Paper 2007/013

Optimised versions of the Ate and Twisted Ate Pairings

Seiichi Matsuda, Naoki Kanayama, Florian Hess, and Eiji Okamoto


The Ate pairing and the twisted Ate pairing for ordinary elliptic curves which are generalizations of the $\eta_T$ pairing for supersingular curves have previously been proposed. It is not necessarily the case that both pairings are faster than the Tate pairing. In this paper we propose optimized versions of the Ate and twisted Ate pairings with the loop reduction method and show that both pairings are always at least as fast as the Tate pairing. We also provide suitable families of elliptic curves that our optimized Ate and optimized twisted Ate pairings can be computed with half the loop length compared to the Tate pairing.

Note: I have modified the definition of $c_S$ and $c_{T^e}$ in section 2 and corrected the abstract.

Available format(s)
Publication info
Published elsewhere. Unknown where it was published
Ordinary curveTate pairingAte pairingTwisted Ate pairing
Contact author(s)
seichi @ cipher risk tsukuba ac jp
2007-01-19: received
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Creative Commons Attribution


      author = {Seiichi Matsuda and Naoki Kanayama and Florian Hess and Eiji Okamoto},
      title = {Optimised versions of the Ate and Twisted Ate Pairings},
      howpublished = {Cryptology ePrint Archive, Paper 2007/013},
      year = {2007},
      note = {\url{}},
      url = {}
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