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Due to the fast progress of the internet technologies, electronic commerce becomes more and more popular. Many people and businesses deal with their transactions via the internet. The technologies of credit cards, electronic tickets, e-cash, and other advanced services have realized the vision of electronic commerce. In this thesis, we propose an off-line e-cash scheme with anonymity, untraceability, double-spending checking, and traceability. Anonymity and untraceability must be possessed in any e-cash scheme. In an off-line e-cash scheme, the bank or the third party (TTP) must be able to revoke the anonymity of a user who doubly spent her/his e-cash(s). In our proposed e-cash scheme, the bank can fast derive the identity of the user who doubly spent her/his e-cash(s) without the participation of TTP. If some illegal transactions are reported, TTP can also directly revoke the anonymity of the user who spent her/his e-cash(s) in the illegal transactions. In addition, the police needs to trace a specific user in some situation, and we propose a process to satisfy this requirement,called traceability.