Commutativity, Associativity, and Public Key Cryptography

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10737)


In this paper, we will study some possible generalizations of the famous Diffie-Hellman algorithm. As we will see, at the end, most of these generalizations will not be secure or will be equivalent to some classical schemes. However, these results are not always obvious and moreover our analysis will present some interesting connections between the concepts of commutativity, associativity, and public key cryptography.


Diffie-Hellman algorithms Chebyshev polynomials New public key algorithms 



The authors want to thank Jérôme Plût, Gerhard Frey and Gérard Maze for very useful comments and particularly Gerhard Frey for his help to improve the presentation of this paper. We have met Gerhard Frey and Gérard Maze at NuTMiC 2017 in Poland.


  1. 1.
    Barsotti, I.: Un Teorema di structura per le variettà di gruppali. Rend. Acc. Naz. Lincei 18, 43–50 (1955)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bergamo, P., D’Arco, P., de Santis, A., Kocarev, L.: Security of Public Key Cryptosystems based on Chebyshev Polynomials. arXiv:cs/0411030v1, 1 February 2008
  3. 3.
    Block, H.D., Thielman, H.P.: Commutative Polynomials. Quart. J. Math. Oxford Ser. 2(2), 241–243 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Couveignes, J.M.: Hard Homogeneous Spaces. Cryptology ePrint archive: 2006/291: Listing for 2006Google Scholar
  5. 5.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theor. 22(6), 644–654 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frey17: Deep Theory, efficient algorithms and surprising applications. In: NuTMiC (2017)Google Scholar
  7. 7.
    Gaudry, P., Lubicz, D.: The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines. Finite Fields Appl. 15(2), 246–260 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hunziker, M., Machiavelo, A., Parl, J.: Chebyshev polynomials over finite fields and reversibility of \(\sigma \)-automata on square grids. Theor. Comput. Sci. 320(2–3), 465–483 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jao, D., De Feo, L.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 19–34. Springer, Heidelberg (2011). CrossRefGoogle Scholar
  10. 10.
    Kocarev, L., Makraduli, J., Amato, P.: Public key encryption based on Chebyshev polynomials. Circ. Syst. Sign. Process. 24(5), 497–517 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, Z., Cui, Y., Jin, Y., Xu, H.: Parameter selection in public key cryptosystem based on Chebyshev polynomials over finite field. J. Commun. 6(5), 400–408 (2011)CrossRefGoogle Scholar
  12. 12.
    Lima, J.B., Panario, D., Campello de Sousa, R.M.: Public-key cryptography based on Chebyshev polynomials over \(G(q)\). Inf. Process. Lett. 111, 51–56 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Maze, G.: Algebraic Methods for Constructing One-Way Trapdoor Functions. Ph.D. thesis - University oof Notre Dame (2003).
  14. 14.
    Rosen, J., Scherr, Z., Weiss, B., Zieve, M.: Chebyshev mappings over finite fields. Amer. Math. Monthly 119, 151–155 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sun, J., Zhao, G., Li, X.: An improved public key encryption algorithm based on Chebyshev polynomials. TELKOMNIKA 11(2), 864–870 (2013)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université de Paris-SaclayVersaillesFrance
  2. 2.Department of Mathematics, University of Cergy-Pontoise, UMR CNRS 8088Cergy-PontoiseFrance

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