We introduce a new concept, theAPN-defect, which can be thought of as measuring the distance of a given function$G:\mathbb {F}_{2^{n}} \rightarrow \mathbb {F}_{2^{n}}$to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity)Gusing the so-calleddifference squares. Indeed, the insight into some structural qualities of S-boxes provided by this new approach is particularly useful in the light of recent refinements of differential cryptanalysis. We describe the relations between the APN-defect and other current concepts of similar nature. Values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. This enables one to identify thequasi-APNones, i.e., those with favourable differential behavior. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending onnis evaluated, the partial quadruple system associated to it is described, and the implications are discussed. In the forthcoming second part of this work we further examine the APN-defect of modifications of the inverse function and address some questions concerning CCZ-equivalence. We also study modifications of classes of functions of low differential uniformity over infinitely many extensions of$\mathbb {F}_{2^{n}}$and present quantitative results on their differential behaviour.

The Walsh transform f̂ of a quadratic function f: F(pn) → Fpsatisfies |f̂| ∈ {0,pn+s/2} for an integer 0 ≤ s ≤ n-1, depending on f. In this paper, quadratic functions of the form Fp,n(x) = Trn(Σi=0kaixpi+1) are studied, with the restriction that ai∈ Fp, 0 ≤ i ≤ k. Three methods for enumeration of such functions are presented when the value for s is prescribed. This paper extends earlier enumeration results significantly, for instance, the generating function for the counting function is obtained, when n is odd and relatively prime to p, or when n = 2 m, for odd m and p = 2. The number of bent and semibent functions for various classes of n is also obtained.