful services. The most important problem in biometrics-based authentication is to prevent a leakage of biological information. This paper focuses on the leakage which enables a spoofing and consider two cases, a leakage from data stored in a server for verification of biological information and a leakage by a cheating detection. This paper proposes a solution by applying a function to biological information and shows the properties required for the function to solve the problem. Moreover, this paper proposes an idea of biometrics-based authentication system with portable devices which is provided a function to capture biological information.