On Semi-bent Functions and Related Plateaued Functions Over the Galois Field \(\mathbb{F}_{2^{n}}\)

  • Sihem Mesnager


Plateaued functions were introduced in 1999 by Zheng and Zhang as good candidates for designing cryptographic functions since they possess desirable various cryptographic characteristics. They are defined in terms of the Walsh–Hadamard spectrum. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions and the semi-bent functions. Bent functions (including their constructions) have been extensively investigated for more than 35 years. Very recently, the study of semi-bent functions has attracted the attention of several researchers. Much progress in the design of such functions has been made. The chapter is devoted to certain plateaued functions. The focus is particularly on semi-bent functions defined over the Galois field \(\mathbb{F}_{2^{n}}\) (n even). We review what is known in this framework and investigate constructions.



The author wishes to thank Claude Carlet for his careful reading and interesting comments.


  1. 1.
    S. Boztas, P.V. Kumar, Binary sequences with Gold-like correlation but larger linear span. IEEE Trans. Inf. Theory 40(2), 532–537 (1994)CrossRefMATHGoogle Scholar
  2. 2.
    A. Canteaut, C. Carlet, P. Charpin, C. Fontaine, On cryptographic properties of the cosets of R(1,m). IEEE Trans. Inf. Theory 47, 1494–1513 (2001)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    C. Carlet, On the secondary constructions of resilient and bent functions, in Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003 (Birkhäuser, Basel, 2004), pp. 3–28Google Scholar
  4. 4.
    C. Carlet, Boolean functions for cryptography and error correcting codes, in Chapter of the monography Boolean Models and Methods in Mathematics, Computer Science, and Engineering, ed. by Y. Crama, P.L. Hammer (Cambridge University Press, Cambridge, 2010), pp. 257–397CrossRefGoogle Scholar
  5. 5.
    C. Carlet, S. Mesnager, On Dillon’s class H of bent functions, niho bent functions and O-polynomials. J. Comb. Theory Ser. A 118(8), 2392–2410 (2011)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    C. Carlet, S. Mesnager, On Semi-bent Boolean Functions. IEEE Trans. Inf. Theory 58(5), 3287–3292 (2012)CrossRefMathSciNetGoogle Scholar
  7. 7.
    C. Carlet, E. Prouff, On plateaued functions and their constructions, in Proceedings of Fast Software Encryption (FSE). Lecture Notes in Computer Science, vol. 2887 (2003), pp. 54–73Google Scholar
  8. 8.
    A. Cesmelioglu, W. Meidl, A construction of bent functions from plateaued functions. Des. Codes Cryptogr. 66(1–3), 231–242 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    P. Charpin, G. Gong, Hyperbent functions, Kloosterman sums and Dickson polynomials. IEEE Trans. Inform. Theory 54(9), 4230–4238 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    P. Charpin, E. Pasalic, C. Tavernier, On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inf. Theory 51(12), 4286–4298 (2005)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    S. Chee, S. Lee, K. Kim, Semi-bent functions, in Advances in Cryptology-ASIACRYPT94. Proceedings of 4th International Conference on the Theory and Applications of Cryptology, Wollongong, ed. by J. Pieprzyk, R. Safavi-Naini. Lecture Notes in Computer Science, vol. 917 (1994), pp. 107–118Google Scholar
  12. 12.
    G. Cohen, S. Mesnager, On constructions of semi-bent functions from bent functions. Journal Contemporary Mathematics 625, Discrete Geometry and Algebraic Combinatorics, Americain Mathematical Society, 141–154 (2014)Google Scholar
  13. 13.
    T.W. Cusick, H. Dobbertin, Some new three-valued crosscorrelation functions for binary m-sequences. IEEE Trans. Inf. Theory 42(4), 1238–1240 (1996)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    J. Dillon, Elementary Hadamard difference sets, Ph.D. dissertation, University of Maryland, 1974Google Scholar
  15. 15.
