Provably secure E-cash system with practical and efficient complete tracing

E-cash system can guarantee the anonymity of the user, but it also can be misused illegally. So the idea of fair E-cash was introduced, i.e., the anonymity of the user can be revoked when a crime is involved in fair E-cash system. However, it is still an unsolved problem to design a practical and efficient fair E-cash scheme, since there are some problems of the impracticality or the incompleteness in the tracing design of existing schemes. First we discuss the principal reasons of it. The reason of impractical tracing is that many schemes have respective unsolved problems in designing the practical tracing. We solve one kind of the problems, then present the practical tracings in various situations, including the bank's double-spending tracing without the trusted third party (TTP), the TTP's unconditional tracing and the bank's loss-coin tracing without TTP, which we call complete tracing. For some schemes, the reason of incomplete tracing is that the unsolved efficiency problems result in the incomplete design of tracing, whereas for other schemes, the reason of incomplete tracing is that the unconditional tracing and the anonymous spending are contradicting properties of E-cash. Considering that, the TTP can be divided into multiple parties, and the TTP is also removable in our system. Moreover, for solving the efficiency problem of practical tracing, we use a nested structure of signature of knowledge in payment protocol, which is proved to be a secure application. Then we provide the detailed security proofs of all security properties of our E-cash scheme and analyze our system efficiency and system functions comparing with the recent schemes.