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.
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.
Every commutative presemifield can be transformed into a commutative semifield.
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 .
Given a finite semifield, the subsets
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.
Hence two quadratic planar functions are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:
- are CCZ-equivalent if and only if the corresponding presemifileds 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.
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
defined over , with is odd and gcd()=1.
In the following the list of some known infinite families of planar functions (and corresponding commutative semifields):
- over (finite field );
- over with odd (Albert's commutative twisted fields);
- over with (Dickson semifields);
over where not square, and for the first one also .
Without loss of generality it is possible to take and fix a value for ;
- over primitive, mod odd;
- over primitive, mod 4, odd;
- , with , over for .
Cases defined for p=3
↑ Coulter R. S., Henderson M. Commutative presemifields and semifields. Advances in Math. 217, pp. 282-304, 2008
↑ 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
- odd (Coulter-Matthews and Ding-Yuan semifields);
- (Ganley semifields);
- (Cohen-Ganley semifileds);
- (Penttila-Williams semifileds);
- (Coulter-Henderson-Kosick semifield);