A Hypercube-based Divisibility Management Framework for e-cash Algorithms

In this paper, we propose a divisibility management framework for e-cash algorithms. The proposed framework is based on the dependency graph of a hypercube, which is constructed from all subcubes of a hypercube, to keep track of partitioned spending of the resource pool represented by the hypercube. On the basis of Eng and Okamoto's recursive key generation equation, we develop an analysis framework to determine the necessary and sufficient conditions to meet the security requirements of authentication, privacy and double spending identification (DSI). We analyze and solve the shared-node DSI problem, in addition to the traditional same-node and route-node DSI problems. Shared-node double spending can occur when a subcube can be divided in different combinations for spending. Such freedom is fundamentally important to distributed resource management, but it is not allowed in the traditional tree-based divisible e-cash schemes. Another critical issue associated with the DSI guarantee is protection of privacy. We show that if the resource pool is allowed to be spent in arbitrary partitions, the user privacy can be violated when shared-node DSI is supported. This problem is solved by using our analysis technique to detect and prevent privacy violation before a particular resource partition can be spent. And any spent token is guaranteed to include a valid secret share of DSI for deciphering of the user identity when any type of double spending occurs.