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A survey on p-ary and generalized bent functions

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Abstract

Boolean bent functions have been introduced by Rothaus in 1966, bent functions in odd characteristic were first considered in 1985 by Kumar, Scholtz, and Welch. Two books on bent functions and some surveys on bent functions and related topics mainly deal with the Boolean case. In this survey, we focus on bent and vectorial bent functions in odd characteristic. Lately, one can observe increasing interest in the bentness of functions from elementary abelian into cyclic groups. Following this development, we also survey recent results on this class of functions.

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Notes

  1. More general, in [72], functions from \(\mathbb {Z}_q^n\) to \(\mathbb {Z}_q\) are considered for arbitrary positive integers q.

  2. Some treatment of the p-ary case can be found e.g. in [92, Chapters 13,14], or in [117, Chapter 15].

  3. In [105], the term perfect nonlinear function is used.

  4. By Parseval’s identity (see [105, Theorem 4]), for a bent function, the maximal value for \(|\sum _{x\in A}\chi (x,f(x))|\) over all such characters (which is \(\sqrt{|A|}\)) is smallest possible.

  5. The parameters of G(f) obviously are \((\mu ,\nu ,k,\lambda ) = (|A|,|B|,|A|,|A|/|B|)\).

  6. Recall that all functions from \(\mathbb {F}_{p^n}\) to \(\mathbb {F}_{p^n}\) (or with values in a subfield) have a unique representation as a univariate polynomial of degree at most \(p^n-1\).

  7. More precisely, the character sum (3) should be of the form \(\mathcal {W}_f(a,b) = \sum _{x\in \mathbb {V}_n^{(p)}}\zeta _p^{af(x)\ominus \langle b,x\rangle _n}\) with \(a\in \mathbb {F}_p^*\). However, since with a p-ary function f, also af is bent for all \(a\in \mathbb {F}_p^*\), in connection with bentness, the Walsh transform of a p-ary function is commonly defined as in (3).

  8. Since the Walsh coefficient \(\mathcal {W}_f(b)\) depends also on the inner product used in (3), strictly speaking, \(f^*\) is the dual of f with respect to \(\langle ,\rangle _n\).

  9. As is shown by Hou in [62], the bent functions \(f:\mathbb {F}_p\rightarrow \mathbb {F}_p\) are the functions \(f(x) = ax^2\oplus bx\oplus c\), \(a\neq 0\).

  10. This trivially applies to Boolean bent functions. Hence we can see Boolean bent functions as regular bent functions.

  11. Recall that by Parseval’s identity we have \(\sum _{b\in \mathbb {V}_n^{(p)}}|\mathcal {W}_f|^2 = p^{2n}\). Hence if f is an s-plateaued function, then \(\mathcal {W}_f(b) \neq 0\) for \(p^{n-s}\) values of b.

  12. The notion of plateaued functions was introduced in [125]. In some literature, the term three valued function is used, and semi-bent functions are called three valued almost optimal, see for instance [15].

  13. If p and n are odd and \(a\in \mathbb {F}_p\) is a nonsquare, then the signs in the Walsh spectra of the bent functions f and af from \(\mathbb {V}_n^{(p)}\) to \(\mathbb {F}_p\) are opposite. Hence in this case e.g. a regular bent function and a weakly regular but not regular bent function can be EA-equivalent. The precise effects of EA-equivalence transformations are listed in [31].

  14. Multiplying \(Q_2\) with a nonsquare \(a\in \mathbb {F}_p\) changes the signs in the Walsh spectrum, \(aQ_2\) can then be transformed to \(Q_1\) with a coordinate transformation.

  15. For \(p=2\), slightly differently, the sum is over i, j with \(i < j\), as the term \(x^{2^j + 2^j} = x^{2^{j+1}}\) is linear.

  16. Boolean Maiorana-McFarland functions were introduced independently by Maiorana (unpublished) and McFarland [84] as a generalization of Rothaus’ bent functions \(x\cdot y \oplus g(x)\) in [110].

  17. Several, but not all, of the primary and secondary Boolean bent function construction procedures work for odd primes p as well. For instance, Niho bent functions, i.e., Dillon’s H-class [25, 49], only exist for \(p=2\). The secondary construction of Boolean bent functions in [20, 91], does not work in this form for odd primes p.

