Abstract: Aiming at the shortcomings of the current electronic cash scheme in anonymity and fairness, two schemes based on elliptic curve and bilinear pair were proposed. The schemes can not only guarantee the anonymity of customers, but also trace the double-spending to ensure the fairness of the transaction. Firstly, the electronic cash scheme based on elliptic curve authentication encryption proposed by Chaudhry et al. was analyzed, which can not guarantee the anonymity of consumption and effectively resolve the transaction disputes. Then the scheme of bank delegating offline electronic cash scheme proposed by Liu et al. was also analyzed, and it was found that the users in it can fake electronic money. Two improved schemes were proposed to modify the above defects and the security of them were analyzed. The analysis show that the new schemes not only inherit the security and efficiency of the previous schemes, and can resist replay attacks and impersonation attacks, but also ensure the anonymity and fairness of the schemes.

Key words: electronic cash, cryptography, anonymity, offline, double-spending

摘要： 针对目前电子货币方案在匿名性和公平性等方面的不足，分别使用椭圆曲线和双线性对构造两个电子货币方案，方案在保证顾客匿名性的同时，又可以追踪二次花费的顾客，保证交易的公平进行。首先分析Chaudhry等提出的基于椭圆曲线认证加密的电子货币方案，其方案不能保证消费的匿名性，同时也无法有效解决交易纠纷；接着分析Liu等提出的银行委托离线电子货币方案，发现其方案中用户可以伪造电子货币。然后针对以上方案中的缺陷分别提出了改进后的新方案，并对新方案进行了安全性分析。分析表明，新方案不仅继承了之前方案的安全性和高效性，可以抵抗重放攻击和假冒攻击，还确保了方案的匿名性和公平性。

关键词: 电子货币, 密码学, 匿名, 离线, 二次花费