Advertisement

On Indifferentiable Hashing into the Jacobian of Hyperelliptic Curves of Genus 2

  • Michel Seck
  • Hortense Boudjou
  • Nafissatou DiarraEmail author
  • Ahmed Youssef Ould Cheikh Khlil
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10239)

Abstract

Many authors have studied the problem of constructing indifferentiable and deterministic hash functions into elliptic and hyperelliptic curves with well-distributed encodings. In this work, we have designed three encodings suitable for indifferentiable hashing for the following hyperellitic curves of genus 2: \(\mathbb {H}^{1}: y^{2}=F_{1}(x)=x^{5}+ax^{4}+cx^{2}+dx, \ \mathbb {H}^{2}: y^{2}=F_{2}(x)=x^{5}+bx^{3}+dx+e; \ \mathbb {H}^{3}: y^{2}=F_{3}(x)=x^{5}+ax^{4}+e\). Since they are well-distributed, our encodings can be used to design indifferentiable and deterministic hash functions into the Jacobian of these hyperelliptic curves, using the technique developed by Farashahi et al. in 2013 (J. Math. Comput). Because of square rooting steps, these new encodings have the same asymptotic complexity as the work of Kammerer et al. at Pairing 2010, namely \(\mathcal {O}(\log ^{2+\circ (1)}q)\).

Keywords

Indifferentiable deterministic hashing Injective encoding Elliptic curve-based cryptography Jacobian Elligator Random bit-string 

References

  1. 1.
    Daniel, J., Bernstein, M., Hamburg, A., Krasnova, T.L.: Elligator: elliptic-curve points indistinguishable from uniform random strings. In: Gligor, V., Yung, M. (eds.) CCS. ACM (2013)Google Scholar
  2. 2.
    Boneh, D., Franklin, M.: Identity-based encryption from the weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001). doi: 10.1007/3-540-44647-8_13 CrossRefGoogle Scholar
  3. 3.
    Brier, E., Coron, J.-S., Icart, T., Madore, D., Randriam, H., Tibouchi, M.: Efficient indifferentiable hashing into ordinary elliptic curves. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 237–254. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14623-7_13 CrossRefGoogle Scholar
  4. 4.
    Mac Kenzie, P.: An efficient two-party public key cryptosystem secure against adaptive chosen ciphertext attack. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 47–61. Springer, Heidelberg (2003). doi: 10.1007/3-540-36288-6_4 CrossRefGoogle Scholar
  5. 5.
    Fouque, P.-A., Joux, A., Tibouchi, M.: Injective encodings to elliptic curves. In: Boyd, C., Simpson, L. (eds.) ACISP 2013. LNCS, vol. 7959, pp. 203–218. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39059-3_14 CrossRefGoogle Scholar
  6. 6.
    Fouque, P.-A., Tibouchi, M.: Deterministic encoding and hashing to odd hyperelliptic curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 265–277. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-17455-1_17 CrossRefGoogle Scholar
  7. 7.
    Farashahi, R.R.: Hashing into Hessian curves. Int. J. Appl. Crypt. 3(2), 139–147 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Farashahi, R.R., Fouque, P.-A., Shparlinski, I.E., Tibouchi, M., Voloch, J.F.: Indifferentiable deterministic hashing to elliptic and hyperelliptic curves. Math. Comput. 82(281), 491–512 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hamburg, M.: Decaf: eliminating cofactors through point compression. In: Proceedings of the 35th Annual Cryptology Conference, Santa Barbara, CA, USA, 16–20 August 2015Google Scholar
  10. 10.
    Haneda, M., Kawazoe, M., Takahashi, T.: Suitable curves for genus-4 HCC over prime fields: point counting formulae for hyperelliptic curves of type y 2=x \(^{\rm 2{k}+1}\)+ax. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 539–550. Springer, Heidelberg (2005). doi: 10.1007/11523468_44 CrossRefGoogle Scholar
  11. 11.
    Horwitz, J., Lynn, B.: Toward hierarchical identity-based encryption. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 466–481. Springer, Heidelberg (2002). doi: 10.1007/3-540-46035-7_31 CrossRefGoogle Scholar
  12. 12.
    Icart, T.: How to hash into elliptic curves. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 303–316. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-03356-8_18 CrossRefGoogle Scholar
  13. 13.
    Kammerer, J.-G., Lercier, R., Renault, G.: Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 278–297. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-17455-1_18 CrossRefGoogle Scholar
  14. 14.
    Menezes, A.J., Wu, Y.-H., Zuccherato, R.J.: An elementary introduction to hyperelliptic curves. In: Koblitz, N. (ed.) Algebraic Aspects of Cryptography. Algorithms and Computation in Mathematics, vol. 3, pp. 155–178. Springer, Heidelberg (1998)Google Scholar
  15. 15.
    Möller, B.: A public-key encryption scheme with pseudo-random ciphertexts. In: Samarati, P., Ryan, P., Gollmann, D., Molva, R. (eds.) ESORICS 2004. LNCS, vol. 3193, pp. 335–351. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-30108-0_21 CrossRefGoogle Scholar
  16. 16.
    Satoh, T.: Generating genus two hyperelliptic curves over large characteristic finite fields. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 536–553. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-01001-9_31 CrossRefGoogle Scholar
  17. 17.
    Shallue, A., Woestijne, C.E.: Construction of rational points on elliptic curves over finite fields. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 510–524. Springer, Heidelberg (2006). doi: 10.1007/11792086_36 CrossRefGoogle Scholar
  18. 18.
    Tibouchi, M.: Hachage vers les courbes elliptiques et cryptanalyse de schémas RSA. Thèse de doctorat de l’Université Paris-Diderot-Luxembourg, Septembre 2011Google Scholar
  19. 19.
    Ulas, M.: Rational points on certain hyperelliptic curves over finite fields. Bull. Pol. Acad. Sci. Math. 55(2), 97–104 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Michel Seck
    • 1
  • Hortense Boudjou
    • 2
  • Nafissatou Diarra
    • 1
    Email author
  • Ahmed Youssef Ould Cheikh Khlil
    • 1
  1. 1.Department of Mathematics-InformaticsCheikh Anta Diop UniversityDakarSenegal
  2. 2.Maroua UniversityMarouaCameroon

Personalised recommendations