  18. For the bentness of g it is sufficient that \(\lambda _1 f_1(x) \oplus \lambda _2 f_2(x) \oplus \lambda _3 f_3(x)\) is bent for all \(\lambda _1, \lambda _2, \lambda _3 \in \mathbb {F}_p\), for which \(\lambda _1\oplus \lambda _2\oplus \lambda _3 = a\oplus b\oplus c\), see [85, Remark 1].

  19. It is conjectured that v has to be a prime power.

  20. As we will see in Section 4.1, there are many possibilities to obtain non-weakly regular bent functions, which never belong to any of the primary classes introduced in this section.

  21. Seen a planar function on \(\mathbb {V}_n^{(p)}\) as an n-dimensional vector space of bent functions, one can consider a projection, i.e., delete some coordinates. This results in a vectorial bent function from \(\mathbb {V}_n^{(p)}\) to \(\mathbb {V}_m^{(p)}\), where \(m < n\) can be chosen arbitrarily, which is a vectorial component function of the planar function.

  22. Spread bent functions with an affine term added, which are also affine on the subspaces of a spread, are excluded.

  23. Note that f is weakly regular only if all of the bent functions \(f_j\) are regular, or all are weakly regular but not regular. Hence, mostly one will obtain a non-weakly regular bent function.

  24. This question is quite opposite to Question 1 on the existence of lonely bent functions.

  25. The exceptions when \((n,p) = (1,3)\), are the (quadratic) bent functions \(cx^2\) on \(\mathbb {F}_3\) (affine term omitted).

  26. Recall that for Boolean bent functions, which are always regular, the existence of not weakly normal functions is confirmed for all (even) dimensions \(n \geq 10\).

  27. In average, Boolean and p-ary functions are deeply non-normal. For details we refer to Theorem 3 and Proposition 1 in [19] for the Boolean case, and to [88, Proposition 1] for the case of odd primes p.

  28. The vertices of this graph are the bent functions from \(\mathbb {V}_n^{(p)}\) to \(\mathbb {F}_p\), and two bent functions f, g are adjacent if they have Hamming distance \(p^{n/2}\).

  29. Note that \(\tilde{\mathcal {C}}_F\) is a punctured version of the subcode of \(\mathcal {C}_F\) with the codewords \(c_{\alpha ,\beta ,0}\).

  30. In [93], the code \(\tilde{\mathcal {C}}_F\) is investigated.

  31. The complementary set, i.e., the set \(\{x\in \mathbb {V}_n^{(2)}\,:\,f(x) = 0\}\), is then a \((2^n,2^{n-1}\mp 2^{n/2-1},2^{n-2}\mp 2^{n/2-1})\)-difference set.

  32. Every finite group G contains the trivial difference sets, G, \(\emptyset\), \(\{z\}\), \(G\setminus \{z\}\), where z is any element of G.

  33. Of course, differently from Theorem 26, the Theorems 27, 28, 30 give only sufficient conditions for PDSs.

  34. Observe that the Cayley graph is k-regular with \(k = |supp(f)|\).

  35. In [95], the bent function a(x) is then called admissible for the partition \(\mathcal {P}\).

  36. As easily confirmed, \(\mathcal {A}(f)\) is a subspace of \(\mathcal {A}(a_{k-1},\mathcal {P})\).

  37. Partitions for spread bent functions are naturally obtained from the spread, see the discussions in [89].

  38. In the case of \(p=2\), \(k=2\), \(f(x) = a_0(x)\oplus a_1(x)2\) is \(\mathbb {Z}_4\)-bent if and only if \(F(x) = (a_0(x),a_1(x))\) is a vectorial bent function.

  39. As pointed out in [87], \(\mathbb {Z}_{p^k}\)-bent functions can more generally be obtained from partial spreads with sufficiently many subspaces.

