Commutative Presemifields and Semifields
Contents
Background
For a prime and a positive integer let be the finite field with elements. Let be a map from the finite field to itself. Such function admits a unique representation as a polynomial of degree at most , i.e.
.
The function is
 linear if ,
 affine if it is the sum of a linear function and a constant,
 DO (DembowskiOstrim) polynomial if ,
 quadratic if it is the sum of a DO polynomial and an affine function.
For a positive integer, the function is called differentially uniform if for any pairs , with , the equation admits at most solutions.
A function is called planar or perfect nonlinear (PN) if . Obviously such functions exist only for an odd prime. In the even case the smallest possible case for is two (APN function).
For planar function we have that the all the nonzero derivatives, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_aF(x)=F(x+a)F(x)} , are permutations.
Equivalence Relations
Two functions and from to itself are called:
 affine equivalent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F'=A_1\circ F\circ A_2} , where are affine permutations;
 EAequivalent (extendedaffine) if , where is affine and is afffine equivalent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} ;
 CCZequivalent if there exists an affine permutation of such that , where .
CCZequivalence is the most general known equivalence relation for functions which preserves differential uniformity. Affine and EAequivalence are its particular cases. For the case of quadratic planar functions the isotopic equivalence is more general than CCZequivalence, 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 , for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p} a prime, a positive integer, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+)} additive group and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\star y} multiplication linear in each variable. Every commutative presemifield can be transformed into a commutative semifield^{[1]}.
Two presemifields Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}_1=(\mathbb{F}_{p^n},+,\star)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T,M,N} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{p^n}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T(x\star y)=M(x)\circ N(y)} , for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x,y\in\mathbb{F}_{p^n}} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+,\star)} let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_\mathbb{S}(x)=\frac{1}{2}(x\star x)} ;
 given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} let defined by .
Given a finite semifield, the subsets
for all
for all
for all
are called left, middle and right nucleus of .
The set is called the nucleus. All these sets are finite field and, when is commutative, . The order of the different nuclei are invariant under isotopism.
Properties
Hence two quadratic planar functions are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:
 are CCZequivalent if and only if the corresponding presemifileds are strongly isotopic^{[2]};
 for odd, isotopic coincides with strongly isotopic;
 if are isotopic equivalent, then there exists a linear map such that is EAequivalent to ;
 any commutative presemifield of odd order can generate at most two CCZequivalence classes of planar DO polynomials;
 if and are isotopic commutative semifields of characteristic with order of middle nuclei and nuclei and respectively, then either one of the following is satisfied:
 is odd and the semifields are strongly isotopic,
 is even and the semifields are strongly isotopic or the only isotopisms are of the form with nonsquare.
Known cases of planar functions and commutative semifields
Among the known example of planar functions, the only ones that are nonquadratic are the power functions defined over , with is odd and gcd()=1.
In the following the list of some known infinite families of planar functions (and corresponding commutative semifields):
 over (finite field );
 over with odd (Albert's commutative twisted fields);
 over with (Dickson semifields);

over where not square, and for the first one also . Without loss of generality it is possible to take and fix a value for ;
 over primitive, mod odd;
 over primitive, mod 4, odd;
 , with , over for .
Cases defined for p=3
 odd (CoulterMatthews and DingYuan semifields);
 (Ganley semifields);
 (CohenGanley semifileds);
 (PenttilaWilliams semifileds);
 (CoulterHendersonKosick semifield);
 .