Commutative Presemifields and Semifields: Difference between revisions

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


<math>F(x)=\sum_{j=0}^{p^n-1}a_jx^j, a_j\in\mathbb{F}_{p^n}</math>.
The function <math>F</math> is
* <span class="definition">linear</span> if <math>F(x)=\sum_{j=0}^{n-1}a_jx^{p^j} </math>,
* <span class="definition">affine</span> if it is the sum of a linear function and a constant,
* <span class="definition">DO</span> (Dembowski-Ostrim) polynomial if <math>F(x)=\sum_{0\le i\le j<n}a_{ij}x^{p^i+p^j} </math>,
* <span class="definition">quadratic</span> if it is the sum of a DO polynomial and an affine function.
For <math>\delta</math> a positive integer, the function <math>F</math> is called <span class="definition">differentially <math>\delta</math>-uniform</span> if for any pairs <math>a,b\in\mathbb{F}_{p^n}</math>, with <math>a\ne0</math>, the equation <math>F(x+a)-F(x)=b</math> admits at most <math>\delta</math> solutions.
A function <math>F</math> is called planar or perfect nonlinear (PN) if <math>\delta_F=1</math>.
Obviously such functions exist only for <math>p</math> an odd prime.
In the even case the smallest possible case for <math>\delta</math> is two ([[differential uniformity|APN]] function).
For planar function we have that the all the nonzero derivatives, <math>D_aF(x)=F(x+a)-F(x)</math>, are permutations.
==Equivalence Relations==
Two functions <math>F</math> and <math>F'</math> from <math>\mathbb{F}_{p^n}</math> to itself are called:
*<span class="definition">affine equivalent</span> if <math>F'=A_1\circ F\circ A_2</math>, where <math>A_1,A_2</math> are affine permutations;
*<span class="definition">EA-equivalent</span> (extended-affine) if <math>F'=F''+A</math>, where <math>A</math> is affine and <math>F''</math> is afffine equivalent to <math>F</math>;
*<span class="definition">CCZ-equivalent</span> if there exists an affine permutation <math>\mathcal{L}</math> of <math>\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}</math> such that <math>\mathcal{L}(G_F)=G_{F'}</math>, where <math>G_F=\lbrace (x,F(x)) : x\in\mathbb{F}_{p^n}\rbrace</math>.
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 <span class="definition">isotopic equivalence</span> is more general than CCZ-equivalence, where two maps are isotopic equivalent if the corresponding presemifields are isotopic.
=On Presemifields and Semifields=
A <span class="definition">presemifield</span> is a ring with left and right distributivity and with no zero divisor.
A <span class="definition">presemifield</span> is a ring with left and right distributivity and with no zero divisor.
A presemifield with a multiplicative identity is called a <span class="definition">semifield</span>.
A presemifield with a multiplicative identity is called a <span class="definition">semifield</span>.
Any finite presemifield can be represented by <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math>,
Any finite presemifield can be represented by <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math>,
for <math>p</math> a prime, <math>n</math> a positive integer, <math>\mathbb{S}=(\mathbb{F}_{p^n},+)</math> additive group and <math>x\star y</math> multiplication linear in each variable.
for <math>p</math> a prime, <math>n</math> a positive integer, <math>\mathbb{S}=(\mathbb{F}_{p^n},+)</math> additive group and <math>x\star y</math> multiplication linear in each variable.
Every commutative presemifield can be transformed into a commutative semifield<ref name="CouHen">Coulter R. S., Henderson M. Commutative presemifields and semifields. Advances in Math. 217, pp. 282-304, 2008</ref>.


