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

Research output: Contribution to journalArticle

4 Citations (Scopus)


It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log2 r/φ(k), where φ(·) 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 ⌊log2 r⌋ to ⌊log2(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.

Original languageEnglish
Pages (from-to)1859-1867
Number of pages9
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number8
Publication statusPublished - Aug 2009



  • Ate pairing
  • Miller's algorithm

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics

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