# Bent Boolean Functions

An π-variable Boolean function π (for even π) is called *bent* if its distance to the set of all π-variable affine functions (the nonlinearity of π) equals 2^{π-1}-2^{π/2-1}.

Equivalently, π is bent if

- π
_{π}(π’) takes only the values Β±2^{π/2}, - π
_{π}(π’)β‘2^{π/2}(mod 2^{π/2+1}), - its distance to any affine function equals 2
^{π-1}Β±2^{π/2-1}, - for any nonzero element π the Boolean function π·
_{π}π(π₯)=π(π₯+π)βπ(π₯) is balanced, - for any π₯βπ½
_{2}^{π}, , - the 2
^{π}Γ2^{π}matrix π»=[(-1)^{π(π₯+π¦)}]_{π₯,π¦βπ½2π}is a Hadamard matrix (i.e. π»Γπ»^{π‘}=2^{π}πΌ, where πΌ is the identity matrix), - the support of π is a difference set of the elementary Abelian 2-group π½
_{2}^{π}.

Bent functions are also called *perfect nonlinear functions*.

The dual of a bent π function is also a bent function, where the dual is defined as

and its own dual is π itself.

# Bent functions and algebraic degree

- For π even and at least 4 the algebraic degree of any bent function is at most π/2.
- The algebraic degree of a bent function and of its dual satisfy the following relation:

- Obviously, no affine function can be bent.
- When π is quadratic, then it is affine equivalent to the function

- The characterisation of cubic bent functions has been done for small dimensions (πβ€8).

# Constructions

## The Maiorana-McFarland Construction

For π=2π, π½_{2}^{π}={(π₯,π¦) : π₯,π¦βπ½_{2}^{π}}, π is of the form

where π is a permutation of π½_{2}^{π} and π is any π-variable Boolean function.
Any such function is bent (the bijectivity of π is a necessary and sufficient condition).
The dual function is .

Such construction contains, up to affine equivalence, all quadratic bent functions and all bent functions in at most 6 variables.