Niho bent functions

Background and Definitions

Definition. A power boolean function 𝑥^𝑑 defined on 𝔽_2^𝑛 is called a Niho power function, if its restriction to 𝔽_2^m is linear, where n=2m.

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∈ 𝔽_2^m, 𝐺 :𝔽_2^m → 𝔽_2^m satisfying the following conditions:

𝐹 : 𝑧 → G(𝑧)+μ𝑧 is a permutation over 𝔽_2^m,

z →𝐹(𝑧)+β𝑧 is 2-to-1 on 𝔽_2^m for any nonzero β∈𝔽_2^m .

Examples

The known Niho type bent functions in univariant form

1. Quadratic function Tr_m (𝑎𝑡^2𝑚+1), where 𝑎∈𝔽_2^m.

2. Binomials of the form 𝑓(𝑡)= Tr_n(α₁𝑡^𝑑₁+α₂𝑡^𝑑₁),

where 2𝑑1\equiv 2m1+1 mod(2n-1) and α1, α2∈𝔽2m are such that (α1+α12m)2=α22m+1.

These functions have algebraic degree $m$ and do not belong to the completed Maiorana-McFarland class.

3. 𝑓(𝑡)= Tr_n（a²t^2m+1+(a+a^2m)[math]\displaystyle{ \sum_{i=1}^{2^{r-1}-1}t^{d_i} }[/math]), where 2^r d_i=(2^m-1)i+2^r and a∈𝔽_2^m is such that a+a^2m≠ 0. This function has algebraic degree r+1 and belongs to the completed Maiorana-McFarland class.