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On the Covering Radius of Second Order Binary Reed-Muller Code in the Set of Resilient Boolean Functions

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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

Let \(\mathcal R_{t,n}\) denote the set of t-resilient Boolean functions of n variables. First, we prove that the covering radius of the binary Reed-Muller code RM(2,6) in the sets \(\mathcal R_{t,6}\), t=0,1,2 is 16. Second, we show that the covering radius of the binary Reed-Muller code RM(2,7) in the set \(\mathcal R_{3,7}\) is 32. We derive a new lower bound for the covering radius of the Reed-Muller code RM(2,n) in the set \(\mathcal R_{n-1,4}\). Finally, we present new lower bounds in the sets \(\mathcal R_{t,7}\), t=0,1,2.

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References

  1. Berlekamp, E., Welch, L.: Weight Distribution of the Cosets of the (32,6) Reed-Muller Code. IEEE IT 18, 203–207 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Carlet, C., Prouff, E.: On Plateaued Functions and their Constructions. In: Johansson, T. (ed.) FSE 2003. LNCS, vol. 2887, pp. 57–78. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  3. Courtois, N.: Higher Order Correlation Attacks, XL Algorithm and Cryptanalysis of Toyocrypt, ePrint Archive 2002/087 (2002)

    Google Scholar 

  4. Courtois, N., Meier, W.: Algebraic Attacks on Stream Ciphers with Linear Feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 345–359. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  5. Courtois, N.: Fast Algebraic Attacks on Stream Ciphers with Linear Feedback. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 176–194. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  6. Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  7. Hou, X.D.: GL(m, 2) Acting on R(r, m)/R(r— 1,m). Discrete Mathematics 149, 99–122 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kurosawa, K., Iwata, T., Yoshiwara, T.: New Covering Radius of Reed-Muller Codes for i-Resilient Functions. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 75–86. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  9. McLoughlin, A.: The Covering Radius of the (m— 3)—rd Order Reed-Muller Codes and a Lower Bound on the (m — 4)—th Order Reed-Muller Codes. SIAM J. Appl. Mathematics 37(2), 419–422 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Carlet, C., Sarkar, P.: Spectral Domain Analysis of Correlation Immune and Resilient Boolean Functions. Finite Fields and Applications 8(1), 120–130 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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 

  13. Schatz, J.: The Second Order Reed-Muller Code of Length 64 has Covering Radius 18. IEEE IT 27, 529–530 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  14. Siegenthaler, T.: Correlation-Immunity of Non-linear Combining Functions for Cryptographic Applications. IEEE IT 30(5), 776–780 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Siegenthaler, T.: Decrypting a Class of Stream Ciphers Using Cyphertext Only. IEEE Trans. Comp. 34(1), 81–85 (1985)

    Article  Google Scholar 

  16. 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 

  17. Guo-Zhen, X., Massey, J.: A Spectral Characterization of Correlation-Immune Combining Functions. IEEE IT 34(3), 569–571 (1988)

    Article  MATH  Google Scholar 

  18. Zheng, Y., Zhang, X.M.: Plateaued Functions. In: Varadharajan, V., Mu, Y. (eds.) ICICS 1999. LNCS, vol. 1726, pp. 284–300. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  19. Zheng, Y., Zhang, X.M.: Improved Upper Bound on the Nonlinearity of High Order Correlation Immune Functions. In: Stinson, D.R., Tavares, S. (eds.) SAC 2000. LNCS, vol. 2012, pp. 262–274. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  20. Zheng, Y., Zhang, X.M.: New Results on Correlation Immunity. In: Won, D. (ed.) ICISC 2000. LNCS, vol. 2015, pp. 49–63. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G.Bonchev, 1113, Sofia, Bulgaria

    Yuri Borissov

  2. Department Electrical Engineering – ESAT/SCD/COSIC, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, B-3001, Leuven, Belgium

    An Braeken, Svetla Nikova & Bart Preneel

Authors
  1. Yuri Borissov
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  2. An Braeken
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  3. Svetla Nikova
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  4. Bart Preneel
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Editor information

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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Borissov, Y., Braeken, A., Nikova, S., Preneel, B. (2003). On the Covering Radius of Second Order Binary Reed-Muller Code in the Set of Resilient Boolean Functions. 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_8

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  • DOI: https://doi.org/10.1007/978-3-540-40974-8_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20663-7

  • Online ISBN: 978-3-540-40974-8

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