# AND THEIR GENERALIZATIONS

Let q be a prime power. If q is odd, a function f: Fq → Fq is planar or perfect nonlinear if, for each nonzero ε ∈ Fq, the function

x → f (x + ε) − f (x) (1)

is a permutation on Fq. Such planar functions can be used to construct finite projective planes, relative difference sets, error-correcting codes, and S-boxes in block ciphers.

If q is even, a function f : Fq → Fq cannot satisfy the above definition of planar functions because x = a and x = a + ε are mapped by (1) to the same image. This is the motivation to define a function f : Fq → Fq for even q to be almost perfect nonlinear (APN) if (1) is a 2-to-1 map. Such functions are highly relevant again for the construction of S-boxes in block ciphers. However, there is no apparent link between APN functions and projective planes. More recently, Zhou defined a natural analogue of planar functions on finite fields of characteristic two: If q is even, a function f : Fq → Fq is planar if, for each nonzero ε ∈ Fq, the function

x → f(x + ε) + f(x) + εx

is a permutation on Fq. As shown by Schmidt and Zhou, such planar functions have similar properties and applications as their counterparts in odd characteristic.

PN and APN functions can be generalized as follows. Given a p-ary (n, m)-function f : Fp^n → Fp^m, and c ∈ Fp^m, the (multiplicative) c-derivative of f with respect to a ∈ Fp^n is the function

cDa f(x) = f(x + a) − cf(x), ∀x ∈ Fp^n.

For an (n, n)-function f, and a, b ∈ Fp^n, let

cΔf (a, b) := |{x ∈ Fpn : f (x + a) − cf (x) = b}|,

and

cΔf := max{cΔf (a, b) : a, b ∈ Fp^n, (a, c)!= (0, 1)},

where |S| is the cardinality of the set S. The quantity cΔf is called c-differential uniformity of f . Note that for c = 1, the above definitions coincide with the usual derivative of f and its differential uniformity.

Crooked functions are particular APN functions. A function f : F2^n → F2^n is crooked if it satisfies the following three properties:

(i) f(0) = 0;

(ii) f(x)+f(y)+f(z)+f(x+y+z)̸!=0 for any x,y,z distinct;

(iii) f(x)+f(y)+f(z)+f(x+a)+f(y+a)+f(z+a)̸!=0 for any x,y,zand a!=0.

Non-existence results for all these type of functions can be obtained via a connection with algebraic surfaces.

Planar functions in characteristic two. In [2019/4] it is proved that no planar functions of low degree in char two exist, apart from linearized polynomials. Planar binomials are investigated in [2020/2].

Planar functions in odd characteristic. Planar polynomials arising from linearized polynomials are considered in [2022/3].

P(c)N functions. These generalizations of PN functions (in both even and odd characteristic) are investigated in [2020/5], [2021/7], [2022/12]

APN and GAPN functions. In [2022/4] it has been proved the infinitness of a particular type of APN function. A conjectur on APN permutations is investigated in [2022/6], whereas [2022/7] deals with a particular generalization of APN functions to any characteristic.

Crooked functions. Non-existence results for crooked functions of exceptional type are obtained in [2022/9] via the investigation of algebraic surfaces.