Self-Bilinear Map from One Way Encoding System and iO

A bilinear map whose domain and target sets are identical is called a self-bilinear map. Original self-bilinear maps are defined over cyclic groups. Since the map itself reveals information about the underlying cyclic group, the Decisional Diffie-Hellman Problem (DDH) and the computational Diffie-Hellman (CDH) problem may be solved easily in some specific groups. This brings a lot of limitations to constructing secure self-bilinear schemes. As a compromise, a self-bilinear map with auxiliary information was proposed in CRYPTO'2014. In this paper, we construct this weak variant of a self-bilinear map from generic sets and indistinguishable obfuscation. These sets should own several properties. A new notion, One Way Encoding System (OWES), is proposed to summarize these properties. The new Encoding Division Problem (EDP) is defined to complete the security proof. The OWES can be built by making use of one level of graded encoding systems (GES). To construct a concrete self-bilinear map scheme, Garg, Gentry, and Halvei(GGH13) GES is adopted in our work. Even though the security of GGH13 was recently broken by Hu et al., their algorithm does not threaten our applications. At the end of this paper, some further considerations for the EDP for concrete construction are given to improve the confidence that EDP is indeed hard.