    H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke, P. Gaborit, Construction of bent functions via Niho Power Functions. J. Comb. Theory Ser. A 113, 779–798 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    D. Dong, L. Qu, S. Fu, C. Li, New constructions of semi-bent functions in polynomial forms. Math. Comput. Model. 57, 1139–1147 (2013)CrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Gold, Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theory 14 (1), 154–156 (1968)CrossRefMATHGoogle Scholar
  18. 18.
    F. Gologlu, Almost bent and almost perfect nonlinear functions, exponential sums, geometries and sequences, Ph.D. dissertation, University of Magdeburg, 2009Google Scholar
  19. 19.
    T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences. Discrete. Math. 16, 209–232 (1976)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    T. Helleseth, Correlation of m-sequences and related topics, in Proceedings of SETAO98, Discrete Mathematics and Theoretical Computer Science, ed. by C. Ding, T. Helleseth, H. Niederreiter (Springer, London, 1999), pp. 49–66Google Scholar
  21. 21.
    T. Helleseth, P.V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, Part 3: Applications, chap. 21, ed. by V.S. Pless, W.C. Huffman, R.A. Brualdi (Elsevier, Amsterdam, 1998), pp. 1765–1853Google Scholar
  22. 22.
    J.Y. Hyun, H. Lee, Y. Lee, Nonexistence of certain types of plateaued functions. Discrete Appl. Math. 161(16–17), 2745–2748 (2013)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    K. Khoo, G. Gong, D.R. Stinson, A new family of Gold-like sequences, in Proceedings IEEE International Symposium on Information Theory, Lausanne (2002)Google Scholar
  24. 24.
    K. Khoo, G. Gong, D.R. Stinson, A new characterization of semi-bent and bent functions on finite fields. J. Design Codes Cryptogr. 38(2), 279–295 (2006)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    G. Leander, A. Kholosha, Bent functions with 2r Niho exponents. IEEE Trans. Inf. Theory 52(12), 5529–5532 (2006)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    G. Leander, G. McGuire, Spectra of functions, subspaces of matrices, and going up versus going down, in International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC). Lecture Notes in Computer Science, vol. 4851 (Springer, Berlin, 2007), pp. 51–66Google Scholar
  27. 27.
    G. Leander, G. McGuire, Construction of bent functions from near-bent functions. J. Comb. Theory Ser. A 116, 960–970 (2009)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    N. Li, T. Helleseth, X. Tang, A. Kholosha, Several new classes of bent functions from Dillon exponents. IEEE Trans. Inf. Theory 59(3), 1818–1831 (2013)CrossRefMathSciNetGoogle Scholar
  29. 29.
    F.J. MacWilliams, N.J. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977)MATHGoogle Scholar
  30. 30.
    A. Maschietti, Difference sets and hyperovals. J. Design Codes Cryptogr. 14(1), 89–98 (1998)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    M. Matsui, Linear cryptanalysis method for DES cipher, in Proceedings of EUROCRYPT’93. Lecture Notes in Computer Science, vol. 765 (1994), pp. 386–397Google Scholar
  32. 32.
    R.L. McFarland, A family of noncyclic difference sets. J. Comb. Theory Ser. A 15, 1–10 (1973)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    W. Meier, O. Staffelbach, Fast correlation attacks on stream ciphers, in Advances in Cryptology, EUROCRYPT’88. Lecture Notes in Computer Science, vol. 330 (1988), 301–314Google Scholar
  34. 34.
    Q. Meng, H. Zhang, M. Yang, J. Cui, On the degree of homogeneous bent functions. Discrete Appl. Math. 155(5), 665–669 (2007)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    S. Mesnager, A new family of hyper-bent boolean functions in polynomial form, in Proceedings of Twelfth International Conference on Cryptography and Coding, IMACC 2009. Lecture Notes in Computer Science, vol. 5921 (Springer, Heidelberg, 2009), pp. 402–417Google Scholar
  36. 36.