  40. The argument is via the algebraic degree of the involved bent functions.

  41. Some questions on bent partitions are collected in section “Perspectives” of [3].

References

  1. A.A. Albert, Generalied twisted fields. Pac. J. Math. 11 (1961), 1–8

    Article  Google Scholar 

  2. N. Anbar, C. Kaşkci, W. Meidl, A. Topuzoǧo, Alev Shifted plateaued functions and their differential properties. Cryptogr. Commun. 12 (2020), 1091–1105

  3. N. Anbar, W. Meidl, Bent partitions. Des. Codes Cryptogr. 90 (2022), 1081–1101

  4. A. Bernasconi, B. Codenotti, Spectral analysis of Boolean functions as a graph eigenvalue problem. IEEE Trans. Comput. 48 (1999), 345–351

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Bernasconi, B. Codenotti, J.M. VanderKam, A characterization of bent functions in terms of strongly regular graphs, IEEE Trans. Comput. 50 (2001), 984–985

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bierbrauer, New semifields, PN and APN functions. Des. Codes Cryptogr. 54 (2010), 189–200

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Bierbrauer, D. Bartoli, G. Faina, S. Marcugini, F. Pambianco, A family of semifields in odd characteristic. Des. Codes Cryptogr. 86 (2018), 611–621

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Budaghyan, C. Carlet, CCZ-equivalence of single and multi output Boolean functions. In; Finite fields: theory and applications, Contemp. Math., 518, pp. 43–54, Amer. Math. Soc., Providence, RI, 2010

  9. L. Budaghyan, C. Carlet, CCZ-equivalence of bent vectorial functions and related constructions. Des. Codes Cryptogr. 59 (2011), 69–87

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha, Generalized bent functions and their relation to Maiorana-McFarland class, in: Proceedings IEEE Int. Symp. on Inform. Theory, 2012, pp. 1217–1220

  11. L. Budaghyan, T. Helleseth, New commutative semifields defined by new PN multinomials. Cryptogr. Commun. 3 (2011), 1–16

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Budaghyan, T. Helleseth, Planar functions and commutative semifields. Tatra Mt. Math. Publ. 45 (2010), 15–25

    MathSciNet  MATH  Google Scholar 

  13. L. Budaghyan, T. Helleseth, New perfect nonlinear multinomials over Fp2k for any odd prime p. In: Sequences and their applications–SETA 2008, Lecture Notes in Comput. Sci., 5203, pp. 403–414, Springer, Berlin, 2008

  14. A. Canteaut, C. Carlet, P. Charpin, C. Fontaine, On cryptographic properties of the cosets of R(1, m). IEEE Trans. Inform. Theory 47 (2001), 1494–1513

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Canteaut, P. Charpin, Decomposing bent functions. IEEE Trans. Inform. Theory 49 (2003), 2004–2019

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Canteaut, M. Daum, H. Dobbertin, G. Leander, Finding nonnormal bent functions. Discrete Appl. Math. 154 (2006), 202–218

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Carlet, A transformation on Boolean functions, its consequences on some problems related to Reed-Muller codes. In: Eurocode ’90 (Udine, 1990), Lecture Notes in Comput. Sci., vol. 514, pp. 42–50, Springer, Berlin, 1991

  18. C. Carlet, Two new classes of bent functions. In: Advances in Cryptology - EUROCRYPT 93, Lecture Notes in Computer Science 765, pp. 77–101, Springer-Verlag, Berlin, 1994

  19. C. Carlet, On the degree, nonlinearity, algebraic thickness, and nonormality of Boolean functions, with developments on symmetric functions. IEEE Trans. Inform. Theory 50 (2004), 2178–2185

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Carlet, On Bent and Highly Non-linear Balanced/Resilient Functions and their Algebraic Immunities. In: M.P.C. Fossorier et al. (Eds.), AAECC, Lecture Notes in Computer Science 3857, pp. 1–28, Springer-Verlag, New York, 2006

    Google Scholar 

  21. C. Carlet, Boolean functions for cryptography and error correcting codes. In: Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397, Cambridge University Press, Cambridge 2010

  22. C. Carlet, Vectorial Boolean functions for cryptography. In: Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 398–469, Cambridge University Press, Cambridge 2010

  23. C. Carlet, Boolean Functions for Cryptography and Coding Theory. Cambridge University Press 2021

  24. C. Carlet, P. Charpin, V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr. 15 (1998), 125–156

    Article  MathSciNet  MATH  Google Scholar 

  25. C. Carlet, S. Mesnager, On Dillon’s class H of bent functions, Niho bent functions and o-polynomials. J. Combin. Theory Ser. A 118 (2011), 2392–2410

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Carlet, S. Mesnager, Four decades of research on bent functions. Des. Codes Cryptogr. 78 (2016), 5–50

