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Abstract

Daniel Bernstein and Tanja Lange [9] proved that two given addition formulas on twisted Edwards elliptic curves ax2 +y2 = 1+dxy are complete (i.e. the sum of any two points on a curve can be computed using one of these formulas). In this paper we give simple verification of completeness of these formulas using a program written in Magma, which is based on the fact that completeness means that some systems of polynomial equations have no solutions. This method may also be useful to verify completeness of additions formulas on other models of elliptic curves.

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Authors and Affiliations

Robert Dryło
Tomasz Kijko
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Bibliography

[1] D. J. Bernstein and T. Lange, “Montgomery curves and the montgomery ladder.” IACR Cryptol. ePrint Arch., vol. 2017, p. 293, 2017.
[2] C. Costello and B. Smith, “Montgomery curves and their arithmetic,” Journal of Cryptographic Engineering, vol. 8, no. 3, pp. 227–240, 2018.
[3] P. L. Montgomery, “Speeding the pollard and elliptic curve methods of factorization,” Mathematics of Computation, vol. 48, pp. 243–264, 1987.
[4] E. Brier and M. Joye, “Weierstraß elliptic curves and side-channel attacks,” in International workshop on public key cryptography. Springer, 2002, pp. 335–345.
[5] R. R. Farashahi and S. G. Hosseini, “Differential addition on twisted edwards curves,” in Australasian Conference on Information Security and Privacy. Springer, 2017, pp. 366–378.
[6] B. Justus and D. Loebenberger, “Differential addition in generalized edwards coordinates,” in International Workshop on Security. Springer, 2010, pp. 316–325.
[7] R. R. Farashahi and M. Joye, “Efficient arithmetic on hessian curves,” in International Workshop on Public Key Cryptography. Springer, 2010, pp. 243–260.
[8] W. Castryck and F. Vercauteren, “Toric forms of elliptic curves and their arithmetic,” Journal of Symbolic Computation, vol. 46, no. 8, pp. 943–966, 2011.
[9] R. Dryło, T. Kijko, and M. Wro´nski, “Determining formulas related to point compression on alternative models of elliptic curves,” Fundamenta Informaticae, vol. 169, no. 4, pp. 285–294, 2019.
[10] K. Okeya and K. Sakurai, “Efficient elliptic curve cryptosystems from a scalar multiplication algorithm with recovery of the y-coordinate on a montgomery-form elliptic curve,” in International Workshop on Cryptographic Hardware and Embedded Systems. Springer, 2001, pp. 126–141.
[11] M. Joye, M. Tibouchi, and D. Vergnaud, “Huff’s model for elliptic curves,” in International Algorithmic Number Theory Symposium. Springer, 2010, pp. 234–250.
[12] H. Wu and R. Feng, “Elliptic curves in huff’s model,” Wuhan University Journal of Natural Sciences, vol. 17, no. 6, pp. 473–480, 2012.
[13] T. Oliveira, J. L´opez, H. Hıs¸ıl, A. Faz-Hern´andez, and F. Rodr´ıguez- Henr´ıquez, “How to (pre-) compute a ladder,” in International Conference on Selected Areas in Cryptography. Springer, 2017, pp. 172–191.
[14] R. R. Farashahi and S. G. Hosseini, “Differential addition on binary elliptic curves,” in International Workshop on the Arithmetic of Finite Fields. Springer, 2016, pp. 21–35.
[15] D. Moody and D. Shumow, “Analogues of v´elu’s formulas for isogenies on alternate models of elliptic curves,” Mathematics of Computation, vol. 85, no. 300, pp. 1929–1951, 2016.
[16] C. Costello and H. Hisil, “A simple and compact algorithm for sidh with arbitrary degree isogenies,” in International Conference on the Theory and Application of Cryptology and Information Security. Springer, 2017, pp. 303–329.
[17] D. Jao, R. Azarderakhsh, M. Campagna, C. Costello, L. Feo, B. Hess, A. Jalali, B. Koziel, B. LaMacchia, P. Longa, M. Naehrig, G. Pereira, J. Renes, V. Soukharev, and D. Urbanik, “Supersingular isogeny key encapsulation,” 04 2019.
[18] D. Jeon, C. H. Kim, and Y. Lee, “Families of elliptic curves over quartic number fields with prescribed torsion subgroups,” Mathematics of Computation, vol. 80, no. 276, pp. 2395–2410, 2011.

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Authors and Affiliations

Robert Dryło
1
Tomasz Kijko
1
Michał Wroński
1

  1. Institute of Mathematics and Cryptology, Faculty of Cybernetics, Military University of Technology, Warsaw, Poland

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