Recent findings confirm that biometric templates derived from electroencephalography (EEG) signals contain sensitive information about registered users, such as age, gender, cognitive ability, mental status and health information. Existing privacy-preserving methods such as hash function and fuzzy commitment are not cancelable, where raw biometric features are vulnerable to hill-climbing attacks. To address this issue, we propose the PolyCosGraph, a system based on Polynomial transformation embedding Cosine functions with Graph features of EEG signals, which is a privacy-preserving and cancelable template design that protects EEG features and system security against multiple attacks. In addition, a template corrupting process is designed to further enhance the security of the system, and a corresponding matching algorithm is developed. Even when the transformed template is compromised, attackers cannot retrieve raw EEG features and the compromised template can be revoked. The proposed system achieves the authentication performance of 1.49% EER with a resting state protocol, 0.68% EER with a motor imagery task, and 0.46% EER under a watching movie condition, which is equivalent to that in the non-encrypted domain. Security analysis demonstrates that our system is resistant to attacks via record multiplicity, preimage attacks, hill-climbing attacks, second attacks and brute force attacks.