    Article  MathSciNet  MATH  Google Scholar 

  27. C. Celerier, D. Joyner, C. Melles, D. Phillips, S. Walsh, Edge-weighted Cayley graphs and p-ary bent functions. Integers 16 (2016), Paper No. A35, 56 pp

  28. A. Çeşmelioğlu, G. McGuire, W. Meidl, A construction of weakly and non-weakly regular bent functions, J. Comb. Theory, Series A 119 (2012), 420–429

  29. A. Çeşmelioğlu, W. Meidl, Bent functions of maximal degree, IEEE Trans. Inform. Theory 58 (2012), 1186–1190

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Çeşmelioğlu, W. Meidl, A. Pott, On the dual of (non)-weakly regular bent functions and self-dual bent functions, Advances in Mathematics of Communications 7 (2013), 425–440

    MATH  Google Scholar 

  32. A. Çeşmelioğlu, W. Meidl, A. Pott, Generalized Maiorana-McFarland class and normality of p-ary bent functions, Finite Fields Appl. 24 (2013), 105–117

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Çeşmelioğlu, W. Meidl, A. Pott, Bent functions, spreads, and o-polynomials, SIAM J. Discrete Math. 29 (2015), 854–867

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Çeşmelioğlu, W. Meidl, A. Pott, There are infinitely many bent functions for which the dual is not bent, IEEE Trans. Inform. Theory 62 (2016), 5204–5208

    Article  MathSciNet  MATH  Google Scholar 

  35. A. Çeşmelioğlu, W. Meidl, A. Pott, Vectorial bent functions and their duals, Linear Algebra and its Applications 548 (2018), 305–320

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Çeşmelioğlu, W. Meidl, Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications 12 (2018), 691–705

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Çeşmelioğlu, W. Meidl, A. Pott, A survey on bent functions and their duals. In: Combinatorics and Finite Fields, Radon Series on Computational and Applied Mathematics, pp. 39–56, de Gruyter, Berlin, 2019

  38. A. Çeşmelioğlu, W. Meidl, A. Pott, Vectorial bent functions in odd characteristic and their components. Cryptogr. Commun. 12 (2020), 899–912

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Çeşmelioğlu, W. Meidl, I. Pirsic, Vectorial bent functions and partial difference sets. Des. Codes Cryptogr. 89 (2021), 2313–2330

    Article  MathSciNet  MATH  Google Scholar 

  40. P. Charpin, Normal Boolean functions. J. Complexity 20 (2004), 245–265

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Charpin, E. Pasalic, C Tavernier, On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inform. Theory 51 (2005), 4286–4298

    Article  MathSciNet  MATH  Google Scholar 

  42. Y.M. Chee, Y. Tan, X.D. Zhang, Strongly regular graphs constructed from p-ary bent functions, J. Algebr. Comb. 34 (2011), 251–266

    Article  MathSciNet  MATH  Google Scholar 

  43. R.S. Coulter, R.W. Matthews, Planar functions and planes of Lenz-Barlotti class II. Des. Codes Cryptogr. 10 (1997), 167–184

    Article  MathSciNet  MATH  Google Scholar 

  44. R.S. Coulter, M. Henderson, Commutative presemifields and semifields. Adv. Math. 217 (2008), no. 1, 282–304

    Article  MathSciNet  MATH  Google Scholar 

  45. T.W. Cusick, P. Stănică, Cryptographic Boolean functions and applications. Second edition. Elsevier/Academic Press, London, 2017

    MATH  Google Scholar 

  46. E. van Dam, Strongly regular decompositions of the complete graph. J. Algebraic Combin. 17 (2003), 181–201

    Article  MathSciNet  MATH  Google Scholar 

  47. J.A. Davis, J. Jedwab, A unifying construction for difference sets, J. Combin. Theory Ser. A 80 (1997), 13–78

    Article  MathSciNet  MATH  Google Scholar 

  48. U. Dempwolff, Automorphisms and isomorphisms of some p-ary bent functions. J. Algebraic Combin. 51 (2020), 527–566

    Article  MathSciNet  MATH  Google Scholar 

  49. J.F. Dillon, Elementary Hadamard difference sets, Ph.D. dissertation, University of Maryland, 1974

  50. C. Ding, Cyclic codes from APN and planar functions, ar**v:1206.4687v1

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

    Article  MathSciNet  MATH  Google Scholar 

  52. H. Dobbertin, Construction of bent functions and balanced boolean functions with high nonlinearity. In: Fast Software Encryption: Proceedings of the 1994 Leuven Workshop on Cryptographic Algorithms, Lecture Notes in Computer Science 1008, pp. 61–74, Berlin, Germany, 1995

