FPGA Implementation of Various Elliptic Curve Pairings over Odd Characteristic Field with Non Supersingular Curves

Yasuyuki NOGAMI  Hiroto KAGOTANI  Kengo IOKIBE  Hiroyuki MIYATAKE  Takashi NARITA  

IEICE TRANSACTIONS on Information and Systems   Vol.E99-D   No.4   pp.805-815
Publication Date: 2016/04/01
Publicized: 2016/01/13
Online ISSN: 1745-1361
DOI: 10.1587/transinf.2015ICP0018
Type of Manuscript: Special Section PAPER (Special Section on Information and Communication System Security)
Category: Cryptography and cryptographic protocols
elliptic curve cryptography,  pairing-based cryptography,  odd characteristic,  FPGA implementation,  

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Pairing-based cryptography has realized a lot of innovative cryptographic applications such as attribute-based cryptography and semi homomorphic encryption. Pairing is a bilinear map constructed on a torsion group structure that is defined on a special class of elliptic curves, namely pairing-friendly curve. Pairing-friendly curves are roughly classified into supersingular and non supersingular curves. In these years, non supersingular pairing-friendly curves have been focused on from a security reason. Although non supersingular pairing-friendly curves have an ability to bridge various security levels with various parameter settings, most of software and hardware implementations tightly restrict them to achieve calculation efficiencies and avoid implementation difficulties. This paper shows an FPGA implementation that supports various parameter settings of pairings on non supersingular pairing-friendly curves for which Montgomery reduction, cyclic vector multiplication algorithm, projective coordinates, and Tate pairing have been combinatorially applied. Then, some experimental results with resource usages are shown.