Existing software proposals for electronic payments can be divided into "on-line" schemes that require participation of a trusted party (the bank) in every transaction and are secure against overspending, and the "off-line" schemes that do not require a third party and guarantee only that overspending is detected when vendors submit their transaction records to the bank (usually at the end of the day). We propose a new hybrid scheme that combines the advantages of both of the above traditional design strategies. It allows for control of overspending at a cost of only a modest increase in communication compared to the off-line schemes. Our protocol is based on probabilistic polling. During each transaction, with some small probability, the vendor forwards information about this transaction to the bank. This enables the bank to maintain an accurate approximation of a customer's spending. The frequency of polling messages is related to the monetary value of transa...
Stanislaw Jarecki, Andrew M. Odlyzko