Abstract
Let E be a non-supersingular elliptic curve over a finite field \(\mathbb{F}_{\!q}\). At CRYPTO 2009, Icart introduced a deterministic function \(\mathbb{F}_{\!q}\to E(\mathbb{F}_{\!q})\) which can be computed efficiently, and allowed him and Coron to define well-behaved hash functions with values in \(E(\mathbb{F}_{\!q})\). Some properties of this function rely on a conjecture which was left as an open problem in Icart’s paper. We prove this conjecture below as well as analogues for other hash functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Boneh, D., Franklin, M.K.: Identity-based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)
Boyd, C., Montague, P., Nguyen, K.Q.: Elliptic curve based password-authenticated key exchange protocols. In: Varadharajan, V., Mu, Y. (eds.) ACISP 2001. LNCS, vol. 2119, pp. 487–501. Springer, Heidelberg (2001)
Boyko, V., MacKenzie, P.D., Patel, S.: Provably secure password-authenticated key exchange using Diffie-Hellman. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 156–171. Springer, Heidelberg (2000)
Brier, E., Coron, J.-S., Icart, T., Madore, D., Randriam, H., Tibouchi, M.: Efficient indifferentiable hashing to elliptic curves. In: Proceedings of CRYPTO 2010. LNCS. Springer, Heidelberg (2010)
Cox, D.A.: Galois theory. Series in Pure Mathematics. Wiley, Chichester (2004)
Fried, M.D., Jarden, M.: Field arithmetic, 2nd edn. Ergebnisse der Mathematik und ihre Grenzgebiete, vol. 11. Springer, Heidelberg (2002)
Gentry, C., Silverberg, A.: Hierarchical ID-based cryptography. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 548–566. Springer, Heidelberg (2002)
Farashahi, R.R., Shparlinski, I., Voloch, J.F.: On hashing into elliptic curves. J. Math. Crypt. 3(10), 353–360 (2009)
Icart, T.: How to hash into elliptic curves. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 303–316. Springer, Heidelberg (2009)
Kappe, L.C., Warren, B.: An elementary test for the Galois group of a quartic polynomial. Amer. Math. Monthly, Mathematical Association of America 96(2), 133–137 (1989)
Shallue, A., van de Woestjine, C.: 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)
Skałba, M.: Points on elliptic curves over finite fields. Acta Arith. impan 117, 293–301 (2005)
Ulas, M.: Rational points on certain hyperelliptic curves over finite fields. Bull. Polish Acad. Sci. Math. impan 55, 97–104 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fouque, PA., Tibouchi, M. (2010). Estimating the Size of the Image of Deterministic Hash Functions to Elliptic Curves. In: Abdalla, M., Barreto, P.S.L.M. (eds) Progress in Cryptology – LATINCRYPT 2010. LATINCRYPT 2010. Lecture Notes in Computer Science, vol 6212. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14712-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-14712-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14711-1
Online ISBN: 978-3-642-14712-8
eBook Packages: Computer ScienceComputer Science (R0)