Commutative Presemifields and Semifields

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

• linear if <math>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>F(x)=\sum_{0\le i\le j<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>\delta</math> a positive integer, the function <math>F</math> is called differentially <math>\delta</math>-uniform 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 (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:

• affine equivalent if <math>F'=A_1\circ F\circ A_2</math>, where <math>A_1,A_2</math> are affine permutations;
• EA-equivalent (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>;
• CCZ-equivalent 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 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>\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.

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 isotopic if there exist three linear permutations <math>T,M,N</math> of <math>\mathbb{F}_{p^n}</math> such that <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 strongly isotopic. 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>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>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math> a finite semifield, the subsets

<math>N_l(\mathbb{S})=\{\alpha\in\mathbb{S} : (\alpha\star x)\star y=\alpha\star(x\star y)</math> for all <math>x,y\in\mathbb{S}\}</math>

<math>N_m(\mathbb{S})=\{\alpha\in\mathbb{S} : (x\star\alpha)\star y=x\star(\alpha\star y)</math> for all <math>x,y\in\mathbb{S}\}</math>

<math>N_r(\mathbb{S})=\{\alpha\in\mathbb{S} : (x\star y)\star \alpha=x\star(y\star \alpha)</math> for all <math>x,y\in\mathbb{S}\}</math>

are called left, middle and right nucleus of <math>\mathbb{S}</math>.

The set <math>N(\mathbb{S})=N_l(\mathbb{S})\cap N_m(\mathbb{S})\cap N_r(\mathbb{S})</math> is called the nucleus. All these sets are finite field and, when <math>\mathbb{S}</math> is commutative, <math>N_l(\mathbb{S})=N_r(\mathbb{S})\subseteq N_m(\mathbb{S})</math>. The order of the different nuclei are invariant under isotopism.

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 the corresponding presemifileds are 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>;
• 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.

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 <math>x^{\frac{3^t+1}{2}}</math> defined over <math>\mathbb{F}_{3^n}</math>, with <math>t</math> is odd and gcd(<math>t,n</math>)=1.

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

• <math>x^2</math> over <math>\mathbb{F}_{p^n}</math> (finite field <math>\mathbb{F}_{p^n}</math>);
• <math>x^{p^t+1}</math> over <math>\mathbb{F}_{p^n}</math> with <math>n/gcd(t,n)</math> odd (Albert's commutative twisted fields);
• <math>L(t^2(x))+\frac{1}{2}x^2</math> over <math>\mathbb{F}_{p^{2km}}</math> with <math>L(x)=\frac{1}{8}(x^{p^k}-x), t(x)=x^{p^{km}}-x</math> (Dickson semifields);
• • <math>(ax)^{p^s+1}-(ax)^{p^k(p^s+1)}+x^{p^k+1}</math>
  • <math>bx^{p^s+1}+(bx^{p^s+1})^{p^k}+cx^{p^k+1}</math>

over <math>\mathbb{F}_{p^{2k}}</math> where <math>a,b\in\mathbb{F}^\star_{2^{2k}}, b</math> not square, <math>c\in\mathbb{F}_{2^{2k}}\setminus\mathbb{F}_{2^k}, gcd(k+s,2k)=gcd(k+s,k)</math> and for the first one also <math>gcd(p^s+1,p^k+1)\ne gcd(p^s+1,(p^k+1)/2)</math>. Without loss of generality it is possible to take <math>a=1</math> and fix a value for <math>c</math>;

• <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>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 p=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>.