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Without sufficient protection during the entire authentication procedure, biometrics cannot supersede traditional authentication methods. In this paper a new method of cancellable biometric transformation is presented. Random finite spaces are utilized to map the original facial biometrics into a secure domain, in which authentication can be accurately performed using PCA. Each face is mapped up using independent random spaces to generate a secure (cancellable) template. Replacing the previous random spaces results in a new cancellable template, this is issued from the same original image. Evaluation has shown significant accuracy and security improvements. Genuine and impostor distributions separation has been improved by 203.78%, leading to a 99.94% success rate, which also means that the error rate has been improved by 99.53%. The cancellable templates do not carry any visual information.