Cryptography and Coding 2007: Cryptography and Coding pp 313-335

Extractors for Jacobian of Hyperelliptic Curves of Genus 2 in Odd Characteristic

• Reza Rezaeian Farashahi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4887)

Abstract

We propose two simple and efficient deterministic extractors for $$J(\mathbb{F}_q)$$, the Jacobian of a genus 2 hyperelliptic curve H defined over $$\mathbb{F}_q$$, for some odd q. Our first extractor, $$\texttt{SEJ}$$, called sum extractor, for a given point D on $$J(\mathbb{F}_q)$$, outputs the sum of abscissas of rational points on H in the support of D, considering D as a reduced divisor. Similarly the second extractor, $$\texttt{PEJ}$$, called product extractor, for a given point D on the $$J(\mathbb{F}_q)$$, outputs the product of abscissas of rational points in the support of D. Provided that the point D is chosen uniformly at random in $$J(\mathbb{F}_q)$$, the element extracted from the point D is indistinguishable from a uniformly random variable in $$\mathbb{F}_q$$. Thanks to the Kummer surface $$\mathcal{K}$$, that is associated to the Jacobian of H over $$\mathbb{F}_q$$, we propose the sum and product extractors, $$\texttt{SEK}$$ and $$\texttt{PEK}$$, for $$\mathcal{K}(\mathbb{F}_q)$$. These extractors are the modified versions of the extractors $$\texttt{SEJ}$$ and $$\texttt{PEJ}$$. Provided a point K is chosen uniformly at random in $$\mathcal{K}$$, the element extracted from the point K is statistically close to a uniformly random variable in $$\mathbb{F}_q$$.

Keywords

Jacobian Hyperelliptic curve Kummer surface Deterministic extractor

References

1. 1.
Artin, E.: Algebraic Numbers and Algebraic Functions. Gordon and Breach, New York (1967)
2. 2.
Avanzi, R.M.: Aspects of Hyperelliptic Curves over Large Prime Fields in Software Implementations. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 148–162. Springer, Heidelberg (2004)Google Scholar
3. 3.
Cantor, D.: Computing in the Jacobian of a Hyperelliptic Curve. Mathematics of Computation 48(177), 95–101 (1987)
4. 4.
Cassels, J.W.S., Flynn, E.V.: Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2. Cambridge University Press, Cambridge (1996)
5. 5.
Chevassut, O., Fouque, P., Gaudry, P., Pointcheval, D.: The Twist-Augmented Technique for Key Exchange. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 410–426. Springer, Heidelberg (2006)
6. 6.
Cohen, H., Frey, G.: Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, New York (2006)Google Scholar
7. 7.
Duquesne, S.: Montgomery Scalar Multiplication for Genus 2 Curves. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 153–168. Springer, Heidelberg (2004)Google Scholar
8. 8.
Farashahi, R.R., Pellikaan, R.: The Quadratic Extension Extractor for (Hyper)Elliptic Curves in Odd Characteristic. In: WAIFI 2007. LNCS, vol. 4547, pp. 219–236. Springer, Heidelberg (2007)Google Scholar
9. 9.
Farashahi, R.R., Pellikaan, R., Sidorenko, A.: Extractors for Binary Elliptic Curves. In: WCC 2007. Workshop on Coding and Cryptography, pp. 127–136 (2007)Google Scholar
10. 10.
Gaudry, P.: An Algorithm for Solving the Discrete Log Problem on Hyperelliptic Curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 3419–3448. Springer, Heidelberg (2000)
11. 11.
Gaudry, P.: Fast genus 2 arithmetic based on Theta functions, Cryptology ePrint Archive, Report 2005/314 (2005), http://eprint.iacr.org/
12. 12.
Gürel, N.: Extracting bits from coordinates of a point of an elliptic curve, Cryptology ePrint Archive, Report 2005/324 (2005), http://eprint.iacr.org/
13. 13.
Juels, A., Jakobsson, M., Shriver, E., Hillyer, B.K.: How to turn loaded dice into fair coins. IEEE Transactions on Information Theory 46(3), 911–921 (2000)
14. 14.
Kaliski, B.S.: A Pseudo-Random Bit Generator Based on Elliptic Logarithms. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 84–103. Springer, Heidelberg (1987)Google Scholar
15. 15.
Koblitz, N.: Hyperelliptic Cryptosystem. J. of Cryptology 1, 139–150 (1989)
16. 16.
Lange, T.: Montgomery Addition for Genus Two Curves. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 307–309. Springer, Heidelberg (2004)Google Scholar
17. 17.
Lange, T.: Formulae for Arithmetic on Genus 2 Hyperelliptic Curves. aaecc 15(1), 295–328 (2005)
18. 18.
Luby, M.: Pseudorandomness and Cryptographic Applications. Princeton University Press, USA (1994)Google Scholar
19. 19.
Mumford, D.: Tata Lectures on Theta II. In: Progress in Mathematics, vol. 43 (1984)Google Scholar
20. 20.
Shaltiel, R.: Recent Developments in Explicit Constructions of Extractors. Bulletin of the EATCS 77, 67–95 (2002)
21. 21.
Smart, N.P., Siksek, S.: A Fast Diffie-Hellman Protocol in Genus 2. Journal of Cryptology 12, 67–73 (1999)
22. 22.
Trevisan, L., Vadhan, S.: Extracting Randomness from Samplable Distributions. In: IEEE Symposium on Foundations of Computer Science, pp. 32–42 (2000)Google Scholar