Bull. Korean Math. Soc. 2009; 46(2): 303-309
Printed March 1, 2009
https://doi.org/10.4134/BKMS.2009.46.2.303
Copyright © The Korean Mathematical Society.
Jung Hee Cheon and Dong Hoon Lee
Seoul National University and ETRI Network and Communications Security Division
Cryptographic protocols depend on the hardness of some computational problems for their security. Joux briefly summarized known 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. Firstly, 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 homomorphic property is impossible. Finally, we remark that a self-bilinear map proposed in [7] is not essentially self-bilinear.
Keywords: cryptography, complexity, elliptic curves, pairing, self-bilinear map
MSC numbers: Primary 94A60, 11Y16, 68Q15
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