Eta pairing computation on general divisors over hyperelliptic curves y2 = xp - x + d

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Recent developments on the Tate or Eta pairing computation over hyperelliptic curves by Duursma-Lee and Barreto et al. have focused on degenerate divisors. We present efficient methods that work for general divisors to compute the Eta paring over divisor class groups of the hyperelliptic curves Hd : y2 = xp - x + d where p is an odd prime. On the curve Hd of genus 3, we provide two efficient methods: The first method generalizes the method of Barreto et al. so that it holds for general divisors, and we call it the pointwise method. For the second method, we take a novel approach using resultant. Our analysis shows that the resultant method is faster than the pointwise method, and our implementation result supports the theoretical analysis. We also emphasize that the Eta pairing technique is generalized to the curve y2 = xp - x + d, p ≡ 1 (mod 4). Furthermore, we provide the closed formula for the Eta pairing computation on general divisors by Mumford representation of the curve Hd of genus 2.

Original languageEnglish
Pages (from-to)452-474
Number of pages23
JournalJournal of Symbolic Computation
Issue number6-7
StatePublished - Jun 2008

Bibliographical note

Funding Information:
We thank the anonymous referees and Cheol-Min Park for their helpful comments. The authors Eunjeong Lee and Hyang-Sook Lee were supported by KOSEF, grant number R01-2005-000-10713-0, and the author Yoonjin Lee was supported by NSERC. The authors also express their gratitude to KIAS. An extended abstract of this paper appears in the proceedings of Pairing 2007, vol. 4575, LNCS, page 349–366, Springer-Verlag, 2007.


  • Divisors
  • Eta pairing
  • Hyperelliptic curves
  • Pairing-based cryptosystem
  • Resultant
  • Tate pairing


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