A Variant of Miller’s Formula and Algorithm

John Boxall (1) , Nadia El Mrabet (2) , Fabien Laguillaumie (3) , Duc-Phong Le (4)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 LIASD - Laboratoire d'Informatique Avancée de Saint-Denis
3 Equipe AMACC - Laboratoire GREYC - UMR6072
4 NUS - National University of Singapore

Miller's algorithm is at the heart of all pairing-based cryp-tosystems since it is used in the computation of pairing such as that of Weil or Tate and their variants. Most of the optimizations of this al-gorithm involve elliptic curves of particular forms, or curves with even embedding degree, or having an equation of a special form. Other im-provements involve a reduction of the number of iterations. In this article, we propose a variant of Miller's formula which gives rise to a generically faster algorithm for any pairing friendly curve. Concretely, it provides an improvement in cases little studied until now, in particular when denominator elimination is not available. It allows for instance the use of elliptic curve with embedding degree not of the form 2 i 3 j , and is suitable for the computation of optimal pairings. We also present a version with denominator elimination for even embedding degree. In our implementations, our variant saves between 10% and 40% in running time in comparison with the usual version of Miller's algorithm without any optimization.

Domaines: Informatique [cs]; Algorithme et structure de données [cs.DS]

John Boxall, Nadia El Mrabet, Fabien Laguillaumie, Duc-Phong Le. A Variant of Miller’s Formula and Algorithm. Proceedings of Pairing 2010, Dec 2010, Ishikawa, Japan. pp.417 - 434, ⟨10.1007/978-3-642-17455-1_26⟩. ⟨hal-01083368⟩