Commutative Presemifields and Semifields

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $F(x)=\sum _{0\leq i\leq j<n}a_{ij}x^{p^{i}+p^{j}}$,
• 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 $a,b\in \mathbb {F} _{p^{n}}$, with $a\neq 0$, the equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x+a)-F(x)=b} admits at most $\delta$ solutions.

A function $F$ is called planar or perfect nonlinear (PN) if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {F} _{p^{n}}\times \mathbb {F} _{p^{n}}} such that ${\mathcal {L}}(G_{F})=G_{F'}$, where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $\mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )$, for $p$ a prime, $n$ a positive integer, $\mathbb {S} =(\mathbb {F} _{p^{n}},+)$ additive group and $x\star y$ multiplication linear in each variable.

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

• given $\mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )$ let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F_{\mathbb {S} }(x)={\frac {1}{2}}(x\star x)} ;
• given $F$ 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 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 $\mathbb {S} _{F},\mathbb {S} _{F'}$ 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 $L$ 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))} .