Commutative Presemifields and Semifields

Background

For a prime Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p^{n}-1} , i.e.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 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 $\delta _{F}=1$. Obviously such functions exist only for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F'} from $\mathbb {F} _{p^{n}}$ to itself are called:

• affine equivalent if $F'=A_{1}\circ F\circ A_{2}$, where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{1},A_{2}} are affine permutations;
• EA-equivalent (extended-affine) if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F'=F''+A} , where $A$ is affine and $F''$ is afffine equivalent to $F$;
• CCZ-equivalent if there exists an affine permutation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\star y=F(x+y)-F(x)-F(y)} .

Given $\mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )$ a finite semifield, the subsets

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N_{l}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(\alpha \star x)\star y=\alpha \star (x\star y)} for all Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x,y\in \mathbb {S} \}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N_{m}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(x\star \alpha )\star y=x\star (\alpha \star y)} for all $x,y\in \mathbb {S} \}$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N_{r}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(x\star y)\star \alpha =x\star (y\star \alpha )} for all $x,y\in \mathbb {S} \}$

are called left, middle and right nucleus of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {S} } .

The set Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N(\mathbb {S} )=N_{l}(\mathbb {S} )\cap N_{m}(\mathbb {S} )\cap N_{r}(\mathbb {S} )} is called the nucleus. All these sets are finite field and, when $\mathbb {S}$ is commutative, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N_{l}(\mathbb {S} )=N_{r}(\mathbb {S} )\subseteq N_{m}(\mathbb {S} )} . The order of the different nuclei are invariant under isotopism.

Properties

Every commutative presemifield can be transformed into a commutative semifield.

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {S} _{F},\mathbb {S} _{F'}} are strongly isotopic;
• for $n$ 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))} .