# Difference between revisions of "Commutative Presemifields and Semifields"

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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== | ||

− | |||

− | |||

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. |

## Revision as of 10:58, 5 September 2019

# Background

For a prime and a positive integer let be the finite field with elements. Let be a map from the finite field to itself. Such function admits a unique representation as a polynomial of degree at most , i.e.

.

The function is

- linear if ,
- affine if it is the sum of a linear function and a constant,
- DO (Dembowski-Ostrim) polynomial if ,
- quadratic if it is the sum of a DO polynomial and an affine function.

For **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta}**
a positive integer, the function is called differentially **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta}**
-uniform if for any pairs , with **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a\ne0}**
, the equation admits at most **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta}**
solutions.

A function **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F}**
is called planar or perfect nonlinear (PN) if .
Obviously such functions exist only for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p}**
an odd prime.
In the even case the smallest possible case for

For planar function we have that the all the nonzero derivatives, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_aF(x)=F(x+a)-F(x)}**
, are permutations.

## Equivalence Relations

Two functions **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F}**
and from **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{p^n}}**
to itself are called:

- affine equivalent if
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F'=A_1\circ F\circ A_2}**, where**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_1,A_2}**are affine permutations; - EA-equivalent (extended-affine) if , where
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A}**is affine and is afffine equivalent to**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F}**; - CCZ-equivalent if there exists an affine permutation
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}}**of such that**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}(G_F)=G_{F'}}**, where**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+,\star)}**
,
for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p}**
a prime, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n}**
a positive integer, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+)}**
additive group and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x\star y}**
multiplication linear in each variable.
Every commutative presemifield can be transformed into a commutative semifield.

Two presemifields **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}_1=(\mathbb{F}_{p^n},+,\star)}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ)}**
are called isotopic if there exist three linear permutations **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T,M,N}**
of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{p^n}}**
such that
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T(x\star y)=M(x)\circ N(y)}**
,
for any **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x,y\in\mathbb{F}_{p^n}}**
. If **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M=N}**
then they are called strongly isotopic.
Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

- given
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{S}=(\mathbb{F}_{p^n},+,\star)}**let**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F_\mathbb{S}(x)=\frac{1}{2}(x\star x)}**; - given

Given a finite semifield, the subsets

for all

for all

for all

are called left, middle and right nucleus of .

The set is called the nucleus. All these sets are finite field and, when is commutative, . The order of the different nuclei are invariant under isotopism.

## Properties

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

- are CCZ-equivalent if and only if are strongly isotopic;
- 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:
- is odd and the semifields are strongly isotopic,
- is even and the semifields are strongly isotopic or the only isotopisms are of the form with non-square.