Full Characterization of Generalized Bent Functions as (Semi)-Bent Spaces, Their Dual, and the Gray Image

A natural generalization of bent functions is a class of functions from <inline-formula> <tex-math notation="LaTeX">${\mathbb F}_{2}^{n}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{2^{k}}$ </tex-math></inline-formula> which is known as generalized bent (gbent) functions. The construction and characterization of gbent functions are commonly described in terms of the Walsh transforms of the associated Boolean functions. Using similar approach, we first determine the dual of a gbent function when <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is even. Then, depending on the parity of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, it is shown that the Gray image of a gbent function is <inline-formula> <tex-math notation="LaTeX">$(k-1)$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$(k-2)$ </tex-math></inline-formula> plateaued, which generalizes previous results for <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> = 2,3, and 4. We then completely characterize gbent functions as algebraic objects. More precisely, again depending on the parity of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, a gbent function is a <inline-formula> <tex-math notation="LaTeX">$(k-1)$ </tex-math></inline-formula>-dimensional affine space of bent functions or semi-bent functions with certain interesting additional properties, which we completely describe. Finally, we also consider a subclass of functions from <inline-formula> <tex-math notation="LaTeX">${\mathbb F}_{2}^{n}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{2^{k}}$ </tex-math></inline-formula>, called <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{q}$ </tex-math></inline-formula>-bent functions (which are necessarily gbent), which essentially gives rise to relative difference sets similarly to standard bent functions. Two examples of this class of functions are provided and it is demonstrated that many gbent functions are not <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{q}$ </tex-math></inline-formula>-bent.

[1]  Patrick Solé,et al.  Connections between Quaternary and Binary Bent Functions , 2009, IACR Cryptol. ePrint Arch..

[2]  A. Pott,et al.  Vectorial bent functions and their duals , 2018, Linear Algebra and its Applications.

[3]  Alexander Pott,et al.  Almost perfect and planar functions , 2016, Des. Codes Cryptogr..

[4]  Enes Pasalic,et al.  Construction methods for generalized bent functions , 2018, Discret. Appl. Math..

[5]  Matthew G. Parker,et al.  On Boolean Functions Which Are Bent and Negabent , 2007, SSC.

[6]  P. Vijay Kumar,et al.  Generalized Bent Functions and Their Properties , 1985, J. Comb. Theory, Ser. A.

[7]  Alexander Pott,et al.  Nonlinear functions in abelian groups and relative difference sets , 2004, Discret. Appl. Math..

[8]  Haiying Liu,et al.  Nonexistence of Generalized Bent Functions From $Z_{2}^{n}$ to $Z_{m}$ , 2015, ArXiv.

[9]  Pantelimon Stanica,et al.  Generalized Bent Functions and Their Gray Images , 2016, WAIFI.

[10]  Enes Pasalic,et al.  Generalized Bent Functions - Some General Construction Methods and Related Necessary and Sufficient Conditions , 2015, Cryptography and Communications.

[11]  Alexander Pott,et al.  Finite Geometry and Character Theory , 1995 .

[12]  Kaisa Nyberg,et al.  Constructions of Bent Functions and Difference Sets , 1991, EUROCRYPT.

[13]  Claude Carlet,et al.  On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities , 2006, AAECC.

[14]  Sugata Gangopadhyay,et al.  Bent and generalized bent Boolean functions , 2013, Des. Codes Cryptogr..

[15]  J. Dillon.  Elementary Hadamard Difference Sets , 1974 .

[16]  Kai-Uwe Schmidt,et al.  Quaternary Constant-Amplitude Codes for Multicode CDMA , 2006, IEEE Transactions on Information Theory.

[17]  Wilfried Meidl.  A secondary construction of bent functions, octal gbent functions and their duals , 2018, Math. Comput. Simul..

[18]  Keqin Feng,et al.  Complete Characterization of Generalized Bent and 2k-Bent Boolean Functions , 2017, IEEE Transactions on Information Theory.

[19]  Pantelimon Stanica,et al.  Octal Bent Generalized Boolean Functions , 2011, IACR Cryptol. ePrint Arch..

[20]  Sihem Mesnager,et al.  Decomposing Generalized Bent and Hyperbent Functions , 2017, IEEE Transactions on Information Theory.

[21]  CarletC..  Z2k-linear codes , 1998 .

[22]  A. Pott,et al.  Semifields, relative difference sets, and bent functions , 2014, 1401.3294.

[23]  Claude Carlet.  Z2k-Linear Codes , 1998, IEEE Trans. Inf. Theory.

[24]  Sugata Gangopadhyay,et al.  A Note on Generalized Bent Criteria for Boolean Functions , 2013, IEEE Transactions on Information Theory.

[25]  Yin Tan,et al.  Strongly regular graphs associated with ternary bent functions , 2010, J. Comb. Theory, Ser. A.

[26]  Sihem Mesnager,et al.  Several New Infinite Families of Bent Functions and Their Duals , 2014, IEEE Transactions on Information Theory.

[27]  Enes Pasalic,et al.  Generalized bent functions - sufficient conditions and related constructions , 2017, Adv. Math. Commun..

[28]  Yue Zhou.  (2n,2n,2n,1)‐Relative Difference Sets and Their Representations , 2013 .

[29]  Yue Zhou,et al.  Planar functions over fields of characteristic two , 2013, ArXiv.

[30]  Haiying Liu,et al.  Nonexistence of generalized bent functions from ℤ2n to ℤm , 2017, Des. Codes Cryptogr..

[31]  Pantelimon Stanica,et al.  Partial spread and vectorial generalized bent functions , 2017, Des. Codes Cryptogr..

[32]  Kaisa Nyberg,et al.  Perfect Nonlinear S-Boxes , 1991, EUROCRYPT.