Abstract

Abstract. One of the central challenges for mathematical cryptogra-phy is to create a payment system that provides the advantages of cash in a digital world. In this expository article we describe two very differ-ent solutions to this problem. The first is an elliptic-curve-based version of a construction of S. Brands, and the second is Bitcoin. We also discuss a generalization of Bitcoin that supports peer-to-peer contracts. AMS subject classifications. 94A60, 68P25, 14G50, 94-02. 1.