# Background

For a prime ${\displaystyle p}$ and a positive integer ${\displaystyle n}$ let ${\displaystyle \mathbb {F} _{p^{n}}}$ be the finite field with ${\displaystyle p^{n}}$ elements. Let ${\displaystyle F}$ be a map from the finite field to itself. Such function admits a unique representation as a polynomial of degree at most ${\displaystyle p^{n}-1}$, i.e.

${\displaystyle F(x)=\sum _{j=0}^{p^{n}-1}a_{j}x^{j},a_{j}\in \mathbb {F} _{p^{n}}}$.

The function ${\displaystyle F}$ is

• linear if ${\displaystyle F(x)=\sum _{j=0}^{n-1}a_{j}x^{p^{j}}}$,
• affine if it is the sum of a linear function and a constant,
• DO (Dembowski-Ostrim) polynomial if ${\displaystyle F(x)=\sum _{0\leq i\leq j,
• quadratic if it is the sum of a DO polynomial and an affine function.

For ${\displaystyle \delta }$ a positive integer, the function ${\displaystyle F}$ is called differentially ${\displaystyle \delta }$-uniform if for any pairs ${\displaystyle a,b\in \mathbb {F} _{p^{n}}}$, with ${\displaystyle a\neq 0}$, the equation ${\displaystyle F(x+a)-F(x)=b}$ admits at most ${\displaystyle \delta }$ solutions.

A function ${\displaystyle F}$ is called planar or perfect nonlinear (PN) if ${\displaystyle \delta _{F}=1}$. Obviously such functions exist only for ${\displaystyle p}$ an odd prime. In the even case the smallest possible case for ${\displaystyle \delta }$ is two (APN function).

For planar function we have that the all the nonzero derivatives, ${\displaystyle D_{a}F(x)=F(x+a)-F(x)}$, are permutations.

## Equivalence Relations

Two functions ${\displaystyle F}$ and ${\displaystyle F'}$ from ${\displaystyle \mathbb {F} _{p^{n}}}$ to itself are called:

• affine equivalent if ${\displaystyle F'=A_{1}\circ F\circ A_{2}}$, where ${\displaystyle A_{1},A_{2}}$ are affine permutations;
• EA-equivalent (extended-affine) if ${\displaystyle F'=F''+A}$, where ${\displaystyle A}$ is affine and ${\displaystyle F''}$ is afffine equivalent to ${\displaystyle F}$;
• CCZ-equivalent if there exists an affine permutation ${\displaystyle {\mathcal {L}}}$ of ${\displaystyle \mathbb {F} _{p^{n}}\times \mathbb {F} _{p^{n}}}$ such that ${\displaystyle {\mathcal {L}}(G_{F})=G_{F'}}$, where ${\displaystyle G_{F}=\lbrace (x,F(x)):x\in \mathbb {F} _{p^{n}}\rbrace }$.

CCZ-equivalence is the most general known equivalence relation for functions which preserves differential uniformity. Affine and EA-equivalence are its particular cases. For the case of quadratic planar functions the isotopic equivalence is more general than CCZ-equivalence, where two maps are isotopic equivalent if the corresponding presemifields are isotopic.

# On Presemifields and Semifields

A presemifield is a ring with left and right distributivity and with no zero divisor. A presemifield with a multiplicative identity is called a semifield. Any finite presemifield can be represented by ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )}$, for ${\displaystyle p}$ a prime, ${\displaystyle n}$ a positive integer, ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+)}$ additive group and ${\displaystyle x\star y}$ multiplication linear in each variable.

Two presemifields ${\displaystyle \mathbb {S} _{1}=(\mathbb {F} _{p^{n}},+,\star )}$ and ${\displaystyle \mathbb {S} _{2}=(\mathbb {F} _{p^{n}},+,\circ )}$ are called isotopic if there exist three linear permutations ${\displaystyle T,M,N}$ of ${\displaystyle \mathbb {F} _{p^{n}}}$ such that ${\displaystyle T(x\star y)=M(x)\circ N(y)}$, for any ${\displaystyle x,y\in \mathbb {F} _{p^{n}}}$. If ${\displaystyle M=N}$ then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

• given ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )}$ let ${\displaystyle F_{\mathbb {S} }(x)={\frac {1}{2}}(x\star x)}$;
• given ${\displaystyle F}$ let ${\displaystyle \mathbb {S} _{F}=(\mathbb {F} _{p^{n}},+,\star )}$ defined by ${\displaystyle x\star y=F(x+y)-F(x)-F(y)}$.

Hence two quadratic planar functions ${\displaystyle F,F'}$ are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:

• ${\displaystyle F,F'}$ are CCZ-equivalent if and only if ${\displaystyle \mathbb {S} _{F},\mathbb {S} _{F'}}$ are strongly isotopic;
• for ${\displaystyle n}$ odd, isotopic coincides with strongly isotopic;
• if ${\displaystyle F,F'}$ are isotopic equivalent, then there exists a linear map ${\displaystyle L}$ such that ${\displaystyle F'}$ is EA-equivalent to ${\displaystyle F(x+L(x))-F(x)-F(L(x))}$.