Advertisement

Nonlinearity Measures of Boolean Functions

  • Chuan-Kun Wu
  • Dengguo Feng
Chapter
  • 948 Downloads
Part of the Advances in Computer Science and Technology book series (ACST)

Abstract

Nonlinearity is an important cryptographic measure to cryptographic Boolean functions, and much study can be found from public literatures (see, e.g., [1, 2, 12, 14]). More generalized cryptographic measures about the nonlinear properties of Boolean functions also include algebraic degree, linear structure property, and higher-order nonlinearity [6, 33, 34]. These properties are extensively studied in this chapter.

Keywords

Boolean Function Linear Structure Linear Complexity Bend Function High Nonlinearity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Beth, T., Ding, C.: On almost perfect nonlinear permutations. In: Advances in Cryptology, Proceedings of Eurocrypt’93. LNCS 765, pp. 65–76. Springer, Berlin/New York (1994)Google Scholar
  2. 2.
    Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complex. 20, 205–244 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carlet, C., Dobbertin, H., Leander, G.: Normal extensions of bent functions. IEEE Trans. Inf. Theory IT-50(11), 2880–2885 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carlet, C., Guillot, P.: An alternate characterization of the bentness of binary functions with uniqueness. Des. Codes Cryptogr. 14, 133–140 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carlet, C., Gouget, A.: An upper bound on the number of m-resilient bent functions. In: Advances in Cryptology, Proceedings of Asiacrypt 2002. LNCS 2501, pp. 484–496. Springer, Berlin/New York (2002)Google Scholar
  6. 6.
    Carlet, C., Tarannikov, Y.: Covering sequences of Boolean functions and their cryptographic significance. Des. Codes Cryptogr. 25, 263–279 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chabaud, F., Vaudenay, S.: Links between differential and linear cryptanalysis. In: Advances in Cryptology, Proceedings of Eurocrypt’94. LNCS 950, pp. 356–365. Springer, Berlin/Heidelberg (1995)Google Scholar
  8. 8.
    Chaum, D., Evertse, J.H.: Cryptanalysis of DES with a reduced number of rounds. In: Advances in Cryptology, Proceedings of Crypto’85. LNCS 218, pp. 192–211. Springer, Berlin (1986)Google Scholar
  9. 9.
    Dobbertin, H.: Construction of bent functions and balanced Boolean functions with high nonlinearity. In: Fast Software Encryption 1994. LNCS 1008, pp. 61–74. Springer, Berlin (1995)Google Scholar
  10. 10.
    Dobbertin, H.: A survey of some recent results on bent functions. In: Sequences and Their Applications – SETA 2004. LNCS 3486, pp. 1–29. Springer, Berlin (2005)Google Scholar
  11. 11.
    Evertse, J.-H.: Linear structures in block ciphers. In: Advances in Cryptology, Proceedings of Eurocrypt’87. LNCS 304, pp. 249–266. Springer, Berlin (1988)Google Scholar
  12. 12.
    Fedorova, M., Tarannikov, Y.: On the construction of highly nonlinear resilient Boolean functions by means of special matrices. In: Proceedings of Indocrypt 2011. LNCS 2247, pp. 254–266. Springer, Heidelberg/New York (2001)Google Scholar
  13. 13.
    Filiol, E., Fontaine, C.: Highly nonlinear balanced Boolean functions with a good correlation immunity. In: Advances in Cryptology, Proceedings Eurocrypt’98. LNCS 1403, pp. 475–488. Springer, Berlin (1998)Google Scholar
  14. 14.
    Fontaine, C.: The nonlinearity of a class of Boolean functions with short representation. In: Proceedings of the 1st International Conference on the Theory and Applications of Cryptology (PRAGOCRYPT’96). CTU Publishing House, Prague, pp. 129–144Google Scholar
  15. 15.
    Kumar, P.V., Scholtz, R.A.: Bounds on the linear span of Bent sequences. IEEE Trans. Inf. Theory IT-29(6), 854–862 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kumar, P.V., Scholtz, R.A.: Generalized bent functions and their properties. J. Combin. Theory (A) 40, 90–107 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lai, X.: Additive and linear structures of cryptographic functions. FSE 1994(1008), 75–85 (1995)zbMATHGoogle Scholar
