Paper 2019/136

Divisible E-Cash from Constrained Pseudo-Random Functions

Florian Bourse, David Pointcheval, and Olivier Sanders


Electronic cash (e-cash) is the digital analogue of regular cash which aims at preserving users' privacy. Following Chaum's seminal work, several new features were proposed for e-cash to address the practical issues of the original primitive. Among them, divisibility has proved very useful to enable efficient storage and spendings. Unfortunately, it is also very difficult to achieve and, to date, quite a few constructions exist, all of them relying on complex mechanisms that can only be instantiated in one specific setting. In addition security models are incomplete and proofs sometimes hand-wavy. In this work, we first provide a complete security model for divisible e-cash, and we study the links with constrained pseudo-random functions (PRFs), a primitive recently formalized by Boneh and Waters. We exhibit two frameworks of divisible e-cash systems from constrained PRFs achieving some specific properties: either key homomorphism or delegability. We then formally prove these frameworks, and address two main issues in previous constructions: two essential security notions were either not considered at all or not fully proven. Indeed, we introduce the notion of clearing, which should guarantee that only the recipient of a transaction should be able to do the deposit, and we show the exculpability, that should prevent an honest user to be falsely accused, was wrong in most proofs of the previous constructions. Some can easily be repaired, but this is not the case for most complex settings such as constructions in the standard model. Consequently, we provide the first construction secure in the standard model, as a direct instantiation of our framework.

Note: correct some errors in the indices of the subsets in the delegatable framework

Available format(s)
Cryptographic protocols
Publication info
A major revision of an IACR publication in ASIACRYPT 2019
E-CashanonymityConstrained PRF
Contact author(s)
Florian Bourse @ ens fr
david pointcheval @ ens fr
olivier sanders @ orange com
2019-11-13: last of 4 revisions
2019-02-13: received
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Creative Commons Attribution


      author = {Florian Bourse and David Pointcheval and Olivier Sanders},
      title = {Divisible E-Cash from Constrained Pseudo-Random Functions},
      howpublished = {Cryptology ePrint Archive, Paper 2019/136},
      year = {2019},
      note = {\url{}},
      url = {}
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