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Characterizations and constructions of plateaued functions on finite abelian groups

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Abstract

Plateaued functions have been studied in many papers. They can be candidates for designing cryptographic functions and have been used to construct linear codes. They also have close connections to combinatorics and design theory. Plateaued functions on finite abelian groups were studied in Xu (J Comb Des 27:756–783, 2019; J Comb Des 2021, https://doi.org/10.1002/jcd.21281). In this paper, we continue the research in Xu (2019, 2021). We will first study the characterizations of plateaued functions in terms of the derivatives and autocorrelation functions. We will also characterize the plateaued-ness of a function by its distance to affine functions. Then we investigate constructions of plateaued functions. In particular, we will give two general methods to construct plateaued functions and prove the existence of plateaued functions on finite abelian groups whose orders are not prime numbers. We will also show how to construct plateaued functions from a finite abelian group to a group of prime order. As applications of plateaued functions to combinatorics, we will show the existence of two new infinite families of directed strongly regular graphs.

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Change history

  • 29 November 2022

    Incorrect page range in Reference 42 has been corrected.

References

  1. Babai L.: The Fourier transform and equations over finite abelian groups, an introduction to the method of trigonometric sums. https://people.cs.uchicago.edu/~laci/reu02/fourier.pdf

  2. Brouwer A.E., Olmez O., Song S.-Y.: Directed strongly regular graphs from \(1\frac{1}{2}\)-designs. Eur. J. Comb. 33, 1174–1177 (2012).

    Article  MATH  Google Scholar 

  3. Carlet C.: Boolean and vectorial plateaued functions and APN functions. IEEE Trans. Inf. Theory 61, 6272–6289 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  4. Carlet C., Ding C.: Nonlinearities of S-boxes. Finite Fields Appl. 13, 121–135 (2007).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Carlet C., Mesnager S., Özbudak F., Sinak A.: Explicit characterizations for plateaued-ness of \(p\)-ary (vectorial) functions. In: International Conference on Codes, Cryptology, and Information Security, pp. 328–345. Springer, Cham (2017).

    Chapter  Google Scholar 

  7. Çeşmelioğlu A., Meidl W.: A construction of bent functions from plateaued functions. Des. Codes Cryptogr. 66, 231–242 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. Çeşmelioğlu A., McGuire G., Meidl W.: A construction of weakly and nonweakly regular bent functions. J. Comb. Theory Ser. A 119, 420–429 (2012).

    Article  MATH  Google Scholar 

  9. Carlet C., Prouff E.: On plateaued functions and their constructions. In: Johansson T. (ed.) Fast Software Encryption, FSE 2003. Lecture Notes in Computer Science, vol. 2887. Springer, Berlin (2003).

    Google Scholar 

  10. Chung H., Kumar P.V.: A new general construction of generalized bent functions. IEEE Trans. Inf. Theory 35, 206–209 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  11. Davis J.A., Olmez O.: A framework for constructing partial geometric difference sets. Des. Codes Cryptogr. 86, 1367–1375 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  12. Davis J.A., Poinsot L.: \(G\)-Perfect nonlinear functions. Des. Codes Cryptogr. 46, 83–96 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  13. Drakakis K., Requena V., McGuire G.: On the nonlinearity of exponential Welch Costas functions. IEEE Trans. Inf. Theory 56, 1230–1238 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan Y., Xu B.: Fourier transforms on finite group actions and bent functions. J. Algebr. Comb. (2021). https://doi.org/10.1007/s10801-021-01057-3.

    Article  MATH  Google Scholar 

  15. Fan Y., Xu B.: Fourier transforms and bent functions on finite groups. Des. Codes Cryptogr. 86, 2091–2113 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  16. Fan Y., Xu B.: Nonlinear functions and difference sets on group actions. Des. Codes Cryptogr. 85, 319–341 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  17. Fan Y., Xu B.: Fourier transforms and bent functions on faithful actions of finite abelian groups. Des. Codes Cryptogr. 82, 543–558 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. Fu S., Feng X., Wang Q., Carlet C.: On the derivative imbalance and ambiguity of functions. IEEE Trans. Inf. Theory 65, 5833–5845 (2019).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Hyun J.Y., Lee J., Lee Y.: Explicit criteria for construction of plateaued functions. IEEE Trans. Inf. Theory 62, 7555–7565 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  21. 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 

  22. Logachev O.A., Salnikov A.A., Yashchenko V.V.: Bent functions over a finite abelian group. Discret. Math. Appl. 7, 547–564 (1997).

    Article  MATH  Google Scholar 

  23. Mesnager S.: Characterizations of plateaued and bent functions in characteristic \(p\). In: International Conference on Sequences and Their Applications, pp. 72–82. Springer, Cham (2014).

