Converting Cryptographic Schemes from Symmetric to Asymmetric Bilinear Groups

We propose a method to convert schemes designed over symmetric bilinear groups into schemes over asymmetric bilinear groups. The conversion assigns variables to one or both of the two source groups in asymmetric bilinear groups so that all original computations in the symmetric bilinear groups go through over asymmetric groups without having to compute isomorphisms between the source groups. Our approach is to represent dependencies among variables using a directed graph, and split it into two graphs so that variables associated to the nodes in each graph are assigned to one of the source groups. Though searching for the best split is cumbersome by hand, our graph-based approach allows us to automate the task with a simple program. With the help of the automated search, our conversion method is applied to several existing schemes including one that has been considered hard to convert.