Two presemifields <math>\mathbb{S}_1=(\mathbb{F}_{p^n},+,\star)</math> and <math>\mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ)</math> are called <span class="definition">isotopic</span>  if there exist three linear permutations <math>T,M,N</math> of <math>\mathbb{F}_{p^n}</math> such that
Two presemifields <math>\mathbb{S}_1=(\mathbb{F}_{p^n},+,\star)</math> and <math>\mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ)</math> are called <span class="definition">isotopic</span>  if there exist three linear permutations <math>T,M,N</math> of <math>\mathbb{F}_{p^n}</math> such that
<math>T(x\star y)=M(x)\circ N(y)</math>,
<math>T(x\star y)=M(x)\circ N(y)</math>,
for any <math>x,y\in\mathbb{F}_{p^n}</math>. If <math>M=N</math> then they are called <span class="definition">strongly isotopic</span>.
for any <math>x,y\in\mathbb{F}_{p^n}</math>. If <math>M=N</math> then they are called <span class="definition">strongly isotopic</span>.
Each commutative presemifields of odd order defines a [[Planar Functions|planar]] DO polynomial and viceversa:
Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:
* given <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math> let <math>F_\mathbb{S}(x)=\frac{1}{2}(x\star x)</math>;
* given <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math> let <math>F_\mathbb{S}(x)=\frac{1}{2}(x\star x)</math>;
* given <math>F</math> let <math>\mathbb{S}_F=(\mathbb{F}_{p^n},+,\star)</math> defined by <math>x\star y=F(x+y)-F(x)-F(y)</math>.
* given <math>F</math> let <math>\mathbb{S}_F=(\mathbb{F}_{p^n},+,\star)</math> defined by <math>x\star y=F(x+y)-F(x)-F(y)</math>.


Hence two quadratic planar functions <math>F,F'</math> are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:
Given <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math> a finite semifield, the subsets
* <math>F,F'</math> are CCZ-equivalent if and only if <math>\mathbb{S}_F,\mathbb{S}_{F'}</math> are strongly isotopic;
 
<math>N_l(\mathbb{S})=\{\alpha\in\mathbb{S} : (\alpha\star x)\star y=\alpha\star(x\star y)</math> for all <math>x,y\in\mathbb{S}\}</math>
 
<math>N_m(\mathbb{S})=\{\alpha\in\mathbb{S} : (x\star\alpha)\star y=x\star(\alpha\star y)</math> for all <math>x,y\in\mathbb{S}\}</math>
 
<math>N_r(\mathbb{S})=\{\alpha\in\mathbb{S} : (x\star y)\star \alpha=x\star(y\star \alpha)</math> for all <math>x,y\in\mathbb{S}\}</math>
 
are called left, middle and right nucleus of <math>\mathbb{S}</math>.
 
The set <math>N(\mathbb{S})=N_l(\mathbb{S})\cap N_m(\mathbb{S})\cap N_r(\mathbb{S})</math> is called the nucleus.
All these sets are finite field and, when <math>\mathbb{S}</math> is commutative,  <math>N_l(\mathbb{S})=N_r(\mathbb{S})\subseteq N_m(\mathbb{S})</math>.
The order of the different nuclei are invariant under isotopism.
 
==Properties==
Hence two quadratic planar functions <math>F,F'</math> are <b>isotopic equivalent</b> if their corresponding presemifields are isotopic. Moreover, we have:
* <math>F,F'</math> are CCZ-equivalent if and only if the corresponding presemifileds are strongly isotopic<ref name="BudHel">Budaghyan L., Helleseth T. On Isotopism of Commutative Presemifields and CCZ-Equivalence of Functions. Special Issue on Cryptography of International Journal of Foundations of Computer Science, v. 22/6), pp- 1243-1258, 2011</ref>;
* for <math>n</math> odd, isotopic coincides with strongly isotopic;
* for <math>n</math> odd, isotopic coincides with strongly isotopic;
* if <math>F,F'</math> are isotopic equivalent, then there exists a linear map <math>L</math> such that <math>F'</math> is EA-equivalent to <math>F(x+L(x))-F(x)-F(L(x))</math>.
* if <math>F,F'</math> are isotopic equivalent, then there exists a linear map <math>L</math> such that <math>F'</math> is EA-equivalent to <math>F(x+L(x))-F(x)-F(L(x))</math>;
* any commutative presemifield of odd order can generate at most two CCZ-equivalence classes of planar DO polynomials;
* if <math>\mathbb{S}_1</math> and <math>\mathbb{S}_2</math> are isotopic commutative semifields of characteristic <math>p</math> with order of middle nuclei and nuclei <math>p^m</math> and <math>p^k</math> respectively, then either one of the following is satisfied:
** <math>m/k</math> is odd and the semifields are strongly isotopic,
** <math>m/k</math> is even and the semifields are strongly isotopic or the only isotopisms are of the form <math>(\alpha\star N,N,L)</math> with <math>\alpha\in N_m(\mathbb{S}_1)</math> non-square.
 
