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Boolean functions with four-valued Walsh spectra

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Abstract

Three classes of Boolean functions with four-valued Walsh spectra are presented and their Walsh spectrum distributions are determined. They are derived from Bent functions of the Maiorana-McFarland and Dillon PS ap types and of the monomial form \(Tr_1^{2m} (\lambda x^{r(2^m - 1)} )\) by complementing the values of the Bent functions at two points.

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References

  1. Rothaus O, On Bent functions, Int. J. Combin. Theory, Series A, 1976, 20: 300–305.

    Article  MATH  MathSciNet  Google Scholar 

  2. Tu Z, Zheng D, Zeng X, and Hu L, Boolean functions with two distinct Walsh coefficients, Applicable Algebra in Engineering, Communication and Computing, 2011, 22(5–6): 359–366.

    Article  MATH  MathSciNet  Google Scholar 

  3. Pei D and Qin W, The correlation of a Boolean function with its variables, in INDOCRYPT 2000, Lecture Notes in Computer Science 2000, 1977: 1–8.

    Article  MathSciNet  Google Scholar 

  4. Dobbertin H, One-to-one highly nonlinear power functions on GF(2n), Appl. Algebra Eng. Comm. Comput., 1998, 9(2): 139–152.

    Article  MATH  MathSciNet  Google Scholar 

  5. Helleseth T and Rosendahl P, New pairs of m-sequences with 4-level cross-correlation, Finite Fields Applic., 2005, 11(4): 674–683.

    Article  MATH  MathSciNet  Google Scholar 

  6. Xie Y, Hu L, Jiang W, and Zeng X, A class of Boolean functions with four-valued Walsh spectra, Proccedings of the 15th Asia-Pacific Conference on Communications (APCC 2009)-209, 2009, 880–883.

    Chapter  Google Scholar 

  7. Lachaud G and Wolfmann J, The weights of the orthogonals of the extended quadratic binary goppa codes, IEEE Trans. Inf. Theory, 1990, 36(3): 686–692.

    Article  MATH  MathSciNet  Google Scholar 

  8. Charpin P and Gong G, Hyperbent functions, Kloosterman sums, and Dickson polynomials, IEEE Trans. Inf. Theory, 2008, 54(9): 4230–4238.

    Article  MATH  MathSciNet  Google Scholar 

  9. Carlet C, Boolean functions for cryptography and error correcting codes, Chapter of the monography, Boolean Models and Methods in Mathematics, Computer Science and Engineering, ed. by Cramma Y and Hammer P, Cambridge University Press, 2010, 257–397.

    Chapter  Google Scholar 

  10. Leander N, Monomial Bent functions, IEEE Trans. Inf. Theory, 2006, 52(2): 738–743.

    Article  MATH  MathSciNet  Google Scholar 

  11. Mesnager S, Semi-Bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials, IEEE Trans. Inf. Theory, 2011, 57(11): 7443–7458.

    Article  MathSciNet  Google Scholar 

  12. Delsarte P and Goethals J, Irreducible binary cyclic codes of even dimension, Combinatorial Mathematics and Its Applications, Proc. 2nd Chapel Hill Conf., Univ. North Carolina, Chapel Hill, NC, 1970, 100–113.

    Google Scholar 

  13. Sole P and Tokareva N, Connections between quaternary and binary Bent functions, http://eprint.iacr.org/2009/544.pdf.

  14. Helleseth T, Hu L, Kholosha A, Zeng X, Li N, and Jiang W, Period-different m-sequences with at most four-valued cross correlation, IEEE Trans. Inf. Theory, 2009, 57(7): 3305–3311.

    Article  MathSciNet  Google Scholar 

  15. Carlet C and Prouff E, On plateaued functions and their constructions, Lecture Notes in Computer Science, 2003, 2887: 54–73.

    Article  Google Scholar 

  16. Dillon J, Elementary hadamard difference sets, Ph. D. Thesis, Univ. of Maryland, City of College Park, 1974.

    Google Scholar 

  17. Zheng Y and Zhang X, On plateaued functions, IEEE Trans. Inf. Theory, 2001, 47(3): 1215–1223.

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhiqiang Sun

  2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Lei Hu

  3. Data Assurance and Communication Security Research Center, Chinese Academy of Sciences, Beijing, 100093, China

    Lei Hu

Authors
  1. Zhiqiang Sun
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  2. Lei Hu
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Corresponding author

Correspondence to Zhiqiang Sun.

Additional information

This research is supported by the National Key Basic Research Program of China under Grant No. 2013CB834203, the National Natural Science Foundation of China under Grant No. 61472417, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDA06010702, and the State Key Laboratory of Information Security, Chinese Academy of Sciences.

This paper was recommended for publication by Editor LI Ziming.

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Sun, Z., Hu, L. Boolean functions with four-valued Walsh spectra. J Syst Sci Complex 28, 743–754 (2015). https://doi.org/10.1007/s11424-014-2224-8

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  • DOI: https://doi.org/10.1007/s11424-014-2224-8

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