Bull. Korean Math. Soc. 46 (2009), No. 2, pp. 303–309 DOI 10.4134/BKMS.2009.46.2.303 A NOTE ON SELF-BILINEAR MAPS Jung Hee Cheon and Dong Hoon Lee Abstract. Cryptographic protocols depend on the hardness of some computational problems for their security. Joux briefly 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 Diffie-Hellman assumption implies the bilinear Diffie-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 fixed 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. 1. Introduction The Weil pairing on an elliptic curve have been used to solve cryptographic problems such as the discrete logarithm (DL) problem, the computational Diffie-Hellman (CDH) problem, the decisional Diffie-Hellman (DDH) problem [8]. After Joux proposed tripartite Diffie-Hellman protocol using the Weil par- ing, however, the Weil (or Tate) pairing is being used as a building block of interesting cryptographic protocols including ID-based schemes, a short signa- ture scheme, self-blindable credentials, and key agreement [5, 1, 4, 2, 11, 9]. The bilinear property of the pairings plays an important role on pairing- based protocols. Given two groups G and H, a map e : G × G → H is said to be bilinear if e(g x 1 1 ,g x 2 2 )= e(g 1 ,g 2 ) x1x2 for all x i ∈ Z and g i ∈ G. Given a quadruple (g,g x ,g y ,g z ) the bilinear Diffie-Hellman (BDH) problem asks to find e(g,g) xyz . Received May 22, 2008. 2000 Mathematics Subject Classification. Primary 94A60, 11Y16, 68Q15. Key words and phrases. cryptography, complexity, elliptic curves, pairing, self-bilinear map. The first author was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) (No. R01-2008-000-11287-0). c 2009 The Korean Mathematical Society 303