=Known cases of planar functions and commutative semifields=
Among the known example of planar functions, the only ones that are non-quadratic are the power functions
<math>x^{\frac{3^t+1}{2}}</math>
defined over <math>\mathbb{F}_{3^n}</math>, with <math>t</math> is odd and gcd(<math>t,n</math>)=1.
 
In the following the list of some known  infinite families of planar functions (and corresponding commutative semifields):
* <math>x^2</math> over <math>\mathbb{F}_{p^n}</math> (finite field <math>\mathbb{F}_{p^n}</math>);
* <math>x^{p^t+1}</math> over <math>\mathbb{F}_{p^n}</math> with <math>n/gcd(t,n)</math> odd (Albert's commutative twisted fields);
* <math>L(t^2(x))+\frac{1}{2}x^2</math> over <math>\mathbb{F}_{p^{2km}}</math> with <math>L(x)=\frac{1}{8}(x^{p^k}-x), t(x)=x^{p^{km}}-x</math> (Dickson semifields);
*
** <math>(ax)^{p^s+1}-(ax)^{p^k(p^s+1)}+x^{p^k+1}</math>
** <math>bx^{p^s+1}+(bx^{p^s+1})^{p^k}+cx^{p^k+1}</math>
over <math>\mathbb{F}_{p^{2k}}</math> where <math>a,b\in\mathbb{F}^\star_{2^{2k}}, b</math> not square, <math>c\in\mathbb{F}_{2^{2k}}\setminus\mathbb{F}_{2^k}, gcd(k+s,2k)=gcd(k+s,k)</math> and for the first one also <math>gcd(p^s+1,p^k+1)\ne gcd(p^s+1,(p^k+1)/2)</math>.
Without loss of generality it is possible to take <math>a=1</math> and fix a value for <math>c</math>;
* <math>x^{p^s+1}-a^{p^t-1}x^{p^t+p^{2t+s}}</math> over <math>\mathbb{F}_{p^{3t}}, a</math> primitive, <math>gcd(3,t)=1, t-s\equiv0</math> mod <math>3, 3t/gcd(s,3t)</math> odd;
* <math>x^{p^s+1}-a^{p^t-1}x^{p^{3t}+p^{t+s}}</math> over <math>\mathbb{F}_{p^{4t}}, a</math> primitive, <math>p^s\equiv p^t\equiv1</math> mod 4, <math>2t/gcd(s,2t)</math> odd;
* <math>a^{1-p}x^2+x^{2p^m}+a^{1-p}T(x)-T(x)^{p^m}</math>, with <math>T(x)=\sum_{i=0}^k(-1)^ix^{p^{2i}(p^2+1)}+a^{p-1}\sum_{j=0}^{k-1}(-1)^{k+j}x^{p^{2j+1}(p^2+1)}</math>, over <math>\mathbb{F}_{p^{2m}}</math> for <math>a\in\mathbb{F}^\star_{p^2}, m=2k+1</math>.
 
==Cases defined for <i>p</i>=3==
* <math>x^{10}\pm x^6-x^2 \mbox{ over } \mathbb{F}_{p^n} \mbox{ with } n</math> odd (Coulter-Matthews and Ding-Yuan semifields);
* <math>L(t^2(x))+D(t(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{2k}} \mbox{ with } k \mbox{ odd, } t(x)=x^{3^k}-x, \beta\in\mathbb{F}_{3^{2k}}\setminus\mathbb{F}_{3^k}, \alpha=t(\beta), L(x)=\alpha^{-5}x^3+x, D(x)=-\alpha^{-10}x^{10}</math> (Ganley semifields);
* <math>L(t^2(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{2k}} \mbox{ with } k \mbox{ odd, } t(x)=x^{3^k}-x, \beta\in\mathbb{F}_{3^{2k}}\setminus\mathbb{F}_{3^k}, \alpha=t(\beta), L(x)=-x^9-\alpha x^3+(1-\alpha^4)x</math> (Cohen-Ganley semifileds);
* <math>L(t^2(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{10}} \mbox{ with } t(x)=x^{243}-x, \beta\in\mathbb{F}_{3^{10}}\setminus\mathbb{F}_{3^5}, \alpha=t(\beta), L(x)=-(\alpha^{-53}x^{27}+\alpha^{-18}x^9-x)</math> (Penttila-Williams semifileds);
* <math>L(t^2(x))+D(t(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{8}} \mbox{ with } t(x)=x^9-x, L(x)=x^{243}+x^9, D(x)=x^{246}+x^{82}-x^{10}</math> (Coulter-Henderson-Kosick semifield);
* <math>x^2+x^{90} \mbox{ over } \mathbb{F}_{3^5}</math>.
 
