On functions of finite fields

Section 2. Planar functions and semifields and other algebraic structures Suppose that f is a function over Fq where pn = q and p is an odd prime. We define about the behavior of the number of solutions of difference equations of f as the following. For a, b ∈ Fq, Nf (a, b) := {x ∈ Fq | f(x+ a)− f(x) = b}, Nf = Maxa,b∈Fq, a =0Nf (a, b). If Nf ≤ δ, a function f is called to be differentially δ-uniform, specially if Nf = 1, then f is called a planar function on Fq. From a planar function f on Fq we can construct an affine plane A(f) of order q as the following(cf.[7]). the set of points: Fq × Fq the set of lines: {g} × Fq (g ∈ Fq), D + (g, h) (g, h ∈ Fq) where D = {(x, f(x)) | x ∈ Fq}. Let P(f) be the projective extention of A(f). Then an additive group Fq × Fq acts on P(f) with the natural action as its orbit lengths are 1, q, q2 of point’s set and also line’s set. The converse is true. A finite algebraic structure E = (E,+, ◦) which is a abelian group with respect to addition and satisfy the following is called a finite semifield(cf.[3],[5]). (1) x ◦ (y + z) = x ◦ y + x ◦ z (∀x, y, z ∈ E) (2) (x+ y) ◦ z = x ◦ z + y ◦ z (∀x, y, z ∈ E) (3) E has the identy element 1 with respect to multiplication. (4) If a ◦ b = 0, then a = 0 or b = 0. Then, semifield (affine) plane A(E) is the following. E × E is the set of points and x = c (c ∈ E), y = x ◦ a+ b (a, b ∈ E) are lines of A(E). A finite commutative semifield E is a vector space over Fp for a prime p. Therefore (E,+) ∼= (Fpn ,+) for some positive integer n. Now if p = 2 we can construct a planar function on