AN ALGORITHM FOR COMPUTING A SEQUENCE OF RICHELOT ISOGENIES

Title & Authors
AN ALGORITHM FOR COMPUTING A SEQUENCE OF RICHELOT ISOGENIES
Takashima, Katsuyuki; Yoshida, Reo;

Abstract
We show that computation of a sequence of Richelot isogenies from specified supersingular Jacobians of genus-2 curves over $\small{\mathbb{F}_p}$ can be executed in $\small{\mathbb{F}_{p2}}$ or $\small{\mathbb{F}_{p4}}$ . Based on this, we describe a practical algorithm for computing a Richelot isogeny sequence.
Keywords
hyperelliptic curve;genus two;Richelot isogeny;isogeny graph;supersingular curve;
Language
English
Cited by
1.
Computing a Sequence of 2-Isogenies on Supersingular Elliptic Curves, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2013, E96.A, 1, 158
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

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

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

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

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

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

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