An Efficient Residue Group Multiplication for the ηT Pairing over \({\mathbb F}_{3^m}\)

  • Yuta Sasaki
  • Satsuki Nishina
  • Masaaki Shirase
  • Tsuyoshi Takagi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5867)


When we implement the η T pairing, which is one of the fastest pairings, we need multiplications in a base field \({\mathbb F}_{3^m}\) and in a group G. We have previously regarded elements in G as those in \({\mathbb F}_{3^{6m}}\) to implement the η T pairing. Gorla et al. proposed a multiplication algorithm in \({\mathbb F}_{3^{6m}}\) that takes 5 multiplications in \({\mathbb F}_{3^{2m}}\), namely 15 multiplications in \({\mathbb F}_{3^{m}}\). This algorithm then reaches the theoretical lower bound of the number of multiplications. On the other hand, we may also regard elements in G as those in the residue group \({\mathbb F}_{3^{6m}}^{\,*}\,/\,{\mathbb F}_{3^{m}}^{\,*}\) in which βa is equivalent to a for \(a \in {\mathbb F}_{3^{6m}}^{\,*}\) and \(\beta \in {\mathbb F}_{3^{m}}^{\,*}\). This paper proposes an algorithm for computing a multiplication in the residue group. Its cost is asymptotically 12 multiplications in \({\mathbb F}_{3^{m}}\) as m → ∞, which reaches beyond the lower bound the algorithm of Gorla et al. reaches. The proposed algorithm is especially effective when multiplication in the finite field is implemented using a basic method such as shift-and-add.


Finite field multiplication pairing residue group Vandermonde matrix 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Yuta Sasaki
    • 1
  • Satsuki Nishina
    • 1
  • Masaaki Shirase
    • 1
  • Tsuyoshi Takagi
    • 1
  1. 1.Future University Hakodate 

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