A fair electronic cash system is a system that allows customers to make payments anonymously. Furthermore the trusted third party can revoke the anonymity when the customers did illegal transactions. In this paper, a new fair electronic cash system based on group signature scheme by using elliptic curve cryptography is proposed, which satisfies properties of secure group signature scheme (correctness, unforgeability, etc.). Moreover, our electronic cash contains group members (users, merchants

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... nd banks) and trusted third party which is acted by central bank as group manager. 1 2 , , , , , A B C D Z Z S to merchant. nerates a pa 2) The merchant ge transaction message of yment for the customer d as challenge and sends to user