Crypto-analyses on "user efficient recoverable off-line e-cashs scheme with fast anonymity revoking"

Recently, Fan et al. proposed a user efficient recoverable off-line e-cash scheme with fast anonymity revoking. They claimed that their scheme could achieve security requirements of an e-cash system such as, anonymity, unlinkability, double spending checking, anonymity control, and rapid anonymity revoking on double spending. They further formally prove the unlinkability and the un-forgeability security features. However, after crypto-analysis, we found that the scheme cannot attain the two proven security features, anonymity and unlinkability. We, therefore, modify it to comprise the two desired requirements which are very important in an e-cash system.