Electronic cash is the most fundamental element in electronic commerce. The finance administration authorities must consider controlling the flow of electronic money differently from the flow of conventional cash. The central bank coordinates the quantity of currency to be printed and coordinates money supply so that inflation does not occur. The money has three functions: information to show its value, evidence to show it is validly issued, and a symbol to show who can use it. The symbolic function is important to prevent duplicate reuse of money. We have been studying an electronic cash system using distributed database of a central bank to develop a new settlement method. In this system, the central bank issues money, and its value is not stored in users hardware such as IC cards. Money and a user are linked by public key cryptography that implements the symbolic function. In this paper, we discuss the conditions for making electronic money a substitute for banknotes. Then we propose an electronic money system with discrete logarithms. In this system, the transactions are less traceable than those in our previous study.