Geodesic
Duffing Oscillator and Elliptic Curve Cryptography
Russian journal of nonlinear dynamics, Tome 14 (2018) no. 2, pp. 235-241.

Voir la notice de l'article provenant de la source Math-Net.Ru

A new approach to exact discretization of the Duffing equation is presented. Integrable discrete maps are obtained by using well-studied operations from the elliptic curve cryptography.
Keywords: integrable maps, divisor arithmetic. 
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A. V. Tsiganov. Duffing Oscillator and Elliptic Curve Cryptography. Russian journal of nonlinear dynamics, Tome 14 (2018) no. 2, pp. 235-241. http://geodesic.mathdoc.fr/item/ND_2018_14_2_a6/

[1] El-Sayed, A. M. A., El-Raheem, Z. F. E., and Salman, S. M., “Discretization of Forced Duffing System with Fractional-Order Damping”, Adv. Diff. Equ., 2014:1 (2014), 66, 12 pp. | DOI

[2] Baker, H. F., Abelian Functions: Abel's Theorem and the Allied Theory of Theta Functions, Cambridge Univ. Press, Cambridge, 1996, 724 pp.

[3] Bos, J. W. , Halderman, J. A., Heninger, N., Moore, J., Naehrig, M., and Wustrow, E., “Elliptic Curve Cryptography in Practice”, Financial Cryptography and Data Security: Proc. of the 18th Internat. Conf. on Financial Cryptography and Data Security (Barbados, March 3–7, 2014), Security and Cryptology, 8437, eds. N. Christin, R. Safavi-Naini, Springer, Berlin, 2014, 157–175

[4] Costello, C. and Lauter, K., “Group Law Computations on Jacobians of Hyperelliptic Curves”, Selected Areas in Cryptography: Proc. of the 18th Internat. Workshop (Toronto, August 11–12, 2011), Lecture Notes in Comput. Sci., 7118, eds. A. Miri, S. Vaudenay, Springer, Berlin, 2012, 92–117 | DOI | Zbl

[5] Costello, C., A Gentle Introduction to Isogeny-Based Cryptography, Tutorial Talk at SPACE'2016 (Hyderabad, India, Dec 15, 2016)

[6] Greenhill, A. G., The Applications of Elliptic Functions, Dover, New York, 1959, xii+357 pp. | Zbl

[7] H. Cohen, G. Frey, R. Avanzi, Ch. Doche, T. Lange, K. Nguyen, F. Vercauteren (eds.), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Math. Appl. (Boca Raton), Chapman Hall/CRC, Boca Raton, Fla., 2006, xxxiv+808 pp. | Zbl

[8] Harley, R., Fast Arithmetic on Genus Two Curves, }, 2000 {\tt http://cristal.inria.fr/h̃arley/hyper/

[9] Hisil, H., Wong, K. K., Carter, G., and Dawson, E., “Jacobi Quartic Curves Revisited”, Information Security and Privacy: Proc. of the 14th Australasian Conference on Information Security and Privacy, ACISP'2009 (Brisbane, Australia, July 1–3, 2009), eds. C. Boyd, J. González Nieto, Springer, Berlin, 2009, 452–468 | Zbl

[10] Kuznetsov, V. and Vanhaecke, P., “Bäcklund Transformations for Finite-Dimensional Integrable Systems: A Geometric Approach”, J. Geom. Phys., 44:1 (2002), 1–40 | DOI | Zbl

[11] Murakami, C., Murakami, W., Hirose, K., and Ichikawa, Y. H., “Integrable Duffing’s Maps and Solutions of the Duffing Equation”, Chaos Solitons Fractals, 15:3 (2003), 425–443 | DOI | Zbl

[12] Murakami, C., Murakami, W., Hirose, K., and Ichikawa, Y. H., “Global Periodic Structure of Integrable Duffing’s Maps”, Chaos Solitons Fractals, 16:2 (2003), 233–244 | DOI | Zbl

[13] Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979, 720 pp. | Zbl

[14] Potts, R. B., “Exact Solution of a Difference Approximation to Duffing's Equation”, J. Austral. Math. Soc. Ser. B, 23:1 (1981/82), 64–77 | DOI | Zbl

[15] Potts, R. B., “Best Difference Equation Approximation to Duffing's Equation”, J. Austral. Math. Soc. Ser. B, 23:4 (1981/82), 349–356 | DOI

[16] Reinhall, P. G., Caughey, T. K., and Storti, D. W., “Order and Chaos in a Discrete Duffing Oscillator: Implications on Numerical Integration”, Trans. ASME J. Appl. Mech., 56:1 (1989), 162–167 | DOI | Zbl

[17] Funktsional. Anal. i Prilozhen., 23:1 (1989), 84–85 (Russian) | DOI | Zbl

[18] Suris, Yu. B., The Problem of Integrable Discretization: Hamiltonian Approach, Progr. Math., 219, Birkhäuser, Boston, Mass., 2003, xxii+1070 pp. | Zbl

[19] Tsiganov, A. V., “Simultaneous Separation for the Neumann and Chaplygin Systems”, Regul. Chaotic Dyn., 20:1 (2015), 74–93 | DOI | Zbl

[20] Tr. Mat. Inst. Steklova, 295, 2016, 261–291 | DOI | Zbl

[21] Tsiganov, A. V., “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179 | DOI | Zbl

[22] Tsiganov, A. V., “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367 | DOI | Zbl

[23] Tsiganov, A. V., “New Bi-Hamiltonian Systems on the Plane”, J. Math. Phys., 58:6 (2017), 062901, 14 pp. | DOI | Zbl

[24] Tsiganov, A. V., “Bäcklund Transformations and Divisor Doubling”, J. Geom. Phys., 126 (2018), 148–158 | DOI | Zbl