Memoirs of the Faculty of Engineering, Okayama University

Published by Faculty of Enginerring, Okayama University <Formerly known as>

Memoirs of the School of Engineering, Okayama University

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Sumo, Taichi

Abstract

Recent efficient pairings such as Ate pairing use two efficient rational point subgroups such that
π(P) = P and π(Q) = [p]Q, where π, p, P, and Q are the Frobenius map for rational point, the
characteristic of definition field, and torsion points for pairing, respectively. This relation accelerates not
only pairing but also pairing–related operations such as scalar multiplications. It holds in the case that
the embedding degree k divides r − 1, where r is the order of torsion rational points. Thus, such a case
has been well studied. Alternatively, this paper focuses on the case that the degree divides r + 1 but
does not divide r − 1. Then, this paper shows a multiplicative representation for r–torsion points based
on the fact that the characteristic polynomial f(π) becomes irreducible over Fr for which π also plays a
role of variable.

Keywords

pairing–friendly curve

torsion point

group structure

rank

ISSN

1349-6115

NCID

AA12014085