To solve the drawback of the present electionic cash(E-cash) schemes that the size of banks transcripts database expands without bound, the concept of validity duration of E-cash is introducedin this paper. Through improving the users blind transforms ofBrands scheme, we design an efficient off-line E-cash system basedon the restrictive blind signature and repressentation problem. The security analysis shows that the users can not forge a valid electronic coinor double spend it , in addition, the bank can not trace an honestusers payment behaviours. In compare with Brands scheme, our scheme reduces storage costs and computation overhead of the usersand the banks. Most of all, the electronic cashes will be authorizedwithin a limit period only, the size of banks transcripts database becomes controllable.