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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

2 Issues per year


CiteScore 2016: 0.35

SCImago Journal Rank (SJR) 2016: 0.372
Source Normalized Impact per Paper (SNIP) 2016: 0.517

Mathematical Citation Quotient (MCQ) 2016: 0.23

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1869-6104
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Cryptography from the tropical Hessian pencil

Jean-Marie Chauvet / Eric Mahé
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/gcc-2017-0002

Abstract

Recent work by Grigoriev and Shpilrain [8] suggests looking at the tropical semiring for cryptographic schemes. In this contribution we explore the tropical analogue of the Hessian pencil of plane cubic curves as a source of group-based cryptography. Using elementary tropical geometry on the tropical Hessian curves, we derive the addition and doubling formulas induced from their Jacobian and investigate the discrete logarithm problem in this group. We show that the DLP is solvable when restricted to integral points on the tropical Hesse curve, and hence inadequate for cryptographic applications. Consideration of point duplication, however, provides instances of solvable chaotic maps producing random sequences and thus a source of fast keyed hash functions.

Keywords: Tropical geometry; discrete dynamical systems; keyed hash function; elliptic curve cryptography

MSC 2010: 14H40; 14H52; 14K25; 37E05; 37F10

References

  • [1]

    M. Artebani and I. Dolgachev, The Hesse pencil of plane cubic curves, preprint (2006), https://arxiv.org/abs/math/0611590.

  • [2]

    P. M. Bellon and C.-M. Viallet, Algebraic entropy, Comm. Math. Phys. 204 (1999), no. 2, 425–437. Google Scholar

  • [3]

    J. Chauvet and E. Mahe, Key agreement under tropical parallels, Groups Complex. Cryptol. 7 (2015), no. 2, 195–198. Google Scholar

  • [4]

    M. Dehli Vigeland, The group law on a tropical elliptic curve, preprint (2004), https://arxiv.org/abs/math/0411485.

  • [5]

    O. Dunkelman, N. Keller and A. Shamir, Minimalism in cryptography: The even-mansour scheme revisited, Technical Report 2011/541, IACR Cryptology ePrint Archive, 2011, http://dblp.org/rec/journals/iacr/DunkelmanKS11a.

  • [6]

    R. R. Farashahi and M. Joye, Efficient arithmetic on Hessian curves, Public Key Cryptography, Lecture Notes in Comput. Sci. 6056, Springer, Berlin (2010), 243–260. Google Scholar

  • [7]

    B. Grammaticos, A. Ramani and C. Viallet, Solvable chaos, Phys. Lett. A 336 (2005), no. 2–3, 152–158. Google Scholar

  • [8]

    D. Grigoriev and V. Shpilrain, Tropical cryptography, Comm. Algebra 42 (2014), no. 6, 2624–2632. Google Scholar

  • [9]

    M. Joye and J.-J. Quisquater, Hessian elliptic curves and side-channel attacks, Proceedings of Cryptographic Hardware and Embedded Systems – CHES 2001, Lecture Notes in Comput. Sci. 2162, Springer, Berlin (2001), 402–410. Google Scholar

  • [10]

    K. Kajiwara, M. Kaneko, A. Nobe and T. Tsuda, Ultradiscretization of a solvable two-dimensional chaotic map associated with the hesse cubic curve, Kyushu J. Math. 63 (2009), no. 2, 315–338. Google Scholar

  • [11]

    M. Kotov and A. Ushakov, Analysis of a key exchange protocol based on tropical matrix algebra, Technical Report 2015/852, Cryptology ePrint Archive, 2015, http://eprint.iacr.org/.

  • [12]

    D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Grad. Stud. Math. 161, American Mathematical Society, Providence, 2015. Google Scholar

  • [13]

    G. Mikhalkin, Tropical geometry and its applications, preprint (2006), https://arxiv.org/abs/math/0601041.

  • [14]

    A. Nobe, A tropical analogue of the Hessian group, preprint (2011), https://arxiv.org/abs/1104.0999.

  • [15]

    A. Nobe, The group law on the tropical Hesse pencil, preprint (2011), http://arxiv.org/abs/1111.0131.

  • [16]

    J. Richter-Gebert, B. Sturmfels and T. Theobald, First steps in tropical geometry, preprint (2003), https://arxiv.org/abs/math/0306366.

  • [17]

    N. Smart, The Hessian form of an elliptic curve, Cryptographic Hardware and Embedded Systems – CHES 2001, Lecture Notes in Comput. Sci. 2162, Springer, Berlin (2001), 118–125. Google Scholar

About the article

Received: 2016-02-27

Published Online: 2017-04-19

Published in Print: 2017-05-01


Citation Information: Groups Complexity Cryptology, Volume 9, Issue 1, Pages 19–29, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2017-0002.

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