\indirect Discourse Proofs": Achieving Eecient Fair Oo-line E-cash

Cryptography has been instrumental in reducing the involvement of overhead third parties in protocols. For example; a digital signature scheme assures a recipient that a judge who is not present at message transmission will nevertheless approve the validity of the signature. Similarly, in oo-line electronic cash the bank (which is oo-line during a purchase) is assured that if a user double spends he will be traced. Here we suggest the notion of Indirect Discourse Proofs with which one can prove indirectly yet eeciently that a third party has a certain future capability (i.e., assure Trustees can trace). The eecient proofs presented here employ algebraic properties of exponentiation (or functions of similar homomorphic nature). Employing this idea we present the concept of \Fair OO-Line e-Cash" (FOLC) system which enables tracing protocols for identifying either the coin or its owner. Recently, the need to trace and identify coins with owners/withdrawals was identiied (to avoid blackmailing and money laundering). Previous solutions that assured this traceability (called fair e-cash as they balance the need for anonymity and the prevention of criminal activities) involved third parties at money withdrawals. In contrast , FOLC keeps any third party uninvolved, thus it is \fully oo-line e-cash" even when law enforcement is added (i.e., it is oo-line w.r.t. law enforcement at withdrawals and oo-line w.r.t. the bank at payments).