Commutative Presemifields and Semifields: Difference between revisions

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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.
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
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==Properties==
==Properties==
Hence two quadratic planar functions <math>F,F'</math> are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:
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 <math>\mathbb{S}_F,\mathbb{S}_{F'}</math> are strongly isotopic;
* <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>;
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** <math>m/k</math> is odd and the semifields are strongly isotopic,
** <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.
** <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 elements. Let 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 is

  • linear if ,
  • affine if it is the sum of a linear function and a constant,
  • DO (Dembowski-Ostrim) 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, , are permutations.

Equivalence Relations

Two functions and from to itself are called:

  • affine equivalent if , where are affine permutations;
  • EA-equivalent (extended-affine) if , where is affine and is afffine equivalent to ;
  • CCZ-equivalent if there exists an affine permutation of such that , where .

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 , for a prime, a positive integer, additive group and multiplication linear in each variable. Every commutative presemifield can be transformed into a commutative semifield[1].

Two presemifields and are called isotopic if there exist three linear permutations of such that , for any . If then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

  • given let ;
  • given let defined by .

Given 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} \}}

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, 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

Hence two quadratic planar functions are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:

  • are CCZ-equivalent 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 EA-equivalent to ;
  • any commutative presemifield of odd order can generate at most two CCZ-equivalence 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:
    • Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m/k} is odd and the semifields are strongly isotopic,
    • Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m/k} is even and the semifields are strongly isotopic or the only isotopisms are of the form with 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 , with is odd and gcd(Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t,n} )=1.

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

  • over (finite field );
  • over with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n/gcd(t,n)} odd (Albert's commutative twisted fields);
  • Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(t^{2}(x))+{\frac {1}{2}}x^{2}} over Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {F} _{p^{2km}}} 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 L(x)=\frac{1}{8}(x^{p^k}-x), t(x)=x^{p^{km}}-x} (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 (ax)^{p^s+1}-(ax)^{p^k(p^s+1)}+x^{p^k+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 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 c} ;

  • 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 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^{3t}}, 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 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 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 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} ;
  • 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}}} 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 } ;
  • 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}{3^L+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}} , 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}}} } 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 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}} 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 \gcd(2n, \ell)=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{5^n-1}{4}+ \frac{5^{\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}_{5^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 } 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{5^{n+1}-1}{2(5^L+1)}+ \frac{5^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}_{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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