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Distribution of Boolean Functions According to the Second-Order Nonlinearity

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Book cover Arithmetic of Finite Fields (WAIFI 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6087))

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Abstract

The nonlinearity of a Boolean function is the minimum number of substitutions required in its truth table to change it into an affine function. Hence, in a cryptographic context, it is used to measure the strength of cryptosystems when facing linear attacks. As for the nonlinearity of order r of a Boolean function, which equals the least number of substitutions needed to change it into a function of degree at most r, it is examined when dealing with low-degree approximation attacks [7,14].

Many studies aimed at the distribution of Boolean functions according to the r-th order nonlinearity. Asymptotically, a lower bound is established in the higher order cases for almost all boolean functions, whereas a concentration point is shown in the (first order) nonlinearity case. We present a more accurate distribution by proving a concentration point in the second-order nonlinearity case.

This work has been done with the support of the Région Provence-Alpes-Côte d’Azur.

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References

  1. Alon Noga, N., Spencer, J.: The probabilistic method. With an appendix on the life and work of Paul Erdös, 3rd edn. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc, Hoboken (2008)

    MATH  Google Scholar 

  2. Carlet, C.: On cryptographic complexity of Boolean functions. In: Mullen, G.L., Stichtenoth, H., Tapia-Recillas, H. (eds.) Proceedings of the Sixth Conference on Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 53–69. Springer, Heidelberg (2002)

    Google Scholar 

  3. Carlet, C.: The complexity of Boolean functions from cryptographic viewpoint (2006), http://drops.dagstuhl.de/volltexte/2006/604

  4. Carlet, C., Mesnager, S.: Improving the upper bounds on the covering radii of binary Reed-Muller codes. IEEE Trans. Inform. Theory 53(1), 162–173 (2007)

    Article  MathSciNet  Google Scholar 

  5. Feller, W.: An introduction to probability theory and its applications, 3rd edn., vol. I. John Wiley & Sons, Inc., New York (1968)

    MATH  Google Scholar 

  6. Helleseth, T., Klove, T., Levenshtein, V.: Error-correction capability of binary linear codes. IEEE Trans. Inform. Theory 51(4), 1408–1423 (2005)

    Article  MathSciNet  Google Scholar 

  7. Knudsen, L., Robshaw, M.: Non-linear approximations in linear cryptanalysis. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 224–236. Springer, Heidelberg (1996)

    Google Scholar 

  8. Litsyn, S., Shpunt, A.: On the distribution of Boolean function nonlinearity. SIAM J. Discrete Math. 23(1), 79–95 (2008/2009)

    Article  MathSciNet  Google Scholar 

  9. MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. I. North-Holland Mathematical Library, vol. 16. North-Holland Publishing Co., Amsterdam (1977)

    Google Scholar 

  10. Olejár, D., Stanek, M.: On cryptographic properties of random Boolean functions. J. UCS 4(8), 705–717 (1998)

    MATH  MathSciNet  Google Scholar 

  11. Rodier, F.: Sur la non-linéarité des fonctions booléennes. In: Acta Arithmetica, vol. 115, pp. 1–22 (2004); prétirage de l’IML n\(^{\hbox{\tiny o}}\) 2002-07, disponible sur ArXiv: math.NT/0306395

    Google Scholar 

  12. Rodier, F.: On the nonlinearity of Boolean functions. In: de Augot, D., Charpin, P., Kabatianski, G. (eds.) Proceedings of WCC 2003, Workshop on coding and cryptography 2003, sous la direction. INRIA, pp. 397–405 (2003)

    Google Scholar 

  13. Rodier, F.: Asymptotic nonlinearity of Boolean functions. Designs, Codes and Cryptography 40, 1 (2006)

    Article  MathSciNet  Google Scholar 

  14. Shimoyama 1, T., Kaneko, T.: Quadratic Relation of S-box and Its Application to the Linear Attack of Full Round DES. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 129–147. Springer, Heidelberg (1998)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Luminy, Marseille, France

    Stéphanie Dib

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  1. Stéphanie Dib
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Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    M. Anwar Hasan

  2. Department of Informatics, HIB, University of Bergen, PB 7803, 5020, Bergen, Norway

    Tor Helleseth

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Dib, S. (2010). Distribution of Boolean Functions According to the Second-Order Nonlinearity. In: Hasan, M.A., Helleseth, T. (eds) Arithmetic of Finite Fields. WAIFI 2010. Lecture Notes in Computer Science, vol 6087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13797-6_7

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  • DOI: https://doi.org/10.1007/978-3-642-13797-6_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13796-9

  • Online ISBN: 978-3-642-13797-6

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