    S. Mesnager, A new class of bent and hyper-bent Boolean functions in polynomial forms. J. Design Codes Cryptogr. 59(1–3), 265–279 (2011)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    S. Mesnager, Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials. IEEE Trans. Inf. Theory 57(9), 5996–6009 (2011)CrossRefMathSciNetGoogle Scholar
  38. 38.
    S. Mesnager, Semi-bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials. IEEE Trans. Inf. Theory 57(11), 7443–7458 (2011)CrossRefMathSciNetGoogle Scholar
  39. 39.
    S. Mesnager, Semi-bent functions with multiple trace terms and hyperelliptic curves, in Proceeding of International Conference on Cryptology and Information Security in Latin America (IACR), Latincrypt 2012. Lecture Notes in Computer Science, vol. 7533 (Springer, Berlin, 2012), pp. 18–36Google Scholar
  40. 40.
    S. Mesnager, Semi-bent functions from oval polynomials, in Proceedings of Fourteenth International Conference on Cryptography and Coding, Oxford, IMACC 2013. Lecture Notes in Computer Science, vol. 8308 (Springer, Heidelberg, 2013), pp. 1–15Google Scholar
  41. 41.
    S. Mesnager, Contributions on boolean functions for symmetric cryptography and error correcting codes, Habilitation to Direct Research in Mathematics (HdR thesis), December 2012Google Scholar
  42. 42.
    S. Mesnager, Bent functions from Spreads. Journal of the American Mathematical Society (AMS), Contemporary Mathematics 632. to appear.Google Scholar
  43. 43.
    S. Mesnager, G. Cohen, On the link of some semi-bent functions with Kloosterman sums, in Proceedings of International Workshop on Coding and Cryptology, IWCC 2011. Lecture Notes in Computer Science, vol. 6639 (Springer, Berlin, 2011), pp. 263–272Google Scholar
  44. 44.
    S. Mesnager, J.P. Flori, Hyper-bent functions via Dillon-like exponents. IEEE Trans. Inf. Theory 59(5), 3215– 3232 (2013)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Y. Niho, Multi-valued cross-correlation functions between two maximal linear recursive sequences, Ph.D. dissertation, University of Sothern California, Los Angeles, 1972Google Scholar
  46. 46.
    O.S. Rothaus, On “bent” functions. J. Comb. Theory Ser. A 20, 300–305 (1976)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    B. Segre, U. Bartocci, Ovali ed altre curve nei piani di Galois di caratteristica due. Acta Arith. 18(1), 423–449 (1971)MATHMathSciNetGoogle Scholar
  48. 48.
    G. Sun, C. Wu, Construction of semi-bent Boolean functions in even number of variables. Chin. J. Electron. 18(2), 231–237 (2009)Google Scholar
  49. 49.
    J. Wolfmann, Cyclic code aspects of bent functions, in Finite Fields Theory and Applications, Contemporary Mathematics Series of the AMS, vol. 518 (American Mathematical Society, Providence, 2010), pp. 363–384Google Scholar
  50. 50.
    J. Wolfmann, Special bent and near-bent functions. Adv. Math. Commun. 8(1), 21–33 (2014)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    J. Wolfmann, Bent and near-bent functions (2013). Scholar
  52. 52.
    T. Xia, J. Seberry, J. Pieprzyk, C. Charnes, Homogeneous bent functions of degree n in 2n variables do not exist for n > 3. Discrete Appl. Math. 142(1–3), 127–132 (2004)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    P. Yuan, C. Ding, Permutation polynomials over finite fields from a powerful lemma. Finite Fields Appl. 17, 560–574 (2011)CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    Y. Zheng, X.M. Zhang, Plateaued functions, in Advances in Cryptology ICICS 1999. Lecture Notes in Computer Science, vol. 1726 (Springer, Berlin, 1999), 284–300Google Scholar
  55. 55.
    Y. Zheng, X.M. Zhang, Relationships between bent functions and complementary plateaued functions. Lecture Notes in Computer Science, vol. 1787 (1999), pp. 60–75CrossRefGoogle Scholar
  56. 56.
    Y. Zheng, X.M. Zhang, On plateaued functions. IEEE Trans. Inform. Theory 47(3), 1215–1223 (2001)CrossRefMATHMathSciNetGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Paris VIII, LAGA (Laboratoire Analyse, Géometrie et Applications), UMR 7539, CNRSParisFrance
  2. 2.University of Paris XIII, Sorbonne Paris CitéSaint-Denis CedexFrance

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