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

    Article  MathSciNet  MATH  Google Scholar 

  54. K. Feng, J. Luo, Value distributions of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inform. Theory 53 (2007), 3035–3041

    Article  MathSciNet  MATH  Google Scholar 

  55. T. Feng, B. Wen, Q. **ang, J. Yin, Partial difference sets from quadratic forms and p-ary weakly regular bent functions. ALM 27, 25–40

  56. G. Gong, T. Helleseth, H. Hu, A. Kholosha, On the dual of certain ternary weakly regular bent functions, IEEE Trans. Inform. Theory 58 (2012), 2237–2243

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Helleseth, H. D. L. Hollmann, A. Kholosha, Z. Wang, Q. **ang, Proofs of two conjectures on ternary weakly regular bent functions, IEEE Trans. Inform. Theory 55 (2009), 5272–5283

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Helleseth, A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory 56 (2010), 4646–4652

    Article  MathSciNet  MATH  Google Scholar 

  60. T. Helleseth, A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun. 3 (2011), 281–291

    Article  MathSciNet  MATH  Google Scholar 

  61. S. Hodžić, W. Meidl, E. Pasalic, Full characterization of generalized bent functions as (semi)-bent spaced, their dual and the Gray image. IEEE Trans. Inform. Theory 64 (2018), 5432–5440

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  63. H. Hu, Q. Zhang, S. Shao, On the dual of the Coulter-Matthews bent functions. IEEE Trans. Inform. Theory 63 (2017), 2454–2463

    Article  MathSciNet  MATH  Google Scholar 

  64. H. Hu, X. Yang, S. Tang, New classes of ternary bent functions from the Coulter-Matthews bent functions. IEEE Trans. Inform. Theory 64 (2018), 4653–4663

    Article  MathSciNet  MATH  Google Scholar 

  65. J.Y. Hyun, Y. Lee, Characterization of p-ary bent functions in terms of strongly regular graphs. IEEE Trans. Inform. Theory 65 (2019), 676–684

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  67. D. Joyner, C. Melles, Perspectives on p-ary bent functions. Elementary theory of groups and group rings, and related topics (New York, Nov. 1–2, 2018), pp. 103–126, de Gruyter, Berlin, 2020

  68. W. Kantor, Exponential numbers of two-weight codes, difference sets and symmetric designs. Discrete Math. 46 (1983), 95–98

    Article  MathSciNet  MATH  Google Scholar 

  69. W. Kantor, Bent functions generalizing Dillon’s partial spread functions, ar**v:1211.2600v1

  70. D.E. Knuth, Finite semifields and projective planes. J. Agebra 2 (1965), 182–217

    Article  MathSciNet  MATH  Google Scholar 

  71. N. Kolomeec, The graph of minimal distances of bent functions and its properties. Des. Codes Cryptogr. 85 (2017), 395–410

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  73. M. Lavrauw, O. Polverino, Finite semifields. In: Current research topics in Galois Geometry, pp. 131–159, Nova Science Publishers, New York, 2012

  74. G. Leander, G. McGuire, Construction of bent functions from near-bent functions. J. Comb. Theory Ser. A 116 (2009), 960–970

    Article  MathSciNet  MATH  Google Scholar 

  75. N. Li, S. Mesnager, Recent results and problems on constructions of linear codes from cryptographic functions. Cryptogr. Commun. 12 (2020), 965–986

    Article  MathSciNet  MATH  Google Scholar 

  76. P. Lisonek, H.Y. Lu, Bent functions on partial spreads, Des. Codes Cryptogr. 73 (2014), 209–216

    Article  MathSciNet  MATH  Google Scholar 

  77. S.L. Ma, Partial difference sets. Discrete Math. 52 (1984), 75–89

    Article  MathSciNet  MATH  Google Scholar 

  78. S.L. Ma, A survey of partial difference sets. Des. Codes Cryptogr. 4 (1994), 221–261

    Article  MathSciNet  MATH  Google Scholar 

  79. B. Mandal, P. Stănică, S. Gangopadhyay, New classes of p-ary bent functions. Cryptogr. Commun. 11 (2019), 77–92

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  82. T. Martinsen, W. Meidl, P. Stănică, Generalized bent functions and their Gray images, in (S. Duquesne, S. Petkova-Nikova, eds.) Arithmetic of Finite Fields, Proceedings of WAIFI 2016, Lecture Notes in Computer Science 10064, pp. 160–173, Springer-Verlag, Berlin Heidelberg, 2017

