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>.
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<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>.

Revision as of 14:08, 29 August 2019

Background

For a prime [math]\displaystyle{ p }[/math] and a positive integer [math]\displaystyle{ n }[/math] let [math]\displaystyle{ \mathbb{F}_{p^n} }[/math] be the finite field with [math]\displaystyle{ p^n }[/math] elements. Let [math]\displaystyle{ 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]\displaystyle{ p^n-1 }[/math], i.e.

[math]\displaystyle{ F(x)=\sum_{j=0}^{p^n-1}a_jx^j, a_j\in\mathbb{F}_{p^n} }[/math].

The function [math]\displaystyle{ F }[/math] is

  • linear if [math]\displaystyle{ F(x)=\sum_{j=0}^{n-1}a_jx^{p^j} }[/math],
  • affine if it is the sum of a linear function and a constant,
  • DO (Dembowski-Ostrim) polynomial if [math]\displaystyle{ F(x)=\sum_{0\le i\le j\lt n}a_{ij}x^{p^i+p^j} }[/math],
  • quadratic if it is the sum of a DO polynomial and an affine function.

For [math]\displaystyle{ \delta }[/math] a positive integer, the function [math]\displaystyle{ F }[/math] is called differentially [math]\displaystyle{ \delta }[/math]-uniform if for any pairs [math]\displaystyle{ a,b\in\mathbb{F}_{p^n} }[/math], with [math]\displaystyle{ a\ne0 }[/math], the equation [math]\displaystyle{ F(x+a)-F(x)=b }[/math] admits at most [math]\displaystyle{ \delta }[/math] solutions.

A function [math]\displaystyle{ F }[/math] is called planar or perfect nonlinear (PN) if [math]\displaystyle{ \delta_F=1 }[/math]. Obviously such functions exist only for [math]\displaystyle{ p }[/math] an odd prime. In the even case the smallest possible case for [math]\displaystyle{ \delta }[/math] is two (APN function).

For planar function we have that the all the nonzero derivatives, [math]\displaystyle{ D_aF(x)=F(x+a)-F(x) }[/math], are permutations.

Equivalence Relations

Two functions [math]\displaystyle{ F }[/math] and [math]\displaystyle{ F' }[/math] from [math]\displaystyle{ \mathbb{F}_{p^n} }[/math] to itself are called:

  • affine equivalent if [math]\displaystyle{ F'=A_1\circ F\circ A_2 }[/math], where [math]\displaystyle{ A_1,A_2 }[/math] are affine permutations;
  • EA-equivalent (extended-affine) if [math]\displaystyle{ F'=F''+A }[/math], where [math]\displaystyle{ A }[/math] is affine and [math]\displaystyle{ F'' }[/math] is afffine equivalent to [math]\displaystyle{ F }[/math];
  • CCZ-equivalent if there exists an affine permutation [math]\displaystyle{ \mathcal{L} }[/math] of [math]\displaystyle{ \mathbb{F}_{p^n}\times\mathbb{F}_{p^n} }[/math] such that [math]\displaystyle{ \mathcal{L}(G_F)=G_{F'} }[/math], where [math]\displaystyle{ 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 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 [math]\displaystyle{ \mathbb{S}=(\mathbb{F}_{p^n},+,\star) }[/math], for [math]\displaystyle{ p }[/math] a prime, [math]\displaystyle{ n }[/math] a positive integer, [math]\displaystyle{ \mathbb{S}=(\mathbb{F}_{p^n},+) }[/math] additive group and [math]\displaystyle{ x\star y }[/math] multiplication linear in each variable.

Two presemifields [math]\displaystyle{ \mathbb{S}_1=(\mathbb{F}_{p^n},+,\star) }[/math] and [math]\displaystyle{ \mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ) }[/math] are called isotopic if there exist three linear permutations [math]\displaystyle{ T,M,N }[/math] of [math]\displaystyle{ \mathbb{F}_{p^n} }[/math] such that [math]\displaystyle{ T(x\star y)=M(x)\circ N(y) }[/math], for any [math]\displaystyle{ x,y\in\mathbb{F}_{p^n} }[/math]. If [math]\displaystyle{ M=N }[/math] then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

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

Hence two quadratic planar functions [math]\displaystyle{ F,F' }[/math] are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:

  • [math]\displaystyle{ F,F' }[/math] are CCZ-equivalent if and only if [math]\displaystyle{ \mathbb{S}_F,\mathbb{S}_{F'} }[/math] are strongly isotopic;
  • for [math]\displaystyle{ n }[/math] odd, isotopic coincides with strongly isotopic;
  • if [math]\displaystyle{ F,F' }[/math] are isotopic equivalent, then there exists a linear map [math]\displaystyle{ L }[/math] such that [math]\displaystyle{ F' }[/math] is EA-equivalent to [math]\displaystyle{ F(x+L(x))-F(x)-F(L(x)) }[/math].