2-Isogenies Between Elliptic Curves in Hesse Model
Resumo
Cryptosystems based on the problem of calculating isogenies between supersingular elliptic curves were recently proposed as strong candidates in the area of Post-Quantum Cryptography. In order to evaluate isogenies applied in cryptography constructions we use the Vèlu formula. However, this formula only applies to elliptic curves in the Weierstrass model. This paper presents morphisms that can be used to construct 2-isogeny formulas for curves in the Hesse model.
Referências
Boneh, D. and Lipton, R. J. (1995). Quantum cryptanalysis of hidden linear functions (extended abstract). In CRYPTO, volume 963 of Lecture Notes in Computer Science, pages 424–437. Springer.
Chen, L., Moody, D., and Liu, Y.-K. (2017). National institute of standards and technology’s post-quantum cryptography standardization.
Jao, D., Azarderakhs, R., Campagna, M., Costello, C., Feo, L. D., Hess, B., Jalali, A., Koziel, B., LaMacchia, B., Longa, P., Naehrig, M., Renes, J., Soukharev, V., and Urbanik, D. (2017). Supersingular isogeny key encapsulation. NIST Post-Quantum Cryptography Standardization,, Round 1 Submission.
Joye, M. and Quisquater, J.-J. (2001). Hessian elliptic curves and side-channel attacks. In Koç, Ç. K., Naccache, D., and Paar, C., editors, Cryptographic Hardware and Embedded Systems — CHES 2001, pages 402–410, Berlin, Heidelberg. Springer Berlin Heidelberg.
Joye, M., Tibouchi, M., and Vergnaud, D. (2010). Huff’s model for elliptic curves.
Moody, D. and Shumow, D. (2011). Analogues of velu’s formulas for isogenies on alternate models of elliptic curves. Cryptology ePrint Archive, Report 2011/430.
Silverman, J. (1986). The arithmetic of elliptic curves. Graduate Texts in Mathematics. Springer-Verlag, first edition.
Vélu, J. (1971). Isogénies entre courbes elliptiques. C.R. Acad. Sc. Paris, Série A(273):238–241.
Washington, L. C. (2008). Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC, second edition.
Xiu Xu, Wei Yu, K. W. X. H. (2016). Constructing isogenies on extended jacobi quartic curves. In International Conference on Information Security and Cryptology, Lecture Notes in Computer Science, pages 416–427. Chen K., Lin D., Yung M. (eds) Information Security and Cryptology, Springer, Cham.
Chen, L., Moody, D., and Liu, Y.-K. (2017). National institute of standards and technology’s post-quantum cryptography standardization.
Jao, D., Azarderakhs, R., Campagna, M., Costello, C., Feo, L. D., Hess, B., Jalali, A., Koziel, B., LaMacchia, B., Longa, P., Naehrig, M., Renes, J., Soukharev, V., and Urbanik, D. (2017). Supersingular isogeny key encapsulation. NIST Post-Quantum Cryptography Standardization,, Round 1 Submission.
Joye, M. and Quisquater, J.-J. (2001). Hessian elliptic curves and side-channel attacks. In Koç, Ç. K., Naccache, D., and Paar, C., editors, Cryptographic Hardware and Embedded Systems — CHES 2001, pages 402–410, Berlin, Heidelberg. Springer Berlin Heidelberg.
Joye, M., Tibouchi, M., and Vergnaud, D. (2010). Huff’s model for elliptic curves.
Moody, D. and Shumow, D. (2011). Analogues of velu’s formulas for isogenies on alternate models of elliptic curves. Cryptology ePrint Archive, Report 2011/430.
Silverman, J. (1986). The arithmetic of elliptic curves. Graduate Texts in Mathematics. Springer-Verlag, first edition.
Vélu, J. (1971). Isogénies entre courbes elliptiques. C.R. Acad. Sc. Paris, Série A(273):238–241.
Washington, L. C. (2008). Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC, second edition.
Xiu Xu, Wei Yu, K. W. X. H. (2016). Constructing isogenies on extended jacobi quartic curves. In International Conference on Information Security and Cryptology, Lecture Notes in Computer Science, pages 416–427. Chen K., Lin D., Yung M. (eds) Information Security and Cryptology, Springer, Cham.
Publicado
25/10/2018
Como Citar
SILVA, João Paulo da; DAHAB, Ricardo; LÓPEZ, Julio.
2-Isogenies Between Elliptic Curves in Hesse Model. In: SIMPÓSIO BRASILEIRO DE SEGURANÇA DA INFORMAÇÃO E DE SISTEMAS COMPUTACIONAIS (SBSEG), 18. , 2018, Natal.
Anais [...].
Porto Alegre: Sociedade Brasileira de Computação,
2018
.
p. 393-400.
DOI: https://doi.org/10.5753/sbseg.2018.4270.
