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On a class of binomial bent functions over the finite fields of odd characteristic

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Abstract

We give a necessary and sufficient condition such that the class of \(p\)-ary binomial functions proposed by Jia et al. (IEEE Trans Inf Theory 58(9):6054–6063, 2012) are regular bent functions, and thus settle the open problem raised at the end of that paper. Moreover, we investigate the bentness of the proposed binomials under the case \(\gcd (\frac{t}{2}, p^{\frac{n}{2}}+1)=1\) for some even integers \(t\) and \(n\). Computer experiments show that the new class contains bent functions that are affinely inequivalent to known monomial and binomial ones.

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References

  1. Canteaut, A., Charpin, P., Kyureghyan, G.: A new class of monomial bent functions. Finite Fields Appl. 14(1), 221–241 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) The Monography Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010)

    Chapter  Google Scholar 

  3. Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complex. 20(2–3), 205–244 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Charpin, P., Kyureghyan, G.: Cubic monomial bent functions: a subclass of \({\cal M}\). SIAM. J. Discret. Math. 22(2), 650–665 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dillon, J.F.: Elementary Hadamard difference sets. Ph. D. these, University Maryland, Collage Park (1974)

  6. Dobbertin, H., Leander, G., Canteaut, A., Gabort, P.: Construction of bent functions via Niho power functions. J. Comb. Theory Ser. 113, 779–798 (2006)

    Article  MATH  Google Scholar 

  7. Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Golomb, S.W., Gong, G.: Signal Designs With Good Correlation: For Wireless Communications. Cryptography and Radar Applications. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  9. Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Helleseth, T., Kholosha, A.: New binomial bent functions over finite fields of odd characteristic. IEEE Trans. Inf. Theory 56(9), 4646–4652 (2010)

    Article  MathSciNet  Google Scholar 

  11. Hou, X.D.: \(p\)-ary and \(q\)-ary versions of certain results about bent functions and resilient functions. Finite Fields Appl. 10(4), 566–582 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jia, W., Zeng, X., Helleseth, T., Li, C.: A class of binomial bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 58(9), 6054–6063 (2012)

    Article  MathSciNet  Google Scholar 

  13. Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40, 90–107 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Leander, G.: Monomial bent functions. IEEE Trans. Inf. Theory 52(2), 738–743 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lidl, R., Niederreiter, H.: Finite fields ser. In: Encyclopedia of Mathematics and its Applications. Addison-Wesley, Amsterdam (1983)

  16. Liu, S.C., Komo, J.J.: Nonbinary Kasami sequence over GF(p). IEEE Trans. Inf. Theory 38(4), 1409–1412 (1983)

    Article  MathSciNet  Google Scholar 

  17. MacWilliams, F.J., Sloane, N.J.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  18. Mesnager, S.: Bent and hyper-bent functions in polynomial form their link with some exponential sums and Dickson polynomials. IEEE Trans. Inf. Theory 57(9), 5996–6009 (2011)

    Article  MathSciNet  Google Scholar 

  19. Rothaus, O.S.: On bent functions. J. Comb. Theory Ser. A 20, 300–305 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  20. Xiang, Q.: Maximally nonlinear functions and bent functions. Des. Codes Cryptogr. 17, 211–218 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors wish to thank Xiangyong Zeng, Xiwang Cao and two anonymous referees for their helpful comments. The work of D. Zheng was supported by National Natural Science Foundation of China (NSFC) under Grant 11101131. The work of L. Hu was supported by the NSFC (61070172 and 10990011), and the National Basic Research Program of China (2013CB834203).

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Correspondence to Dabin Zheng.

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Zheng, D., Yu, L. & Hu, L. On a class of binomial bent functions over the finite fields of odd characteristic. AAECC 24, 461–475 (2013). https://doi.org/10.1007/s00200-013-0202-3

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  • DOI: https://doi.org/10.1007/s00200-013-0202-3

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