Abstract
We present an algorithm based on the birthday paradox, which is a low-memory parallel counterpart to the algorithm of Matsuo, Chao and Tsujii. This algorithm computes the group order of the Jacobian of a genus 2 curve over a finite field for which the characteristic polynomial of the Frobenius endomorphism is known modulo some integer. The main tool is a 2-dimensional pseudo-random walk that allows to heuristically choose random elements in a 2-dimensional space. We analyze the expected running time based on heuristics that we validate by computer experiments. Compared with the original algorithm by Matsuo, Chao and Tsujii, we lose a factor of about 3 in running time, but the memory requirement drops from several GB to almost nothing. Our method is general and can be applied in other contexts to transform a baby-step giant-step approach into a low memory algorithm.
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
MPICH: A portable implementation of MPI, http://www-unix.mcs.anl.gov/mpi/mpich/
Adleman, L., Huang, M.-D.: Counting points on curves and abelian varieties over finite fields. J. Symbolic Comput. 32, 171–189 (2001)
Bauer, M., Teske, E., Weng, A.: Point counting on Picard curves in large characteristic (2003) (preprint)
Blake, I., Seroussi, G., Smart, N.: Elliptic curves in cryptography. London Math. Soc. Lecture Note Ser., vol. 265. Cambridge University Press, Cambridge (1999)
Bosma, W., Cannon, J.: Handbook of Magma functions (1997), http://www.maths.usyd.edu.au:8000/u/magma/
Bostan, A., Gaudry, P., Schost, É.: Linear recurrences with polynomial coefficients and computation of the Cartier-Manin operator on hyperelliptic curves. To appear in Proceedings Fq’7 (2003)
Delescaille, J.-P., Quisquater, J.-J.: How easy is collision search? Application to DES. In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 429–434. Springer, Heidelberg (1990)
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)
Gaudry, P., Schost, É.: Construction of secure random curves of genus 2 over prime fields. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 239–256. Springer, Heidelberg (2004) (to appear)
Huang, M.-D., Ierardi, D.: Counting points on curves over finite fields. J. Symbolic Comput. 25, 1–21 (1998)
Izadi, F., Murty, K.: Counting points on an abelian variety over a finite field (2003) (preprint)
Kedlaya, K.S.: Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. J. Ramanujan Math. Soc. 16(4), 323–338 (2001)
Lenstra Jr., H.W., Pila, J., Pomerance, C.: A hyperelliptic smoothness test, II. Proc. London Math. Soc. 84, 105–146 (2002)
Matsuo, K., Chao, J., Tsujii, S.: An improved baby step giant step algorithm for point counting of hyperelliptic curves over finite fields. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 461–474. Springer, Heidelberg (2002)
Mestre, J.-F.: Utilisation de l’AGM pour le calcul de E(\(\mathbb{F}\) \(_{2^{n}}\)). Letter to Gaudry and Harley (December 2000)
Pila, J.: Frobenius maps of abelian varieties and finding roots of unity in finite fields. Math. Comp. 55(192), 745–763 (1990)
Pollard, J.M.: Monte Carlo methods for index computation mod p. Math. Comp. 32(143), 918–924 (1978)
Satoh, T.: The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc. 15, 247–270 (2000)
Shoup, V.: NTL: A library for doing number theory, http://www.shoup.net/ntl/
Stein, A., Teske, E.: Explicit bounds and heuristics on class numbers in hyperelliptic function fields. Math. Comp. 71, 837–861 (2002)
Stein, A., Teske, E.: The parallelized Pollard kangaroo method in real quadratic function fields. Math. Comp. 71, 793–814 (2002)
Teske, E.: Speeding up Pollard’s rho method for computing discrete logarithms. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 541–554. Springer, Heidelberg (1998)
Teske, E.: Computing discrete logarithms with the parallelized kangaroo method. Discrete Appl. Math. 130, 61–82 (2003)
van Oorschot, P.C., Wiener, M.J.: Parallel collision search with cryptanalytic applications. J. of Cryptology 12, 1–28 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gaudry, P., Schost, É. (2004). A Low-Memory Parallel Version of Matsuo, Chao, and Tsujii’s Algorithm. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-24847-7_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22156-2
Online ISBN: 978-3-540-24847-7
eBook Packages: Springer Book Archive