Epi062: Created page with "An (n, n)-function F is called crooked if, for every nonzero a, the set
$\{D_aF(x) \colon x\in\mathbb{F}_2^n\}$
is an affine hyperplane (i.e. a linear..."

2022-12-23T09:01:35Z

Created page with "An (n, n)-function F is called crooked if, for every nonzero a, the set <div><math>\{D_aF(x) \colon x\in\mathbb{F}_2^n\}</math></div> is an affine hyperplane (i.e. a linear..."

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An (n, n)-function F is called crooked if, for every nonzero a,
the set
<div><math>\{D_aF(x) \colon x\in\mathbb{F}_2^n\}</math></div>
is an affine hyperplane (i.e. a linear hyperplane or its
complement).

Conversely, crooked functions are strongly plateaued and APN.

The component functions of a crooked function are all partially-bent.

CCZ equivalence does not preserve crookedness

= Characterization of Crooked Functions =
For n odd, F is crooked if and only if F is almost bent (AB).

F is crooked if and only if, for every nonzero a, there exists
a unique nonzero v such that
<div><math>W_{D_aF}(0,v)\neq 0</math></div>
This characterization can be expressed by means of the Walsh transform of F since
<div><math>W_{D_aF}(0,v)=\Delta_{v\cdot F}(a)=2^{-n}\sum_{u\in\mathbb{F}_2^n}(-1)^{u\cdot a}W_{F}^2(u,v)</math></div>

= Known Crooked Functions =
All quadratic APN functions are crooked

If a monomial is crooked, then it is quadratic

If a binomial is crooked, then it is quadratic

An open problem is to find a crooked function that is not quadratic

Crooked Functions - Revision history

Epi062: Created page with "An (n, n)-function F is called crooked if, for every nonzero a, the set \{D_aF(x) \colon x\in\mathbb{F}_2^n\} is an affine hyperplane (i.e. a linear..."

Epi062: Created page with "An (n, n)-function F is called crooked if, for every nonzero a, the set
$\{D_aF(x) \colon x\in\mathbb{F}_2^n\}$
is an affine hyperplane (i.e. a linear..."