 
=Known cases of APN functions in odd characteristic=
*<math>x^3</math> over <math>\mathbb{F}_{p^n}</math>, <math> p \neq 3 </math>;
*<math>x^{p^n-2}</math> over <math>\mathbb{F}_{p^n}</math> with <math> p^n \equiv 2 \pmod 3 </math>;
*<math>x^{\frac{p^n-3}{2}}</math> over <math>\mathbb{F}_{p^n}</math> with <math> p^n \equiv 3,7 \pmod {20},~ p^n>7,~ p^n \neq 27, n </math> odd;
*<math>x^{\frac{p^n+1}{4}+\frac{p^n-1}{2}}</math> over <math>\mathbb{F}_{p^n}</math> with <math> p^n \equiv 3 \pmod 8 </math>;
*<math>x^{\frac{p^n+1}{4}}</math> over <math>\mathbb{F}_{p^n}</math> with <math> p^n \equiv 7 \pmod 8,~ n>1 </math>;
*<math>x^{\frac{2p^n-1}{3}}</math> over <math>\mathbb{F}_{p^n}</math> with <math> p^n \equiv 2 \pmod 3 </math>;
*<math>x^{p^m+2}</math> over <math>\mathbb{F}_{p^n}</math> with <math> p^m \equiv 1 \pmod 3,~ n=2m </math>;
*<math>x^{3^n-3}</math> over <math>\mathbb{F}_{3^n}</math> with <math> n>1</math> odd;
*<math>x^{\frac{3^\frac{n+1}{2}-1}{2}}</math> over <math>\mathbb{F}_{3^n}</math> with <math> n \equiv 3 \pmod 4,~ n>3</math>;
*<math>x^{\frac{3^\frac{n+1}{2}-1}{2}+\frac{3^n-1}{2}}</math> over <math>\mathbb{F}_{3^n}</math> with <math> n \equiv 1 \pmod 4,~ n>1</math>;
*<math>x^{\frac{3^{n+1}-1}{8}}</math> over <math>\mathbb{F}_{3^n}</math> with <math> n \equiv 3 \pmod 4 </math>;
*<math>x^{\frac{3^{n+1}-1}{8}+\frac{3^{n}-1}{4}}</math> over <math>\mathbb{F}_{3^n}</math> with <math> n \equiv 1 \pmod 4 </math>;
*<math>x^{\frac{3^{n+1}-1}{3^L+1}}</math> over <math>\mathbb{F}_{3^n}</math>, where <math> L={\frac{n+1}{2^{\ell}}} </math> with <math> n \equiv -1 \pmod {2^\ell} </math>;
*<math>x^{\frac{5^\ell+1}{2}}</math> over <math>\mathbb{F}_{5^n}</math> with <math> \gcd(2n, \ell)=1 </math>;
*<math>x^{\frac{5^n-1}{4}+ \frac{5^{\frac{n+1}{2}}-1}{2}}</math> over <math>\mathbb{F}_{5^n}</math> with <math> n </math> odd;
*<math>x^{\frac{5^{n+1}-1}{2(5^L+1)}+ \frac{5^n-1}{4}}</math> over <math>\mathbb{F}_{5^n}</math>, where <math> L={\frac{n+1}{2^{\ell}}},~n \equiv -1 \pmod {2^\ell} </math> and <math> \ell \geq 2 </math>;

Latest revision as of 12:20, 6 March 2023

Background

For a prime and a positive integer let be the finite field with 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^n} elements. 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} 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 (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^n-1} , i.e.

