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Chaos-Based Public-Key Cryptography

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Chaos-Based Cryptography

Part of the book series: Studies in Computational Intelligence ((SCI,volume 354))

Abstract

In this chapter we give an overview and the state of the art in the field of Chaos-based cryptography. The public key cryptosystems based on Chebyshev polynomials enjoy some nice chaotic properties, which makes them suitable for use in both encryption and digital signature. The cryptosystem can work either on real or integer numbers. The cryptosystem that works on real numbers is not secure and permits to recover the corresponding plaintext from a given ciphertext. In addition, it also allows forgeries if the cryptosystem is used for signing messages. On the other hand, ElGamal-like and RSA-like algorithms when using Chebyshev polynomials on integer numbers are secure as the aforementioned encryption algorithms. The chaos-based cryptography is discussed from a point of view which we believe is closer to the spirit of both cryptography and chaos theory than the way the subject has been treated recently by many researchers.

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References

  1. Kolumban, G., Kennedy, M.P., Kis, G., Jako, Z.: FM-DCSK: A novel method for chaotic communications. In: Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, ISCAS 1998, vol. 4, pp. 477–480 (1998)

    Google Scholar 

  2. Sushchik, M., Rulkov, N., Larson, L., Tsimring, L., Abarbanel, H., Yao, K., Volkovskii, A.: Chaotic pulse position modulation: A robust method of communicating with chaos. IEEE Commun. Lett. 4(4), 128–130 (2000)

    Article  Google Scholar 

  3. Kocarev, L.: Chaos-based cryptography: A brief overview. IEEE Circuits Systems Magazine 1(3), 6–21 (2001)

    Article  Google Scholar 

  4. Dachselt, F., Schwarz, W.: Chaos and cryptography. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 48(12), 1498–1509 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Menezes, A., van Oorschot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)

    MATH  Google Scholar 

  6. Shanon, C.E.: Communication theory of secrecy systems. Bell Sys. Tech. J. 28, 656–715 (1949)

    Google Scholar 

  7. Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos based cryptosystems. International Journal of Bifurcation and Chaos 16, 2129–2151 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Pichler, F., Scharinger, J.: Finite dimensional generalized Baker dynamical systems for cryptographic applications, Lect. In: Albrecht, R., Moreno-Díaz, R., Pichler, F. (eds.) EUROCAST 1995. LNCS, vol. 1030, pp. 465–476. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  9. Fridrich, J.: Symmetric ciphers based on two-dimensional chaotic maps. Internat. J. Bifur. Chaos 8(6), 1259–1284 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jakimoski, G., Kocarev, L.: Chaos and cryptography: block encryption ciphers based on chaotic maps. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 48(2), 163–169 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jakimoski, G., Kocarev, L.: Differential and linear probabilities of a block-encryption cipher. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 50(1), 121–123 (2003)

    Article  MathSciNet  Google Scholar 

  12. Masuda, N., Aihara, K.: Cryptosystems with discretized chaotic maps. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 49(1), 28–40 (2002)

    Article  MathSciNet  Google Scholar 

  13. Lawande, Q.V., Ivan, B.R., Dhodapkar, S.D.: Chaos Based Cryptography: A New Approach to Secure Communications, BARC Newsletter, Bombay (2005)

    Google Scholar 

  14. Matthews, R.A.J.: On the derivation of a ‘chaotic’ encryption algorithm. Cryptologia 13, 29–42 (1989); Wheeler, D.D.: Problems with chaotic cryptosystems. Cryptologia 13, 243–250 (1989); Wheeler, D.D., Matthews, R.A.J.: Supercomputer investigations of a chaotic encryption algorithm. Cryptologia 15(2), 140–152 (1991)

    Article  MathSciNet  Google Scholar 

  15. Kohda, T., Tsuneda, A.: Statistics of chaotic binary sequences. IEEE Trans. Inform. Theory 43, 104–112 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kocarev, L., Jakimoski, G.: Pseudorandom bits generated by chaotic maps. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 50(1), 123–126 (2003)

    Article  MathSciNet  Google Scholar 

  17. Petrie, C.S., Connelly, J.A.: A noise-based IC random number generator for applications in cryptography. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 47(5), 615–621 (2000)

    Article  Google Scholar 

  18. Stojanovski, T., Kocarev, L.: Chaos-based random number generators-part I: Analysis. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 48(3), 281–288 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Stojanovski, T., Pihl, J., Kocarev, L.: Chaos-based random number generators PART II: Practical realization. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 48(3), 382–385 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gerosa, A., Bernardini, R., Pietri, S.: A fully integrated chaotic system for the generation of truly random numbers. IEEE Trans. Circuits Systems I: Fund. Theory Appl. 49(7), 993–1000 (2002)

