Commutative Presemifields and Semifields

Background

For a prime Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and a positive integer $n$ let $\mathbb {F} _{p^{n}}$ be the finite field with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $F(x)=\sum _{0\leq i\leq j<n}a_{ij}x^{p^{i}+p^{j}}$,
• 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $\delta$ is two (APN function).

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

Equivalence Relations

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

• affine equivalent if $F'=A_{1}\circ F\circ A_{2}$, where $A_{1},A_{2}$ are affine permutations;
• EA-equivalent (extended-affine) if $F'=F''+A$, where $A$ is affine and $F''$ is afffine equivalent to $F$;
• CCZ-equivalent if there exists an affine permutation ${\mathcal {L}}$ of $\mathbb {F} _{p^{n}}\times \mathbb {F} _{p^{n}}$ such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}(G_F)=G_{F'}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+,\star)} , for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} a prime, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} a positive integer, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+)} additive group and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\star y} multiplication linear in each variable.

Two presemifields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}_1=(\mathbb{F}_{p^n},+,\star)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ)} are called isotopic if there exist three linear permutations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T,M,N} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{p^n}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(x\star y)=M(x)\circ N(y)} , for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y\in\mathbb{F}_{p^n}} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=N} then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

• given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+,\star)} let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\mathbb{S}(x)=\frac{1}{2}(x\star x)} ;
• given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}_F=(\mathbb{F}_{p^n},+,\star)} defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\star y=F(x+y)-F(x)-F(y)} .

Given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+,\star)} a finite semifield, the subsets

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_l(\mathbb{S})=\{\alpha\in\mathbb{S} : (\alpha\star x)\star y=\alpha\star(x\star y)} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y\in\mathbb{S}\}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_m(\mathbb{S})=\{\alpha\in\mathbb{S} : (x\star\alpha)\star y=x\star(\alpha\star y)} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y\in\mathbb{S}\}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_r(\mathbb{S})=\{\alpha\in\mathbb{S} : (x\star y)\star \alpha=x\star(y\star \alpha)} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y\in\mathbb{S}\}}

are called left, middle and right nucleus of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}} .

Properties

Hence two quadratic planar functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F,F'} are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F,F'} are CCZ-equivalent if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}_F,\mathbb{S}_{F'}} are strongly isotopic;
• for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} odd, isotopic coincides with strongly isotopic;
• if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F,F'} are isotopic equivalent, then there exists a linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F'} is EA-equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x+L(x))-F(x)-F(L(x))} .