Implementing Cryptography Pairings over Ordinary Pairing-Friendly Curves of Type \( y^2 = x^5 +a \, x\)

  • Mohammed ZitouniEmail author
  • Farid Mokrane
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12001)


In this paper, we describe an efficient implementation in Sage of the Tate pairing over ordinary hyperelliptic curves of type \( y^2 = x^5 +a \, x\). First, we describe a method of construction of these curves according to Kawazoe and Takahashi [8]. Then, we describe an efficient formula for computing pairings on such curves over prime fields, and develop algorithms to compute Tate pairing. We provide a faster optimisation of the final exponentiation in particular for the embedding degree \(k = 28\).


Hyperelliptic curve Tate pairing Finite field Embedding degree Final exponentiation 


  1. 1.
    Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comput. 48(177), 95–101 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Galbraith, S.D., Hess, F., Vercauteren, F.: Hyperelliptic pairings. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 108–131. Springer, Heidelberg (2007). Scholar
  3. 3.
    Gaudry, P., Harley, R.: Counting points on hyperelliptic curves over finite fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 313–332. Springer, Heidelberg (2000). Scholar
  4. 4.
    Granger, R., Hess, F., Oyono, R., Thériault, N., Vercauteren, F.: Ate pairing on hyperelliptic curves. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 430–447. Springer, Heidelberg (2007). Scholar
  5. 5.
    Granger, R., Page, D., Smart, N.P.: High security pairing-based cryptography revisited. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 480–494. Springer, Heidelberg (2006). Scholar
  6. 6.
    Joux, A.: A one round protocol for Tripartite Diffie–Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–393. Springer, Heidelberg (2000). Scholar
  7. 7.
    Kachisa, E.J.: Generating more kawazoe-takahashi genus 2 pairing-friendly hyperelliptic curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 312–326. Springer, Heidelberg (2010). Scholar
  8. 8.
    Kawazoe, M., Takahashi, T.: Pairing-friendly hyperelliptic curves with ordinary Jacobians of Type \(y^2=x^5+ax\). In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 164–177. Springer, Heidelberg (2008). Scholar
  9. 9.
    Koblitz, N.: Hyperelliptic cryptosystems. J. Cryptol. 1(3), 139–150 (1989)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Manin, J.I.: The Hasse-Witt matrix of an algebraic curve. In: Selected Papers of Yu I Manin, pp. 3–22. World Scientific (1996)Google Scholar
  11. 11.
    Menezes, A.J., Okamoto, T., Vanstone, S.A.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inf. Theory 39(5), 1639–1646 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Miller, V., et al.: Short programs for functions on curves. 97(101–102), 44 (1986, Unpublished manuscript)Google Scholar
  13. 13.
    Mumford, D.: Tata Lectures on Theta i, ii. Birkhäuser, Boston (1984)zbMATHGoogle Scholar
  14. 14.
    Pollard, J.M.: Monte carlo methods for index computation. Math. Comput. 32(143), 918–924 (1978)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Van Oorschot, P.C., Wiener, M.J.: Parallel collision search with cryptanalytic applications. J. Cryptol. 12(1), 1–28 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yui, N.: On the Jacobian varieties of hyperelliptic curves over fields of characteristic \(p> 2\). J. Algebra 52(2), 378–410 (1978)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratory Analysis, Geometry and Applications CNRS (UMR 7539), Galilee InstituteParis 13 UniversityVilletaneuseFrance

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