Abstract
Recently, some cryptographic primitives have been described that are based on the supposed hardness of finding an isogeny between two (supersingular) elliptic curves. As a part of such a primitive, Charles et al. proposed an algorithm for computing sequences of 2-isogenies.However, their method involves several redundant computations. We construct simple algorithms without such redundancy, based on very compact descriptions of the 2-isogenies. For that, we use some observations on 2-torsion points.
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References
Auer, R., Top, J.: Legendre elliptic curves over finite fields. J. Number Theory 95, 303–312 (2002)
Charles, D.X., Goren, E.Z., Lauter, K.E.: Cryptographic hash functions from expander graphs. To appear in Journal of Cryptology, electronically (2007), http://www.springerlink.com/
Cohen, H., Frey, G., et al.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman and Hall, Boca Raton (2006)
Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. AMS 43(4), 439–561 (2006)
Goldreich, O.: Candidate one-way functions based on expander graphs. Elect. Colloq. on Computational Complexity (ECCC) 7(090) (2000)
Goldreich, O.: Randomized Methods in Computation - Lecture Notes (2001), http://www.wisdom.weizmann.ac.il/~oded/rnd.html
Pizer, A.K.: Ramanujan graphs and Hecke operators. Bull. AMS 23(1), 127–137 (1990)
Rostovtsev, A., Stolbunov, A.: Public-key cryptosystem based on isogenies (preprint, 2006), http://eprint.iacr.org
Schoof, R.: Nonsingular plane cubic curves over finite fields. J. Comb. Th., Series A 46, 183–211 (1990)
Silverman, J.H.: The Arithmetic of Elliptic Curves, GTM 106. Springer, Heidelberg (1986)
Tate, J.: Endomorphisms of Abelian varieties over finite fields. Inv. Math. 2, 134–144 (1966)
Vélu, J.: Isogénies entre courbes elliptiques. Comptes-Rendus de l’Académie des Sciences 273, 238–241 (1971)
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Yoshida, R., Takashima, K. (2009). Simple Algorithms for Computing a Sequence of 2-Isogenies. In: Lee, P.J., Cheon, J.H. (eds) Information Security and Cryptology – ICISC 2008. ICISC 2008. Lecture Notes in Computer Science, vol 5461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00730-9_4
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DOI: https://doi.org/10.1007/978-3-642-00730-9_4
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