A Note on the Pairing Computation Using Normalized Miller Functions

Naoki OGURA  Shigenori UCHIYAMA  Naoki KANAYAMA  Eiji OKAMOTO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E95-A   No.1   pp.196-203
Publication Date: 2012/01/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E95.A.196
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
Category: Mathematics
Keyword: 
pairing,  Miller's algorithm,  rational functions,  normalization,  

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Summary: 
This paper considers the normalization of Miller functions for computing “point-evaluation” pairings on an elliptic curve E over a finite field Fq, where the characteristic of Fq is neither 2 nor 3. It is shown that the normalized Miller functions for computing point-evaluation pairings on G2G1 when (i) the embedding degree k is even, or (ii) 3|k and E/Fq(q ≡ (1 mod 3)) is a curve of the form Y2=X3+b. Thus, there is no need to consider the normalization for computing pairings on many pairing-friendly elliptic curves.