    Article  Google Scholar 

  21. Kocarev, L., Tasev, Z.: Public-key encryption based on Chebyshev maps. In: 2003 IEEE International Symposium on Circuits and Systems, ISCAS 2003, Bangkok, Thailand, May 25-28 (2003)

    Google Scholar 

  22. Kocarev, L., Tasev, Z., Makraduli, J.: Public-key encryption and digital-signature schemes using chaotic maps. In: 16th European Conference on Circuits Theory and Design, ECCTD 2003, Krakow, Poland, September 1-4 (2003)

    Google Scholar 

  23. Bergamo, P., D’Arco, P., De Santis, A., Kocarev, L.: Security of Public-Key Cryptosystems Based on Chebyshev Polynomials. IEEE Transactions on Circuits and Systems—I: Regular Papers 52(7) (2005)

    Google Scholar 

  24. Kocarev, L., Makraduli, J.: Public-key Encryption Based on Chebyshev Polynomials. Circuits Systems Signal Processing 24(5), 497–517 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shannon, C.E.: Bell Syst. Tech. J., 27(379) (1948); 27(623) (1948)

    Google Scholar 

  26. Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inform. Theory 22, 644–654 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rivlin, T.J.: Chebyshev Polynomials. Wiley, New York (1990)

    MATH  Google Scholar 

  28. Percival, I., Vivaldi, F.: Arithmetical properties of strongly chaotic motions. Physica D 25(1-3), 105–130 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Cohn, H.: A Second Course in Number Theory. Wiley, New York (1962)

    MATH  Google Scholar 

  30. Hasse, H.: Number Theory. Springer, Berlin (2002)

    MATH  Google Scholar 

  31. El Gamal, T.: A public-key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inf. Theory IT-31(4), 469–472 (1985)

    Google Scholar 

  32. Fateman, R.J.: Lookup tables, recurrences, and complexity. In: Proc. Int. Symp. Symbolic and Algebraic Computation. ISSAC 1989, pp. 68–73 (1989)

    Google Scholar 

  33. Umeno, K.: Method of constructing exactly solvable chaos. Phys. Rev. E 55, 5280–5284 (1997)

    Article  Google Scholar 

  34. Kohda, T., Fujisaki, T.: Jacobian elliptic Chebyshev rational maps. Phys. D 148, 242–254 (2001)

    Article  MathSciNet  Google Scholar 

  35. Yacas (Yet Another Computer Algebra System), http://homepage.mac.com/yacas/manual/Algochapter4.html

  36. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1970)

    Google Scholar 

  37. Wachspress, E.L.: Evaluating elliptic functions and their inverses. Computers and Mathematics with Applications (39), 131–136 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wachspress, E.L.: System, apparatus, and method for outputting pseudorandom noise sequences, and data recording medium, Japanese Patent no. 3 455 483 (July 25, 2003), see also the US Patent no. 6 654 404, http://www.uspto.gov

  39. Xiao, D., Liao, X., Tang, G., Li, C.: Using chebyshev chaotic map to construct infinite length hash chains. In: Proc. Int. Conf. Communications, Circuits and Systems, vol. 1, pp. 11–12 (2004)

    Google Scholar 

  40. Knuth, D.E.: The Art of Computer Programming, vol. 2. Addison-Wesley, Reading (1998)

    Google Scholar 

  41. http://java.sun.com

  42. http://www.swox.com/gmp/

  43. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge Univ. Press, Cambridge (1935)

    Google Scholar 

  44. Umeno, K.: Exactly solvable chaos and addition theorems of elliptic functions, RIMS Kokyuroku no. 1098 (1999)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electronics, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129, Turin, Italy

    Igor Mishkovski

  2. Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0402, USA

    Ljupco Kocarev

Authors
  1. Igor Mishkovski
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  2. Ljupco Kocarev
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Editors and Affiliations

  1. Macedonain Academy of Sciences and Arts, bul. Krste Misirkov 2, 428, 1000, Skopje, Republic of Macedonia

    Ljupco Kocarev

  2. France Telecom R&D Beijing, 2 Science Institute South Rd,Haidian District, 100080, Beijing, China

    Shiguo Lian

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Mishkovski, I., Kocarev, L. (2011). Chaos-Based Public-Key Cryptography. In: Kocarev, L., Lian, S. (eds) Chaos-Based Cryptography. Studies in Computational Intelligence, vol 354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20542-2_2

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  • DOI: https://doi.org/10.1007/978-3-642-20542-2_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20541-5

  • Online ISBN: 978-3-642-20542-2

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