Self-bilinear Map from One Way Encoding System and Indistinguishability Obfuscation

The bilinear map whose domain and range are identical is called self-bilinear map. Once such kind of bilinear map exists, the multilinear map can be constructed easily by using self bilinear map as a component. Yamakawa et al. have introduced the first secure self-bilinear map with auxiliary information based on the integer factoring assumption in Crypto 2014. Inspired by their work, we find that any encoding system with particular properties could be used to build self-bilinear map. We generalize them as one way encoding system and propose a generic construction of self-bilinear map. For cryptographic use, we define a new encoding division assumption to make the analog DDHP hard. We show that one level encoding of graded encoding system which is used to build multilinear map nowadays satisfy all the properties of one way encoding system. We also present an instance that is build on GGH graded encoding scheme and analyze how hard the encoding division problem is. Our self-bilinear map is believed to be quantum resistance. It seems more secure than the scheme of Yamakawa et al. Moreover, the encoding size of n-multilinear built on our self-bilinear map is smaller than that of GGH scheme.