  18. 18.
    Lempel, A., Cohn, M.: Maximal families of Bent sequences. IEEE Trans. Inf. Theory IT-28(6), 865–868 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, New York (1977)zbMATHGoogle Scholar
  20. 20.
    Meier, W., Staffelbach, O.: Nonlinearity criteria for cryptographic functions. In: Advances in Cryptology, Proceedings of Eurocrypt’89. LNCS 434, pp. 549–562. Springer, Berlin (1990)Google Scholar
  21. 21.
    Nyberg, K.: Construction of Bent functions and different sets. In: Advances in Cryptology, Proceedings of Eurocrypt’90. LNCS 473, pp. 151–160. Springer, Berlin (1991)Google Scholar
  22. 22.
    Patterson, N.J., Wiedemann, D.H.: The covering radius of the [215, 16] Reed-Muller code is at least 16276. IEEE Trans. Inf. Theory IT-29(3), 354–356 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Qu, C., Seberry, J., Pieprzyk, J.P.: Homogeneous bent functions. Discret. Appl. Math. 102, 133–139 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Reeds, J.A., Manferdeli, J.L.: DES has no per round linear factors. In: Advances in Cryptology, Proceedings of Crypto’84. LNCS 196, pp. 377–389. Springer, Berlin/Heidelberg (1985)Google Scholar
  25. 25.
    Rothaus, O.S.: On ‘bent’ functions. J. Combin. Theory (A) 20, 300–305 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sarkar, P., Maitra, S.: Nonlinearity bounds and constructions of resilient Boolean functions. In: Advances in Cryptology, Proceedings of Crypto’2000. LNCS 1880, pp. 515–532. Springer, Berlin (2000)Google Scholar
  27. 27.
    Sarkar, P., Maitra, S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Advances in Cryptology, Proceedings of Eurocrypt’2000. LNCS 1807, pp. 485–506. Springer, Berlin (2000)Google Scholar
  28. 28.
    Sarkar, P., Maitra, S.: Construction of nonlinear resilient Boolean functions using ‘small’ affine functions. IEEE Trans. Inf. Theory IT-50(1), 2185–2193 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schnoor, C.P.: The multiplicative complexity of Boolean functions. In: Proceedings of AAECC-6, pp. 45–58. Springer, Berlin (1989)Google Scholar
  30. 30.
    Seberry, J., Zhang, X.M., Zheng, Y.: On construction and nonlinearity of correlation immune functions, (extended abstract). In: Advances in Cryptology, Proceedings of Eurocrypt’93. LNCS 765, pp. 181–199. Springer, Berlin (1994)Google Scholar
  31. 31.
    Seberry, J., Zhang, X.M., Zheng, Y.: Nonlinearly balanced Boolean functions and their propagation characteristics (extended abstract). In: Advances in Cryptology, Proceedings of Crypto’93. LNCS 773, pp. 49–60. Springer, Berlin (1994)Google Scholar
  32. 32.
    Seberry, J., Zhang, X.M., Zheng, Y.: Relationships among nonlinearity criteria (extended abstract). In: Advances in Cryptology, Proceeding of Eurocrypt’94. LNCS 950, pp. 376–388. Springer, Berlin (1995)Google Scholar
  33. 33.
    Sun, G., Wu, C.: The lower bound on the second order nonlinearity of a class of Boolean functions with high nonlinearity. Appl. Algebra Eng. Commun. Comput. (AAECC) 22(1), 37–45 (2011)Google Scholar
  34. 34.
    Tang, D., Carlet, C., Tang, X.: On the second-order nonlinearities of some bent functions. Inf. Sci. 223, 322–330 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wu, C.K.: Boolean functions in cryptology. Ph.D. Thesis, Xidian University, Xian (1993) (in Chinese)Google Scholar
  36. 36.
    Yu, N.Y., Gong, G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory IT-52(2), 3291–3299 (2006)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zheng, Y., Xhang, X.M., Imai, H.: Restriction, terms and nonlinearity of Boolean functions. Theor. Comput. Sci. 226, 207–223 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zheng, Y., Zhang, X.M.: On plateaued functions. IEEE Trans. Inf. Theory IT-47(3), 1215–1223 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Chuan-Kun Wu
    • 1
  • Dengguo Feng
    • 2
  1. 1.Chinese Academy of SciencesState Key Lab of Information Security Institute of Information EngineeringBeijingChina
  2. 2.Chinese Academy of SciencesInstitute of SoftwareBeijingChina

Personalised recommendations