integer variable χ-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve, it reduces the number of loops to ⌊log 2χ⌋. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results." />


Integer Variable χ-Based Cross Twisted Ate Pairing and Its Optimization for Barreto-Naehrig Curve

Yasuyuki NOGAMI  Yumi SAKEMI  Hidehiro KATO  Masataka AKANE  Yoshitaka MORIKAWA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E92-A   No.8   pp.1859-1867
Publication Date: 2009/08/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E92.A.1859
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
Category: Theory
Keyword: 
Ate pairing,  Miller's algorithm,  

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Summary: 
It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log 2r/(k), where () is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from ⌊log 2r⌋ to ⌊ log 2(t-1)⌋, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "χ." For such a curve, this paper gives integer variable χ-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve, it reduces the number of loops to ⌊log 2χ⌋. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.