  83. T. Martinsen, W. Meidl, P. Stănică, Partial spread and vectorial generalized bent functions. Designs, Codes, Cryptogr. 85 (2017), 1–13

    Article  MathSciNet  MATH  Google Scholar 

  84. R.L. McFarland, A family of noncyclic difference sets, J. Combin. Theory Ser. A 15 (1973), 1–10

    Article  MATH  Google Scholar 

  85. W. Meidl, Generalized Rothaus construction and non-weakly regular bent functions, J. Combin. Theory Ser. A 141 (2016), 78–89

    Article  MathSciNet  MATH  Google Scholar 

  86. W. Meidl, A secondary construction of bent functions, octal gbent functions and their duals. Math. Comput. Simulation 143 (2018), 57–64

    Article  MathSciNet  MATH  Google Scholar 

  87. W. Meidl, I. Pirsic, Bent and Z2k-bent functions from spread-like partitions. Des. Codes Cryptogr. 89 (2021), 75–89

    Article  MathSciNet  MATH  Google Scholar 

  88. W. Meidl, I. Pirsic, On the normality of p-ary bent functions. Cryptogr. Commun. 10 (2018), 1037–1049

    Article  MathSciNet  MATH  Google Scholar 

  89. W. Meidl, A. Pott, Generalized bent functions into Z2k from the partial spread and the Maiorana-McFarland class, Cryptogr. Commun. 11 (2019), 1233–1245

    Article  MathSciNet  MATH  Google Scholar 

  90. C. Melles, D. Joyner, On p-ary bent functions and strongly regular graphs, ar**v:1904.09359v1

  91. S. Mesnager, Several new infinite families of bent functions and their duals. IEEE Trans. Inform. Theory 60 (2014), 4397–4407

    Article  MathSciNet  MATH  Google Scholar 

  92. S. Mesnager, Bent functions. Fundamentals and results. Springer 2016

    Book  MATH  Google Scholar 

  93. S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction. Cryptogr. Commun. 9 (2017), 71–84

    Article  MathSciNet  MATH  Google Scholar 

  94. S. Mesnager, C. Tang, Y. Qi, L. Wang, B. Wu, K. Feng, Further results on generalized bent functions and their complete characterization. IEEE Trans. Inform. Theory 64 (2018), 5441–5452

    Article  MathSciNet  MATH  Google Scholar 

  95. S. Mesnager, C. Tang, Y. Qi, Generalized plateaued functions and admissible (plateaued) functions. IEEE Trans. Inform. Theory 63 (2017), 6139–6148

    Article  MathSciNet  MATH  Google Scholar 

  96. K. Nyberg, Perfect nonlinear S-boxes, In: Advances in cryptology–EUROCRYPT ’91 (Brighton, 1991), Lecture Notes in Comput. Sci., 547, pp. 378–386, Springer, Berlin, 1991

  97. K. Nyberg, Construction of bent functions and difference sets, In: Advances in cryptology–EUROCRYPT ’90 (Aarhus, 1990), Lecture Notes in Comput. Sci., 473, pp. 151–160, Springer, Berlin, 1991

  98. R.M. Pelen, F. Özbudak, Duals of non weakly regular bent functions are not weakly regular and generalization to plateaued functions, Finite Fields Appl. 64 (2020), 101668, 16 pp

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

    Article  MathSciNet  MATH  Google Scholar 

  100. L. Poinsot, Non Abelian bent functions. Cryptogr. Commun. 4 (2012), 1–23

    Article  Google Scholar 

  101. L. Poinsot, A. Pott, Non-Boolean almost perfect nonlinear functions on non-Abelian groups. Internat. J. Found. Comput. Sci. 22 (2011), 1351–1367

    Article  MathSciNet  MATH  Google Scholar 

  102. V. Potapov, On minimal distance between p-ary bent functions. In: Problems of redundancy in information and control systems, pp. 115–116, IEEE, 2016

