Commutative Presemifields and Semifields

Background

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

$F(x)=\sum _{j=0}^{p^{n}-1}a_{j}x^{j},a_{j}\in \mathbb {F} _{p^{n}}$.

The function $F$ is

• linear if $F(x)=\sum _{j=0}^{n-1}a_{j}x^{p^{j}}$,
• affine if it is the sum of a linear function and a constant,
• DO (Dembowski-Ostrim) polynomial if $F(x)=\sum _{0\leq i\leq j<n}a_{ij}x^{p^{i}+p^{j}}$,
• quadratic if it is the sum of a DO polynomial and an affine function.

For $\delta$ a positive integer, the function $F$ is called differentially $\delta$-uniform if for any pairs $a,b\in \mathbb {F} _{p^{n}}$, with $a\neq 0$, the equation $F(x+a)-F(x)=b$ 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 $p$ an odd prime. In the even case the smallest possible case for $\delta$ is two (APN function).

For planar function we have that the all the nonzero derivatives, $D_{a}F(x)=F(x+a)-F(x)$, are permutations.

Equivalence Relations

Two functions $F$ and $F'$ from $\mathbb {F} _{p^{n}}$ to itself are called:

• affine equivalent if $F'=A_{1}\circ F\circ A_{2}$, where $A_{1},A_{2}$ are affine permutations;
• EA-equivalent (extended-affine) if $F'=F''+A$, where $A$ is affine and $F''$ is afffine equivalent to $F$;
• CCZ-equivalent if there exists an affine permutation ${\mathcal {L}}$ of $\mathbb {F} _{p^{n}}\times \mathbb {F} _{p^{n}}$ such that ${\mathcal {L}}(G_{F})=G_{F'}$, where $G_{F}=\lbrace (x,F(x)):x\in \mathbb {F} _{p^{n}}\rbrace$.

CCZ-equivalence is the most general known equivalence relation for functions which preserves differential uniformity. Affine and EA-equivalence are its particular cases. For the case of quadratic planar functions the isotopic equivalence is more general than CCZ-equivalence, where two maps are isotopic equivalent if the corresponding presemifields are isotopic.

On Presemifields and Semifields

A presemifield is a ring with left and right distributivity and with no zero divisor. A presemifield with a multiplicative identity is called a semifield. Any finite presemifield can be represented by $\mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )$, for $p$ a prime, $n$ a positive integer, $\mathbb {S} =(\mathbb {F} _{p^{n}},+)$ additive group and $x\star y$ multiplication linear in each variable. Every commutative presemifield can be transformed into a commutative semifield.

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 $F_{\mathbb {S} }(x)={\frac {1}{2}}(x\star x)$;
• given $F$ let $\mathbb {S} _{F}=(\mathbb {F} _{p^{n}},+,\star )$ defined by $x\star y=F(x+y)-F(x)-F(y)$.

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

$N_{l}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(\alpha \star x)\star y=\alpha \star (x\star y)$ for all $x,y\in \mathbb {S} \}$

$N_{m}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(x\star \alpha )\star y=x\star (\alpha \star y)$ for all $x,y\in \mathbb {S} \}$

Failed to parse (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 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(\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 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, $N_{l}(\mathbb {S} )=N_{r}(\mathbb {S} )\subseteq N_{m}(\mathbb {S} )$. 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;
• for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} odd, isotopic coincides with strongly isotopic;
• if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F,F'} are isotopic equivalent, then there exists a linear map 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 $F'$ is EA-equivalent to $F(x+L(x))-F(x)-F(L(x))$;
• any commutative presemifield of odd order can generate at most two CCZ-equivalence classes of planar DO polynomials;
• if $\mathbb {S} _{1}$ 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 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} and $p^{k}$ respectively, then either one of the following is satisfied:
  • $m/k$ is odd and the semifields are strongly isotopic,
  • $m/k$ 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 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^t+1}{2}}} 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($t,n$)=1.

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

• $x^{2}$ over $\mathbb {F} _{p^{n}}$ (finite field $\mathbb {F} _{p^{n}}$);
• $x^{p^{t}+1}$ 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^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);
• 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} 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^{2km}}} with $L(x)={\frac {1}{8}}(x^{p^{k}}-x),t(x)=x^{p^{km}}-x$ (Dickson semifields);
• • $(ax)^{p^{s}+1}-(ax)^{p^{k}(p^{s}+1)}+x^{p^{k}+1}$
  • $bx^{p^{s}+1}+(bx^{p^{s}+1})^{p^{k}}+cx^{p^{k}+1}$

over $\mathbb {F} _{p^{2k}}$ where $a,b\in \mathbb {F} _{2^{2k}}^{\star },b$ not square, $c\in \mathbb {F} _{2^{2k}}\setminus \mathbb {F} _{2^{k}},gcd(k+s,2k)=gcd(k+s,k)$ and for the first one also $gcd(p^{s}+1,p^{k}+1)\neq gcd(p^{s}+1,(p^{k}+1)/2)$. Without loss of generality it is possible to take $a=1$ 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 $\mathbb {F} _{p^{n}}$ with $p^{n}\equiv 3{\pmod {8}}$;
• $x^{\frac {p^{n}+1}{4}}$ over $\mathbb {F} _{p^{n}}$ with $p^{n}\equiv 7{\pmod {8}},~n>1$;
• $x^{\frac {2p^{n}-1}{3}}$ over $\mathbb {F} _{p^{n}}$ with $p^{n}\equiv 2{\pmod {3}}$;
• $x^{p^{m}+2}$ over $\mathbb {F} _{p^{n}}$ with $p^{m}\equiv 1{\pmod {3}},~n=2m$;
• $x^{3^{n}-3}$ over $\mathbb {F} _{3^{n}}$ 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 $n\equiv 1{\pmod {4}},~n>1$;
• 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 } ;
• $x^{\frac {3^{n+1}-1}{3^{L}+1}}$ 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 $L={\frac {n+1}{2^{\ell }}}$ with $n\equiv -1{\pmod {2^{\ell }}}$;
• $x^{\frac {5^{\ell }+1}{2}}$ over $\mathbb {F} _{5^{n}}$ with $\gcd(2n,\ell )=1$;
• $x^{{\frac {5^{n}-1}{4}}+{\frac {5^{\frac {n+1}{2}}-1}{2}}}$ 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 } ;