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)=\sum_{j=0}^{p^n-1}a_jx^j, a_j\in\mathbb{F}_{p^n}} .

The function 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

  • linear if ,
  • affine if it is the sum of a linear function and a constant,
  • DO (Dembowski-Ostrim) polynomial 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(x)=\sum_{0\le i\le j<n}a_{ij}x^{p^i+p^j} } ,
  • quadratic if it is the sum of a DO polynomial and an affine function.

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

A function 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 called planar or perfect nonlinear (PN) 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 \delta_F=1} . Obviously such functions exist only 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} 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 (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 D_aF(x)=F(x+a)-F(x)} , are permutations.

Equivalence Relations

Two 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} 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 F'} from 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}} to itself are called:

  • affine equivalent 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'=A_1\circ F\circ A_2} , where are affine permutations;
  • EA-equivalent (extended-affine) 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''+A} , 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 A} is affine 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 F''} is afffine 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} ;
  • CCZ-equivalent if there exists an affine permutation 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}\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, 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. Every commutative presemifield can be transformed into a commutative semifield[1].

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 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 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 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}} .

The set is called the nucleus. All these sets are finite field and, when 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}} is commutative, 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})=N_r(\mathbb{S})\subseteq N_m(\mathbb{S})} . The order of the different nuclei are invariant under isotopism.

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 the corresponding presemifileds are strongly isotopic[2];
  • 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 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))} ;
  • any commutative presemifield of odd order can generate at most two CCZ-equivalence classes of planar DO polynomials;
  • 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}_1} 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} are isotopic commutative semifields of characteristic 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} with order of middle nuclei and nuclei 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 p^k} respectively, then either one of the following is satisfied:
    • 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/k} is odd and the semifields are strongly isotopic,
    • 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/k} is even and the semifields are strongly isotopic or the only isotopisms are of the form 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 (\alpha\star N,N,L)} with 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 \alpha\in N_m(\mathbb{S}_1)} non-square.

Known cases of planar functions and commutative semifields

Among the known example of planar functions, the only ones that are non-quadratic are the power functions defined over 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}_{3^n}} , with 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} is odd and gcd(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,n} )=1.

In the following the list of some known infinite families of planar functions (and corresponding commutative semifields):

  • 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^2} over 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}} (finite field 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}} );
  • over 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}} with 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/gcd(t,n)} odd (Albert's commutative twisted fields);
  • over with (Dickson semifields);
    • 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 bx^{p^s+1}+(bx^{p^s+1})^{p^k}+cx^{p^k+1}}

over 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^{2k}}} 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 a,b\in\mathbb{F}^\star_{2^{2k}}, b} not square, 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 c\in\mathbb{F}_{2^{2k}}\setminus\mathbb{F}_{2^k}, gcd(k+s,2k)=gcd(k+s,k)} and for the first one also 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 gcd(p^s+1,p^k+1)\ne gcd(p^s+1,(p^k+1)/2)} . Without loss of generality it is possible to take 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 a=1} and fix a value 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 x^{p^s+1}-a^{p^t-1}x^{p^t+p^{2t+s}}} over primitive, 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 gcd(3,t)=1, t-s\equiv0} mod 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 3, 3t/gcd(s,3t)} odd;
  • 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^{p^s+1}-a^{p^t-1}x^{p^{3t}+p^{t+s}}} over 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^{4t}}, a} primitive, 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^s\equiv p^t\equiv1} mod 4, 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 2t/gcd(s,2t)} odd;
  • 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 a^{1-p}x^2+x^{2p^m}+a^{1-p}T(x)-T(x)^{p^m}} , with 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)=\sum_{i=0}^k(-1)^ix^{p^{2i}(p^2+1)}+a^{p-1}\sum_{j=0}^{k-1}(-1)^{k+j}x^{p^{2j+1}(p^2+1)}} , over 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^{2m}}} 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 a\in\mathbb{F}^\star_{p^2}, m=2k+1} .

