Commutative Presemifields and Semifields - Revision history

Mpal: /* Known cases of APN functions in odd characteristic */

2023-03-06T12:20:55Z

Known cases of APN functions in odd characteristic

← Older revision		Revision as of 12:20, 6 March 2023
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	*<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}{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{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>;

Mpal at 11:15, 6 March 2023

2023-03-06T11:15:56Z

← Older revision		Revision as of 11:15, 6 March 2023
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	* <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>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>.		* <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>;

Ivi062: /* Known cases od planar functions and commutative semifields */

2019-11-05T15:40:52Z

Known cases od planar functions and commutative semifields

← Older revision		Revision as of 15:40, 5 November 2019
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	** <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 od planar functions and commutative semifields=		=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		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>		<math>x^{\frac{3^t+1}{2}}</math>

Ivi062: /* Known cases od planar functions and commutative semifields */

2019-09-23T09:13:59Z

Known cases od planar functions and commutative semifields

← Older revision		Revision as of 09:13, 23 September 2019
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	* <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>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>.		* <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>.

Ivi062: /* Properties */

2019-09-23T08:50:47Z

Properties

← Older revision		Revision as of 08:50, 23 September 2019
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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<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>;		* <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>;

Ivi062: /* Known cases od planar functions and commutative semifields */

2019-09-18T07:22:22Z

Known cases od planar functions and commutative semifields

← Older revision		Revision as of 07:22, 18 September 2019
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	* <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^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>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}~~\sum_{i=0}^k~~(-1)~~^ix^{p^{2i}(p^2+1)}+\sum_{j=0}^{k~~-1}(-1)~~^{k+j}x~~^{p^~~{2j+1~~}~~(p^2+1)}+~~</math>		* <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>.
	<math>-(\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)~~})^{p^m~~}</math> over <math>\mathbb{F}_{p^{2m}}</math> for <math>a\in\mathbb{F}^star_{p^2}, m=2k+1</math>.

Ivi062 at 15:22, 17 September 2019

2019-09-17T15:22:54Z

@@ Line 33: / Line 33: @@
 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.
-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
@@ Line 58: / Line 58: @@
 ==Properties==
 Hence two quadratic planar functions <math>F,F'</math> are isotopic equivalent 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 <math>\mathbb{S}_F,\mathbb{S}_{F'}</math> 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;
 * 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>;
@@ Line 65: / Line 65: @@
 ** <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.

Ivi062: Undo revision 359 by Ivi062 (talk)

2019-09-17T12:56:00Z

Undo revision 359 by Ivi062 (talk)

← Older revision		Revision as of 12:56, 17 September 2019
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	==Equivalence Relations==		==Equivalence Relations==
	Two functions <math>F</math> and <math>F'</math> from <math>\mathbb{F}_{p^n}</math> to itself are called:		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, A_1,A_2</math> are affine permutations;		*<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">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>.		*<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>.

Ivi062 at 12:55, 17 September 2019

2019-09-17T12:55:09Z

← Older revision		Revision as of 12:55, 17 September 2019
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	==Equivalence Relations==		==Equivalence Relations==
	Two functions <math>F</math> and <math>F'</math> from <math>\mathbb{F}_{p^n}</math> to itself are called:		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">affine equivalent</span> if <math>F'=A_1\circ F\circ A_2, 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">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>.		*<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>.

Ivi062 at 08:58, 5 September 2019

2019-09-05T08:58:24Z

← Older revision		Revision as of 08:58, 5 September 2019
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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.

	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==
	~~Every commutative presemifield can be transformed into a commutative semifield.~~

	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 isotopic equivalent 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 <math>\mathbb{S}_F,\mathbb{S}_{F'}</math> are strongly isotopic;
	* 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.