  103. V. Potapov, On q-ary bent and plateaued functions. Des. Codes Cryptogr. 88 (2020), 2037–2049

    Article  MathSciNet  MATH  Google Scholar 

  104. A. Pott, A survey on relative difference sets. Groups, difference sets, and the Monster. In: Ohio State Univ. Math. Res. Inst. Publ., 4, pp. 195–232, de Gruyter, Berlin, 1996

  105. A. Pott, Nonlinear functions in abelian groups and relative difference sets. Discrete Appl. Math. 138 (2004), 177–193

    Article  MathSciNet  MATH  Google Scholar 

  106. A. Pott, Almost perfect and planar functions. Des. Codes Cryptogr. 78 (2016), 141–195

    Article  MathSciNet  MATH  Google Scholar 

  107. A. Pott, Y. Tan, T. Feng, S. Ling, Association schemes arising from bent functions. Des. Codes Cryptogr. 59 (2011), 319–331

    Article  MathSciNet  MATH  Google Scholar 

  108. Y. Qi, C. Tang, D Huang, Explicit characterization of two classes of regular bent functions. Appl. Algebra Engrg. Comm. Comput. 29 (2018), 529–544

    Article  MathSciNet  MATH  Google Scholar 

  109. Y. Qi, Yanfeng, C.Tang, Z. Zhou, C. Fan, Several infinite families of p-ary weakly regular bent functions. Adv. Math. Commun. 12 (2018), 303–315

    Article  MathSciNet  MATH  Google Scholar 

  110. O.S. Rothaus, On bent functions, J. Combin. Theory Ser. A 20 (1976), 300–305

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  112. Y. Tan, A. Pott, T. Feng, Strongly regular graphs associated with ternary bent functions. J. Comb. Theory, Series A 117 (2010), 668–682

  113. Y. Tan, J. Yang, X. Zhang, A recursive approach to construct p-ary bent functions which are not weakly regular, In: Proceedings of IEEE International Conference on Information Theory and Information Security, pp. 156–159, Bei**g, 2010

  114. C. Tang, Y. Qi, D. Huang, Regular p-ary bent functions with five terms and Kloosterman sums. Cryptogr. Commun. 11 (2019), 1133–1144

    Article  MathSciNet  MATH  Google Scholar 

  115. C. Tang, C. **ang, Y. Qi, K. Feng, Complete characterization of generalized bent and 2k-bent Boolean functions. IEEE Trans. Inform. Theory 63 (2017), 4668–4674

    Article  MathSciNet  MATH  Google Scholar 

  116. C. Tang, M. Xu, Y. Qi, M. Zhou, A new class of p-ary regular bent functions. Adv. Math. Commun. 15 (2021), 55–64

    Article  MathSciNet  MATH  Google Scholar 

  117. N. Tokareva, Bent Functions, Results and Applications to Cryptography, Academic Press, San Diego, CA, 2015

    Book  MATH  Google Scholar 

  118. G. Weng, X. Zeng, Further results on planar DO functions and commutative semifields. Des. Codes Cryptogr. 63 (2012), 413–423

    Article  MathSciNet  MATH  Google Scholar 

  119. Y. Wu, N. Li, X. Zeng, Linear codes from perfect nonlinear functions over finite fields. IEEE Trans. Commun. 68 (2020), 3–11

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  122. G. Xu, X. Cao, S. Xu, Constructing new APN functions and bent functions over finite fields of odd characteristic via the switching method. Cryptogr. Commun. 8 (2016), 155–171

    Article  MathSciNet  MATH  Google Scholar 

  123. G. Xu, X. Cao, S. Xu, Two classes of p-ary bent functions and linear codes with three or four weights. Cryptogr. Commun. 9 (2017), 117–131

    Article  MathSciNet  MATH  Google Scholar 

  124. D. Zheng, L. Yu, L. Hu, On a class of binomial bent functions over the finite fields of odd characteristic. Appl. Algebra Engrg. Comm. Comput. 24 (2013), 461–475

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thanks Sabancı University for the hospitality during several research visits. The author also wishes to thank the reviewer and the associate editor for valuable comments, which helped to improve the paper.

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    Wilfried Meidl

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Meidl , W. A survey on p-ary and generalized bent functions. Cryptogr. Commun. 14, 737–782 (2022). https://doi.org/10.1007/s12095-022-00570-x

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