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Degree Optimized Resilient Boolean Functions from Maiorana-McFarland Class

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Cryptography and Coding (Cryptography and Coding 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2898))

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Abstract

In this paper we present a construction method of degree optimized resilient Boolean functions with very high nonlinearity. We present a general construction method valid for any n ≥ 4 and for order of resiliency t satisfying t ≤ n-3. The construction is based on the modification of the famous Marioana-McFarland class in a controlled manner such that the resulting functions will contain some extra terms of high algebraic degree in its ANF including one term of highest algebraic degree. Hence, the linear complexity is increased, the functions obtained reach the Siegentheler’s bound and furthermore the nonlinearity of such a function in many cases is superior to all previously known construction methods. This construction method is then generalized to the case of vectorial resilient functions, that is {F} : \(\mathbb F{^n_2}\)↦\(\mathbb F{^m_2}\), providing functions of very high algebraic degree almost reaching the Siegenthaler’s upper bound.

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References

  1. Camion, P., Carlet, C., Charpin, P., Sendrier, N.: On correlation-immune functions. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 86–100. Springer, Heidelberg (1991)

    Google Scholar 

  2. Carlet, C.: On the coset weight divisibility and nonlinearity of resilient and correlation-immune functions. Discrete Mathematics and Theoretical Computer Science (2001)

    Google Scholar 

  3. Carlet, C.: A larger class of cryptographic Boolean functions via a study of the Maiorana-McFarland constructions. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 549–564. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  4. Carlet, C., Sarkar, P.: Spectral domain analysis of correlation immune and resilient Boolean functions. Finite Fields and Their Applications 8(1), 120–130 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Charpin, P., Pasalic, E.: Disjoint linear codes in construction of nonlinear resilient functions (2003) (Preprint to be submitted)

    Google Scholar 

  6. Charpin, P., Pasalic, E.: On propagation properties of resilient functions. In: Selected Areas in Cryptography—SAC 2002. LNCS, vol. 2595, pp. 356–365. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Chee, S., Lee, S., Lee, D., Sung, H.S.: On the correlation immune functions and their nonlinearity. In: Kim, K.-c., Matsumoto, T. (eds.) ASIACRYPT 1996. LNCS, vol. 1163, pp. 232–243. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  8. Cheon, J.H.: Nonlinear vector resilient functions. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 181–195. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  9. Courtois, N.: Higher order correlation attacks, XL algorithm and cryptoanalysis of Toyocrypt. In: Lee, P.J., Lim, C.H. (eds.) ICISC 2002. LNCS, vol. 2587, pp. 182–199. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  10. Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 346–359. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  11. Friedman, J.: On the bit extraction problem. In: 33rd IEEE Symposium on Foundations of Computer Science, pp. 314–319 (1982)

    Google Scholar 

  12. Gupta, K.C., Sarkar, P.: Improved constructions of nonlinear resilient S-boxes. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 466–483. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  13. Johansson, T., Pasalic, E.: A construction of resilient functions with high non-linearity. IEEE Trans. on Inform. Theory IT-49(2) (February 2003)

    Google Scholar 

  14. Kurosawa, K., Satoh, T., Yamamoto, K.: Highly nonlinear i-resilient functions. Journal of Universal Computer Science 3(6), 721–729 (1997)

    MATH  MathSciNet  Google Scholar 

  15. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  16. Maitra, S., Pasalic, E.: Further constructions of resilient Boolean functions with very high nonlinearity. IEEE Trans. on Inform. Theory IT-48(7), 1825–1834 (2002)

    Article  MathSciNet  Google Scholar 

  17. Nyberg, K.: On the construction of highly nonlinear permutations. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 92–98. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  18. Pasalic, E., Johansson, T., Maitra, S., Sarkar, P.: New constructions of resilient and correlation immune Boolean functions achieving upper bounds on nonlinearity. In: Workshop on Coding and Cryptography Proceedings, vol. 6, pp. 425–435. Elsevier Science, Amsterdam (2001)

    Google Scholar 

  19. Pasalic, E., Maitra, S.: Linear codes in generalized construction of resilient functions with very high nonlinearity. IEEE Trans. on Inform. Theory IT-48(8), 2182–2191 (2002)

    Article  MathSciNet  Google Scholar 

  20. Sarkar, P., Maitra, S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 485–506. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  21. Sarkar, P., Maitra, S.: Nonlinearity bounds and constructions of resilient Boolean functions. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 515–532. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  22. Seberry, J., Zhang, X.M., Zheng, Y.: On constructions and nonlinearity of correlation immune Boolean functions. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 181–199. Springer, Heidelberg (1994)

    Google Scholar 

  23. Siegenthaler, T.: Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Trans. on Inform. Theory IT-30, 776–780 (1984)

    Article  MathSciNet  Google Scholar 

  24. Tarannikov, Y.: On resilient Boolean functions with maximal possible nonlinearity. In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 19–30. Springer, Heidelberg (2000)

    Google Scholar 

  25. Xiao, G.-Z., Massey, J.L.: A spectral characterization of correlation-immune combining functions. IEEE Trans. on Inform. Theory IT-34, 569–571 (1988)

    Article  MathSciNet  Google Scholar 

  26. Zhang, X.M., Zheng, Y.: Cryptographically resilient functions. IEEE Trans. on Inform. Theory IT-43(5), 1740–1747 (1997)

    Article  MathSciNet  Google Scholar 

  27. Zheng, Y., Zhang, X.M.: Improving upper bound on nonlinearity of high order correlation immune functions. In: Stinson, D.R., Tavares, S. (eds.) SAC 2000. LNCS, vol. 2012, pp. 264–274. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. INRIA, Domaine de Voluceau, BP 105, 78153, Le Chesnay Cedex, FRANCE

    Enes Pasalic

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  1. Enes Pasalic
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Editors and Affiliations

  1. Information Security Group, Royal Holloway, University of London, TW20 0EX, Egham, Surrey, U.K.

    Kenneth G. Paterson

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Pasalic, E. (2003). Degree Optimized Resilient Boolean Functions from Maiorana-McFarland Class. In: Paterson, K.G. (eds) Cryptography and Coding. Cryptography and Coding 2003. Lecture Notes in Computer Science, vol 2898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40974-8_9

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  • DOI: https://doi.org/10.1007/978-3-540-40974-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20663-7

  • Online ISBN: 978-3-540-40974-8

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