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Some semi-bent functions with polynomial trace form

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Abstract

This paper is devoted to the study of semi-bent functions with several parameters flexible on the finite field \(\mathbb{F}_{2^n } \). Boolean functions defined on \(\mathbb{F}_{2^n } \) of the form

$f_{a,b}^{(r)} (x) = Tr_1^n (ax^{r(2^m - 1)} ) + Tr_1^4 (bx^{\tfrac{{2^n - 1}} {5}} ) $

and the form

$g_{a,b,c,d}^{(r,s)} (x) = Tr_1^n (ax^{r(2^m - 1)} ) + Tr_1^4 (bx^{\tfrac{{2^n - 1}} {5}} ) + Tr_1^n (cx^{(2^m - 1)\tfrac{1} {2} + 1} ) + Tr_1^n (dx^{(2^m - 1)s + 1} ) $

where n = 2m, m ≡ 2 (mod 4), a, c\(\mathbb{F}_{16} \), and b\(\mathbb{F}_2 \), d\(\mathbb{F}_2 \), are investigated in constructing new classes of semi-bent functions. Some characteristic sums such as Kloosterman sums and Weil sums are employed to determine whether the above functions are semi-bent or not.

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References

  1. Rothus O, On “bent” functions. J. Combin. Theory Ser. A, 1976, 20: 300–305.

    Article  Google Scholar 

  2. Chee S, Lee S, and Kim K, Semi-bent Functions. Advances in Cryptology-ASIACRYPT’94, 4th Int. Conf. on the Theory and Applications of Cryptology, Wollongong, Australia (eds. by Pieprzyk J and Safavi-Naini R), Lecture Notes on Computer Science, 1994, 917: 107–118.

    Google Scholar 

  3. Zeng X, Carlet C, Shan J, and Hu L, More balanced Boolean functions with optimal algebraic immunity and nonlinearity and resistance to fast algebraic attacks, IEEE Transactions on Information Theory, 2011, 57(9): 6310–6320.

    Article  MathSciNet  Google Scholar 

  4. Zheng Y and Zhang X, Plateaued functions. Advances in Cryptology-ICICS 1999, Lecture Notes in Computer Science, 1726, Springer-Verlag, Berlin, Germany, 1999.

  5. Mesnager S, Semi-bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials, IEEE Transactions on Information Theory, 2011, 57(11): 7443–7458.

    Article  MathSciNet  Google Scholar 

  6. Wang B, Tang C, Qi Y, Yang Y, and Xu M, A new class of hyper-bent Boolean functions in binomial forms, CoRR, abs/1112.0062, 2011.

    Google Scholar 

  7. Wang B, Tang C, Qi Y, Yang Y, and Xu M, A generalization of the class of hyper-bent Boolean functions in binomial forms. Cryptology ePrint Archive, Report 2011/698, http://eprint.iacr.org/, 2011.

    Google Scholar 

  8. Charpin P, Pasalic E, and Tavernier C, On bent and semi-bent quadratic Boolean functions, IEEE Transactions on Information Theory, 2005, 51(12): 4286–4298.

    Article  MATH  MathSciNet  Google Scholar 

  9. Lachaud G and Wolfmann J, The weights of the orthogonal of the extended quadratic binary Goppa codes, IEEE Transactions on Information Theory, 1990, 36(3): 686–692.

    Article  MATH  MathSciNet  Google Scholar 

  10. Lidl R, Mullen G, and Turnwald G, Dickson Polynomials. Pitman Monographs in Pure and Applied Mathematics, Addison-Wesley, 1993, 65.

  11. Dobbertin H, Leander G, Canteaut A, Carlet C, Felke P, and Gaborit P, Construction of bent functions via Niho power functions, J. Combin.Theory, Ser. A, 2006, 113: 779–798.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Xiwang Cao.

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This research was supported by the National Natural Science Foundation of China under Grant No. 11371011.

This paper was recommended for publication by Editor HU Lei.

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Chen, H., Cao, X. Some semi-bent functions with polynomial trace form. J Syst Sci Complex 27, 777–784 (2014). https://doi.org/10.1007/s11424-014-2090-4

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  • DOI: https://doi.org/10.1007/s11424-014-2090-4

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