Difference between revisions of "Commutative Presemifields and Semifields"

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=Background=
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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.
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Let <math>F</math> be a map from the finite field to itself.
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Such function admits a unique representation as a polynomial of degree at most <math>p^n-1</math>, i.e.
  
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<math>F(x)=\sum_{j=0}^{p^n-1}a_jx^j, a_j\in\mathbb{F}_{p^n}</math>.
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The function <math>F</math> is
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* <span class="definition">linear</span> if <math>F(x)=\sum_{j=0}^{n-1}a_jx^{p^j} </math>,
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* <span class="definition">affine</span> if it is the sum of a linear function and a constant,
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* <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>,
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* <span class="definition">quadratic</span> if it is the sum of a DO polynomial and an affine function.
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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.
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A function <math>F</math> is called planar or perfect nonlinear (PN) if <math>\delta_F=1</math>.
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Obviously such functions exist only for <math>p</math> an odd prime.
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In the even case the smallest possible case for <math>\delta</math> is two ([[differential uniformity|APN]] function).
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For planar function we have that the all the nonzero derivatives, <math>D_aF(x)=F(x+a)-F(x)</math>, are permutations.
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==Equivalence Relations==
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Two functions <math>F</math> and <math>F'</math> from <math>\mathbb{F}_{p^n}</math> to itself are called:
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*<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;
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*<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>;
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*<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>.
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CCZ-equivalence is the most general known equivalence relation for functions which preserves differential uniformity. Affine and EA-equivalence are its particular cases.
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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.
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=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:
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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 15:08, 29 August 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 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.

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 .

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 .