New nonexistence results on (mn)-generalized bent functions

Abstract

In this paper, we present some new nonexistence results on (mn)-generalized bent functions, which improved recent results. More precisely, we derive new nonexistence results for general n and m odd or \(m \equiv 2 \pmod {4}\), and further explicitly prove nonexistence of (m, 3)-generalized bent functions for all integers m odd or \(m \equiv 2 \pmod {4}\). The main tools we utilized are certain exponents of minimal vanishing sums from applying characters to group ring equations that characterize (mn)-generalized bent functions.

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References

  1. 1.

    Carlet C.: Boolean functions for cryptography and error correcting codes. Boolean Models Methods Math. Comput. Sci. Eng. 2, 257–397 (2010).

    Article  Google Scholar 

  2. 2.

    Conway J.H., Jones A.J.: Trigonometric diophantine equations (on vanishing sums of roots of unity). Acta Arith. 30(3), 229–240 (1976).

    MathSciNet  Article  Google Scholar 

  3. 3.

    Kumar P.V., Scholtz R.A., Welch L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40(1), 90–107 (1985).

    MathSciNet  Article  Google Scholar 

  4. 4.

    Lam T.Y., Leung K.H.: On vanishing sums of roots of unity. J. Algebra 224(1), 91–109 (2000).

    MathSciNet  Article  Google Scholar 

  5. 5.

    Lenstra Jr., H.W.: Vanishing sums of roots of unity. In: Proceedings of the Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978), Part II, Math. Centre Tracts, vol. 101, pp. 249–268. Math. Centrum, Amsterdam (1979)

  6. 6.

    Leung K.H., Schmidt B.: Nonexistence results on generalized bent functions \({\mathbb{Z}}_q^m\rightarrow {\mathbb{Z}}_q\) with odd \(m\) and \(q\equiv 2\)  (mod 4). J. Comb. Theory Ser. A 163, 1–33 (2019).

    Article  Google Scholar 

  7. 7.

    Liu H., Feng K., Feng R.: Nonexistence of generalized bent functions from \({\mathbb{Z}}_2^n\) to \({\mathbb{Z}}_m\). Des. Codes Cryptogr. 82(3), 647–662 (2017).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Mann H.B.: Difference sets in elementary Abelian groups. Ill. J. Math. 9, 212–219 (1965).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Martinsen T., Meidl W., Mesnager S., Stǎnicǎ P.: Decomposing generalized bent and hyperbent functions. IEEE Trans. Inform. Theory 63(12), 7804–7812 (2017).

    MathSciNet  Article  Google Scholar 

  10. 10.

    Rothaus O.S.: On “bent” functions. J. Comb. Theory Ser. A 20(3), 300–305 (1976).

    Article  Google Scholar 

  11. 11.

    Schmidt K.U.: \({\mathbb{Z}}_4\)-valued quadratic forms and quaternary sequence families. IEEE Trans. Inform. Theory 55(12), 5803–5810 (2009).

    MathSciNet  Article  Google Scholar 

  12. 12.

    Schmidt K.U.: Quaternary constant-amplitude codes for multicode CDMA. IEEE Trans. Inform. Theory 55(4), 1824–1832 (2009).

    MathSciNet  Article  Google Scholar 

  13. 13.

    Stănică P., Martinsen T., Gangopadhyay S., Singh B.K.: Bent and generalized bent Boolean functions. Des. Codes Cryptogr. 69(1), 77–94 (2013).

    MathSciNet  Article  Google Scholar 

  14. 14.

    Tang C., Xiang C., Qi Y., Feng K.: Complete characterization of generalized bent and \(2^k\)-bent Boolean functions. IEEE Trans. Inform. Theory 63(7), 4668–4674 (2017).

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors are very grateful to the two anonymous reviewers for all their detailed comments that improved the quality and the presentation of this paper.

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Correspondence to Qi Wang.

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K.H. Leung’s research was supported by Grant R-146-000-158-112, Ministry of Education, Singapore. Q. Wang’s research was supported by the National Natural Science Foundation of China under Grant 61672015.

Communicated by G. Kyureghyan.

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Leung, K.H., Wang, Q. New nonexistence results on (mn)-generalized bent functions. Des. Codes Cryptogr. 88, 755–770 (2020). https://doi.org/10.1007/s10623-019-00708-8

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Keywords

  • Exponent
  • Generalized bent function
  • Minimal relation
  • Nonexistence
  • Vanishing sum

Mathematics Subject Classification

  • 11A07
  • 16S34
  • 05B10
  • 94A15