Cases defined for p=3

  • 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^{10}\pm x^6-x^2 \mbox{ over } \mathbb{F}_{p^n} \mbox{ with } n} odd (Coulter-Matthews and Ding-Yuan semifields);
  • 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(t^2(x))+D(t(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{2k}} \mbox{ with } k \mbox{ odd, } t(x)=x^{3^k}-x, \beta\in\mathbb{F}_{3^{2k}}\setminus\mathbb{F}_{3^k}, \alpha=t(\beta), L(x)=\alpha^{-5}x^3+x, D(x)=-\alpha^{-10}x^{10}} (Ganley semifields);
  • 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(t^2(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{2k}} \mbox{ with } k \mbox{ odd, } t(x)=x^{3^k}-x, \beta\in\mathbb{F}_{3^{2k}}\setminus\mathbb{F}_{3^k}, \alpha=t(\beta), L(x)=-x^9-\alpha x^3+(1-\alpha^4)x} (Cohen-Ganley semifileds);
  • 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(t^2(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{10}} \mbox{ with } t(x)=x^{243}-x, \beta\in\mathbb{F}_{3^{10}}\setminus\mathbb{F}_{3^5}, \alpha=t(\beta), L(x)=-(\alpha^{-53}x^{27}+\alpha^{-18}x^9-x)} (Penttila-Williams semifileds);
  • 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(t^2(x))+D(t(x))+\frac{1}{2}x^2, \mbox{ over } \mathbb{F}_{3^{8}} \mbox{ with } t(x)=x^9-x, L(x)=x^{243}+x^9, D(x)=x^{246}+x^{82}-x^{10}} (Coulter-Henderson-Kosick semifield);
  • 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^2+x^{90} \mbox{ over } \mathbb{F}_{3^5}} .


Known cases of APN functions in odd characteristic

  • 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^3} over 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}} , 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 \neq 3 } ;
  • 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^{p^n-2}} over 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}} with 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^n \equiv 2 \pmod 3 } ;
  • 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^{\frac{p^n-3}{2}}} over 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}} with 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^n \equiv 3,7 \pmod {20},~ p^n>7,~ p^n \neq 27, n } odd;
  • 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^{\frac{p^n+1}{4}+\frac{p^n-1}{2}}} over 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}} with 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^n \equiv 3 \pmod 8 } ;
  • 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^{\frac{p^n+1}{4}}} over with 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^n \equiv 7 \pmod 8,~ n>1 } ;
  • 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^{\frac{2p^n-1}{3}}} over 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}} with 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^n \equiv 2 \pmod 3 } ;
  • 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^{p^m+2}} over 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}} with 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^m \equiv 1 \pmod 3,~ n=2m } ;
  • 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^{3^n-3}} over 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}_{3^n}} with 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>1} odd;
  • 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^{\frac{3^\frac{n+1}{2}-1}{2}}} over 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}_{3^n}} with 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 \equiv 3 \pmod 4,~ n>3} ;
  • 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^{\frac{3^\frac{n+1}{2}-1}{2}+\frac{3^n-1}{2}}} over 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}_{3^n}} with 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 \equiv 1 \pmod 4,~ n>1} ;
  • over 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}_{3^n}} with 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 \equiv 3 \pmod 4 } ;
  • 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^{\frac{3^{n+1}-1}{8}+\frac{3^{n}-1}{4}}} over 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}_{3^n}} with 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 \equiv 1 \pmod 4 } ;
  • over , where with 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 \equiv -1 \pmod {2^\ell} } ;
  • 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^{\frac{5^\ell+1}{2}}} over with ;
  • over with odd;
  • over 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}_{5^n}} , 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 L={\frac{n+1}{2^{\ell}}},~n \equiv -1 \pmod {2^\ell} } 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 \ell \geq 2 } ;
  1. ↑ Coulter R. S., Henderson M. Commutative presemifields and semifields. Advances in Math. 217, pp. 282-304, 2008
  2. ↑ Budaghyan L., Helleseth T. On Isotopism of Commutative Presemifields and CCZ-Equivalence of Functions. Special Issue on Cryptography of International Journal of Foundations of Computer Science, v. 22/6), pp- 1243-1258, 2011
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