# Background

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

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

The function $F$ is

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

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

A function $F$ is called planar or perfect nonlinear (PN) if $\delta _{F}=1$ . Obviously such functions exist only for $p$ 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, $\displaystyle D_aF(x)=F(x+a)-F(x)$ , 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 $\mathbb {S} _{F}=(\mathbb {F} _{p^{n}},+,\star )$ defined by $x\star y=F(x+y)-F(x)-F(y)$ .

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

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