Enumeration of 9-Variable Rotation Symmetric Boolean Functions Having Nonlinearity > 240

  • Selçuk Kavut
  • Subhamoy Maitra
  • Sumanta Sarkar
  • Melek D. Yücel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4329)


The existence of 9-variable Boolean functions having nonlinearity strictly greater than 240 has been shown very recently (May 2006) by Kavut, Maitra and Yücel; a few functions with nonlinearity 241 have been identified by a heuristic search in the class of Rotation Symmetric Boolean Functions (RSBFs). In this paper, using combinatorial results related to the Walsh spectra of RSBFs, we efficiently perform the exhaustive search to enumerate the 9-variable RSBFs having nonlinearity > 240 and found that there are 8 ×189 many functions with nonlinearity 241 and there is no RSBF having nonlinearity > 241. We further prove that among these functions, there are only two which are different up to the affine equivalence. This is found by utilizing the binary nonsingular circulant matrices and their variants. Finally we explain the coding theoretic significance of these functions. This is the first time orphan cosets of R(1, n) having minimum weight 241 are demonstrated for n = 9. Further they provide odd weight orphans for n = 9; earlier these were known for certain n ≥11.


Boolean Functions Covering Radius Reed-Muller Code Idempotents Nonlinearity Rotational Symmetry Walsh Transform 


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  1. 1.
    Berlekamp, E.R., Welch, L.R.: Weight distributions of the cosets of the (32, 6) Reed-Muller code. IEEE Transactions on Information Theory IT-18(1), 203–207 (1972)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Brualdi, R.A., Pless, V.S.: Orphans of the first order Reed-Muller codes. IEEE Transactions on Information Theory 36(2), 399–401 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brualdi, R.A., Cai, N., Pless, V.: Orphan structure of the first order Reed-Muller codes. Discrete Mathematics (102), 239–247 (1992)Google Scholar
  4. 4.
    Clark, J., Jacob, J., Maitra, S., Stănică, P.: Almost Boolean Functions: The Design of Boolean Functions by Spectral Inversion. Computational Intelligence 20(3), 450–462 (2004)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cusick, T.W., Stănică, P.: Fast Evaluation, Weights and Nonlinearity of Rotation-Symmetric Functions. Discrete Mathematics 258, 289–301 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dalai, D.K., Gupta, K.C., Maitra, S.: Results on Algebraic Immunity for Cryptographically Significant Boolean Functions. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 92–106. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Dalai, D.K., Maitra, S., Sarkar, S.: Results on rotation symmetric Bent functions. In: Second International Workshop on Boolean Functions: Cryptography and Applications, BFCA 2006 (March 2006)Google Scholar
  8. 8.
    Ding, C., Xiao, G., Shan, W.: The Stability Theory of Stream Ciphers. LNCS, vol. 561. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  9. 9.
    Filiol, E., Fontaine, C.: Highly nonlinear balanced Boolean functions with a good correlation-immunity. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 475–488. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Fontaine, C.: On some cosets of the First-Order Reed-Muller code with high minimum weight. IEEE Transactions on Information Theory 45(4), 1237–1243 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hell, M., Maximov, A., Maitra, S.: On efficient implementation of search strategy for rotation symmetric Boolean functions. In: Ninth International Workshop on Algebraic and Combinatoral Coding Theory, ACCT 2004, June 19–25. Black Sea Coast, Bulgaria (2004)Google Scholar
  12. 12.
    Helleseth, T., Kløve, T., Mykkeltveit, J.: On the covering radius of binary codes. IEEE Transactions on Information Theory IT-24, 627–628 (1978)CrossRefGoogle Scholar
  13. 13.
    Hou, X.-d.: On the norm and covering radius of the first order Reed-Muller codes. IEEE Transactions on Information Theory 43(3), 1025–1027 (1997)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kavut, S., Maitra, S., Yucel, M.D.: Autocorrelation spectra of balanced boolean functions on odd number input variables with maximum absolute value \(<2^{\frac{n+1}{2}}\). In: Second International Workshop on Boolean Functions: Cryptography and Applications, BFCA 2006, March 13-15, LIFAR, University of Rouen, France (2006)Google Scholar
  15. 15.
    Kavut, S., Maitra, S., Yücel, M.D.: There exist Boolean functions on n (odd) variables having nonlinearity \(> 2^{n-1} - 2^{\frac{n-1}{2}}\) if and only if n > 7,
  16. 16.
    Langevin, P.: On the orphans and covering radius of the Reed-Muller codes. In: Mattson, H.F., Rao, T.R.N., Mora, T. (eds.) AAECC 1991. LNCS, vol. 539, pp. 234–240. Springer, Heidelberg (1991)Google Scholar
  17. 17.
    MacWillams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. North-Holland, Amsterdam (1977)Google Scholar
  18. 18.
    Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994)Google Scholar
  19. 19.
    Maximov, A., Hell, M., Maitra, S.: Plateaued Rotation Symmetric Boolean Functions on Odd Number of Variables. In: First Workshop on Boolean Functions: Cryptography and Applications, BFCA 2005, March 7–9, LIFAR, University of Rouen, France (2005)Google Scholar
  20. 20.
    Maximov, A.: Classes of Plateaued Rotation Symmetric Boolean functions under Transformation of Walsh Spectra. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 325–334. Springer, Heidelberg (2006); See also IACR eprint server, no. 2004/354 Google Scholar
  21. 21.
    Mykkeltveit, J.J.: The covering radius of the (128, 8) Reed-Muller code is 56. IEEE Transactions on Information Theory IT-26(3), 359–362 (1980)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Patterson, N.J., Wiedemann, D.H.: The covering radius of the (215, 16) Reed-Muller code is at least 16276. IEEE Transactions on Information Theory IT-29(3), 354–356 (1983); See also the correction in IEEE Transactions on Information Theory IT-36(2), 443 (1990)Google Scholar
  23. 23.
    Pieprzyk, J., Qu, C.X.: Fast Hashing and Rotation-Symmetric Functions. Journal of Universal Computer Science 5, 20–31 (1999)MathSciNetGoogle Scholar
  24. 24.
    Preneel, B., et al.: Propagation characteristics of Boolean functions. In: Damgård, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 161–173. Springer, Heidelberg (1991)Google Scholar
  25. 25.
    Rothaus, O.S.: On bent functions. Journal of Combinatorial Theory, Series A 20, 300–305 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Stănică, P., Maitra, S.: Rotation Symmetric Boolean Functions – Count and Cryptographic Properties. In: R. C. Bose Centenary Symposium on Discrete Mathematics and Applications. Electronic Notes in Discrete Mathematics, vol. 15. Elsevier, Amsterdam (2002)Google Scholar
  27. 27.
    Stănică, P., Maitra, S., Clark, J.: Results on Rotation Symmetric Bent and Correlation Immune Boolean Functions. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 161–177. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  28. 28.
    Zhang, X.M., Zheng, Y.: GAC - the criterion for global avalanche characteristics of cryptographic functions. Journal of Universal Computer Science 1(5), 316–333 (1995)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Selçuk Kavut
    • 1
  • Subhamoy Maitra
    • 2
  • Sumanta Sarkar
    • 2
  • Melek D. Yücel
    • 1
  1. 1.Department of Electrical Engineering and Institute of Applied MathematicsMiddle East Technical University(METU – ODTÜ)Ankara, Türkiye
  2. 2.Applied Statistics UnitIndian Statistical InstituteKolkataIndia

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