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 are bent functions with Niho exponents.

Niho bent functions in bivariant form: $g(x,y)=\left\{{\begin{aligned}Tr_{m}{\Big (}xG{\Big (}{\frac {y}{x}}{\Big )}{\Big )},&{\text{ if }}x\neq 0;\\Tr_{m}(\mu y),&{\text{ if }}x=0,\end{aligned}}\right.$

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, (1)

z →𝐹(𝑧)+β𝑧 is 2-to-1 on 𝔽_2^m for any nonzero β∈𝔽_2^m . (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+t^2m and y=αt +(αt)^2m, where α is a primitive element of 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

2^𝑑₁≡ 2^m₁+1 mod(2ⁿ-1) and α₁, α₂∈𝔽^*_2ⁿ are such that (α₁+α₁^2m)²=α₂^2m+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(𝑡)= Tr_n（a²t^2m+1+(a+a^2m)$\sum _{i=1}^{2^{r-1}-1}t^{d_{i}}$),

where 2^r d_i=(2^m-1)i+2^r and a∈𝔽_2ⁿ is such that a+a^2m≠ 0. This function has algebraic degree r+1 and belongs to the completed Maiorana-McFarland class.