Towards Ideal Self-bilinear Map

Bilinear maps (also called pairings) have been used for constructing various kinds of cryptographic primitives including (but not limited to) short signatures, identity-based encryption, attribute-based encryption, and non-interactive zero-knowledge proof systems. In known instantiations of cryptographic bilinear maps based on eliptic curves, source and target groups are different groups, which may restrict applications of bilinear maps. Cheon and Lee studied self-bilinear maps, which are bilinear maps whose source and target groups are identical. They showed huge potential of self-bilinear maps by showing that self-bilinear maps can be transformed into multilinear maps, which give further more cryptographic applications including (but not limited to) multiparty non-interactive key exchange, broadcast encryption, attribute-based encryption, homomorphic signatures, and obfuscation. However, they also showed a strong negative result on the existence of cryptographic self-bilinear maps. Namely, they showed that if there exists an efficiently computable self-bilinear map on a known order group, then the computational Diffie-Hellman (CDH) assumption does not hold on the group. This means that cryptographically useful self-bilinear maps do not exist on groups of known order. On the other hand, there is no negative result for self-bilinear maps on groups of unknown order. Indeed, Yamakawa et al. gave a partial positive result for self-bilinear maps on unknown order groups. Namely, they constructed self-bilinear maps with auxiliary information, which is a weaker variant of self-bilinear maps based on indistinguishability obfuscation. Though they showed that they are sufficient for some applications of self-bilinear maps, they are not as useful as "ideal" self-bilinear maps, which do not need auxiliary information. In this talk, we first review the construction of self-bilinear maps with auxiliary information given by Yamakawa et al. Then we consider the possibility of constructing ideal self-bilinear maps.