A NOTE ON SELF-BILINEAR MAPS

Cryptographic protocols depend on the hardness of some computational problems for their security. Joux brie∞y summarized kno- wn relations between assumptions related bilinear map in a sense that if one problem can be solved easily, then another problem can be solved within a polynomial time (6). In this paper, we investigate additional relations between them. First- ly, we show that the computational Di-e-Hellman assumption implies the bilinear Di-e-Hellman assumption or the general inversion assumption. Secondly, we show that a cryptographic useful self-bilinear map does not exist. If a self-bilinear map exists, it might be used as a building block for several cryptographic applications such as a multilinear map. As a corollary, we show that a flxed inversion of a bilinear map with homomor- phic property is impossible. Finally, we remark that a self-bilinear map proposed in (7) is not essentially self-bilinear.