The importance of electronic payment methods has been widely recognized, triggered by the proliferation of electronic services, such as services available over the Internet or cellular systems. Electronic cash, introduced as a concept by David Chaum in 1982, is among the most important of these methods due to its guarantee of user anonymity; hence the parallelism with physical cash. Privacy of transactions is progressively viewed to be a highly desirable feature in financial services. We present our results in efficient and secure e-cash; our systems can be implemented in current smart cards and are provably secure.

[Providing exact payments]
Efficiency for practical payment systems requires the ability to conduct payments of exact amounts. We investigate two viable approaches, each optimal for different system requirements:
1) Using electronic coins that can be divided to allow for exact payments (divisible e-cash). We improve existing systems by three orders of magnitude, without sacrificing security.
2) Keeping a multitude of coins in users' "electronic wallets": we construct an optimal algorithm for withdrawing coins that suffice for a fixed number of exact payments.

[Controlling user anonymity]
The anonymity of e-cash could be used for criminal activities such as money laundering, committing perfect (anonymous) crimes or making illegal purchases. We propose to solve this problem by allowing an escrow agent (Trustee/Judge) to remove anonymity of users when necessary. Earlier similar proposals required involvement of Trustee(s) on a per-coin basis, preventing the systems from being used in practice. Our system allows Trustee(s) to be Off-Line, and ranks among the most efficient e-cash systems to date while remaining provably secure.

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