  24. Mesnager S., Özbudak F., Sınak A., Cohen G.: On \(q\)-ary plateaued functions over \({{\mathbb{F} }_q}\) and their explicit characterizations. Eur. J. Comb. 80, 71–81 (2019).

    Article  MATH  Google Scholar 

  25. Mesnager S., Özbudak F., Sınak A.: Secondary constructions of (non)-weakly regular plateaued functions over finite fields. Turk. J. Math. 45, 2295–2306 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  26. Mesnager S., Özbudak F., Sınak A.: Linear codes from weakly regular plateaued functions and their secret sharing schemes. Des. Codes Cryptogr. 87, 463–480 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  27. Mesnager S., Özbudak F., Sınak A.: On the p-ary (cubic) bent and plateaued (vectorial) functions. Des. Codes Cryptogr. 86, 1865–1892 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  28. Mesnager S., Özbudak F., Sınak A.: Results on characterizations of plateaued functions in arbitrary characteristic. In: International Conference on Cryptography and Information Security in the Balkans, pp. 17–30. Springer, Cham (2015).

  29. Nowak K., Olmez O., Song S.-Y.: Partial geometric difference families. J. Comb. Des. 24, 112–131 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  30. Olmez O.: Plateaued functions and one-and-half difference sets. Des. Codes Cryptogr. 76, 537–549 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  31. Olmez O.: Symmetric \(1\frac{1}{2}\)-designs and \(1\frac{1}{2}\)-difference sets. J. Comb. Des. 22, 252–269 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  32. Panario D., Sakzad A., Stevens B., Wang Q.: Two new measures for permutations: ambiguity and deficiency. IEEE Trans. Inf. Theory 57, 7648–7657 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  33. Panario D., Sakzad A., Stevens B., Thomson D., Wang Q.: Ambiguity and deficiency of permutations over finite fields with linearized difference map. IEEE Trans. Inf. Theory 59, 5616–5626 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  34. Poinsot L.: Non abelian bent functions. Cryptogr. Commun. 4, 1–23 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  35. Poinsot L.: A new characterization of group action-based perfect nonlinearity. Discret. Appl. Math. 157, 1848–1857 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  36. Poinsot L.: Bent functions on a finite nonabelian group. J. Discret. Math. Sci. Cryptogr. 9, 349–364 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  37. Poinsot L., Harari S.: Group actions based perfect nonlinearity. GESTS Int. Trans. Comput. Sci. Eng. 12, 1–14 (2005).

    Google Scholar 

  38. Poinsot L., Pott A.: Non-boolean almost perfect nonlinear functions on non-abelian groups. Int. J. Found. Comput. Sci. 22, 1351–1367 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  39. Pott A.: Nonlinear functions in abelian groups and relative difference sets, Optimal Discrete Structures and Algorithms, ODSA 2000. Discret. Appl. Math. 138, 177–193 (2004).

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

    Article  MATH  Google Scholar 

  41. Solodovnikov V.I.: Bent functions from a finite abelian group to a finite abelian group. Diskret. Mat. 14, 99–113 (2002).

    MathSciNet  MATH  Google Scholar 

  42. Xu B.: Plateaued functions on finite abelian groups and partial geometric difference sets. J. Comb. Des. 30, 220–250 (2022).

  43. Xu B.: A new framework for identifying absolute maximum nonlinear functions. J. Comb. Theory Ser. A 187, 105560 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  44. Xu B.: Absolute maximum nonlinear functions on finite nonabelian groups. IEEE Trans. Inf. Theory 66, 5167–5181 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  45. Xu B.: Plateaued functions, partial geometric difference sets, and partial geometric designs. J. Comb. Des. 27, 756–783 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  46. Xu B.: Bentness and nonlinearity of functions on finite groups. Des. Codes Cryptogr. 76, 409–430 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu B.: Multidimensional Fourier transforms and nonlinear functions on finite groups. Linear Algebra Appl. 452, 89–105 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  48. Zheng Y., Zhang X.-M.: Plateaued functions. In: Varadharajan V., Mu Y. (eds.) Information and Communication Security. ICICS 1999. Lecture Notes in Computer Science, vol. 1726, Springer, Berlin (1999).

  49. Zheng Y., Zhang X.-M.: On plateaued functions. IEEE Trans. Inf. Theory 47, 1215–1223 (2001).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank the referees for useful comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY, 40475, USA

    Bangteng Xu

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  1. Bangteng Xu
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Correspondence to Bangteng Xu.

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Communicated by Y. Zhou.

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Xu, B. Characterizations and constructions of plateaued functions on finite abelian groups. Des. Codes Cryptogr. 91, 1261–1292 (2023). https://doi.org/10.1007/s10623-022-01151-y

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