Electronic payments of small amounts

This note considers the application of electronic cash to transactions in which many small amounts must be paid to the same payee and in which it is not possible to just pay the total amount afterwards. The most notable example of such a transaction is payment for phone calls. If currently published electronic cash systems are used and a full payment protocol is executed for each of the small amounts, the overall complexity of the system will be prohibitively large (time, storage and communication). This note describes how such payments can be handled in a wide class of payment systems. The solution is very easy to adapt as it only influences the payment and deposit transactions involving such payments. Furthermore, making and verifying each small payment requires very little computation and communication, and the total complexity of both transactions is comparable to that of a payment of a fixed amount.