Niho bent functions

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Background and Definitions

Definition. A power boolean function π‘₯𝑑 defined on 𝔽2𝑛 is called a Niho power function, if its restriction to 𝔽2m is linear, where n=2m.

Niho bent functions are bent functions with Niho exponents.

Niho bent functions in bivariant form: [math]\displaystyle{ g(x,y)= \left\{ \begin{aligned} Tr_m\Big(xG\Big(\frac{y}{x}\Big)\Big), & \text{ if } x\neq 0; \\ Tr_m(\mu y), & \text{ if } x=0, \end{aligned} \right. }[/math]

where μ∈ 𝔽2m, 𝐺 :𝔽2m β†’ 𝔽2m satisfying the following conditions:

𝐹 : 𝑧 β†’ G(𝑧)+μ𝑧 is a permutation over 𝔽2m, (1)

z →𝐹(𝑧)+β𝑧 is 2-to-1 on 𝔽2m for any nonzero Ξ²βˆˆπ”½2m . (2)


Here condition (2) implies condition (1) and it is necessary and suffcient for g being bent.

One can get a univariate representation of Niho bent function from bivariant one making the following substitutions: x=t+t2m and y=Ξ±t +(Ξ±t)2m, where Ξ± is a primitive element of 𝔽2m .

Examples

The known Niho type bent functions in univariant form

1. Quadratic function Trm (π‘Žπ‘‘2π‘š+1), where π‘Žβˆˆπ”½2m.

2. Binomials of the form 𝑓(𝑑)= Trn(Ξ±1𝑑𝑑1+Ξ±2𝑑𝑑1),where

2𝑑1≑ 2m1+1 mod(2n-1) and Ξ±1, Ξ±2βˆˆπ”½*2n are such that (Ξ±1+Ξ±12m)2=Ξ±22m+1. These functions have algebraic degree m and do not belong to the completed Maiorana-McFarland class.

3. Let 1 < r < m with gcd(r,m) = 1,

𝑓(𝑑)= Trn(a2t2m+1+(a+a2m)[math]\displaystyle{ \sum_{i=1}^{2^{r-1}-1}t^{d_i} }[/math]),

where 2r di=(2m-1)i+2r and aβˆˆπ”½2n is such that a+a2mβ‰  0. This function has algebraic degree r+1 and belongs to the completed Maiorana-McFarland class.