Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

AN ALGORITHM FOR COMPUTING A SEQUENCE OF RICHELOT ISOGENIES

  • Takashima, Katsuyuki (INFORMATION TECHNOLOGY R&D CENTER MITSUBISHI ELECTRIC) ;
  • Yoshida, Reo (DEPARTMENT OF SOCIAL INFORMATICS GRADUATE SCHOOL OF INFORMATICS KYOTO UNIVERSITY)
  • Published : 2009.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

We show that computation of a sequence of Richelot isogenies from specified supersingular Jacobians of genus-2 curves over $\mathbb{F}_p$ can be executed in $\mathbb{F}_{p2}$ or $\mathbb{F}_{p4}$ . Based on this, we describe a practical algorithm for computing a Richelot isogeny sequence.

Keywords

References

  1. P. R. Bending, Curves of genus 2 with p2 multiplication, Ph. D. Thesis, University of Oxford, 1998
  2. J.-B. Bost and J.-F. Mestre, Moyenne arithmetico-geometrique et periodes des courbes de genre 1 et 2, Gaz. Math. No. 38 (1988), 36–64
  3. J. W. S. Cassels and E. V. Flynn, Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996
  4. H. Cohen and G. Frey et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall, 2006
  5. D. X. Charles, E. Z. Goren, and K. E. Lauter, Cryptographic hash functions from expander graphs, to appear in Journal of Cryptology https://doi.org/10.1007/s00145-007-9002-x
  6. D. X. Charles, E. Z. Goren, and K. E. Lauter, Families of Ramanujan graphs and quaternion algebras, to appear in AMSCRM volume 'Groups and Symmetries' in honor of John McKay
  7. Y. J. Choie, E. K. Jeong, and E. J. Lee, Supersingular hyperelliptic curves of genus 2 over finite fields, Appl. Math. Comput. 163 (2005), no. 2, 565–576 https://doi.org/10.1016/j.amc.2004.03.030
  8. T. Ibukiyama, T. Katsura, and F. Oort, Supersingular curves of genus two and class numbers, Compositio Math. 57 (1986), no. 2, 127–152
  9. S. Paulus and H.-G. Ruck, Real and imaginary quadratic representations of hyperelliptic function fields, Math. Comp. 68 (1999), no. 227, 1233–1241 https://doi.org/10.1090/S0025-5718-99-01066-2
  10. S. Paulus and A. Stein, Comparing real and imaginary arithmetics for divisor class groups of hyperelliptic curves, Algorithmic number theory (Portland, OR, 1998), 576–591, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998 https://doi.org/10.1007/BFb0054894
  11. A. Rostovtsev and A. Stolbunov, Public-key cryptosystem based on isogenies, preprint, IACR ePrint 2006/145
  12. B. Smith, Explicit endomorphisms and correspondences, Ph. D. Thesis, The Univ. of Sydney, 2005
  13. J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144 https://doi.org/10.1007/BF01404549
  14. J. Velu, Isogenies entre courbes elliptiques, C. R. Acad. Sci. Paris Ser. A-B 273 (1971), A238–A241
  15. C. Xing, On supersingular abelian varieties of dimension two over finite fields, Finite Fields Appl. 2 (1996), no. 4, 407–421 https://doi.org/10.1006/ffta.1996.0024
  16. R. Yoshida and K. Takashima, Simple algorithms for computing a sequence of 2- isogenies, ICISC 2008, LNCS No. 5461, pp. 52–65, Springer Verlag, 2009 https://doi.org/10.1007/978-3-642-00730-9_4

Cited by

  1. Computing a Sequence of 2-Isogenies on Supersingular Elliptic Curves vol.E96.A, pp.1, 2013, https://doi.org/10.1587/transfun.E96.A.158