We introduce a new concept, the APN-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) $G$ using the so-called difference squares. We describe the relations between the APN-defect and other recent concepts of similar nature. Upper and lower bounds for the values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. Its exact values in some cases are also calculated. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending on $n$ is evaluated and the implications are discussed. In the forthcoming second part of this work we further examine modifications of the inverse function. We also study modifications of classes of functions of low uniformity over infinitely many extensions of $\mathbb{F}_{2^n}$. We present quantitative results on their differential behaviour, especially in connection with their APN-defects.

翻译：本文引入了一个新概念——APN缺陷，可将其视为衡量给定函数$G:\mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n}$与几乎完美非线性（APN）函数集之间距离的度量。该概念的提出源于利用所谓差分平方对非APN函数（具有低差分均匀度）$G$的差分特性进行详细分析。我们阐述了APN缺陷与近期其他类似性质概念之间的关系。针对包括Dembowski-Ostrom多项式在内的多类重要函数，给出了APN缺陷值的上下界估计，并计算了部分情形下的精确值。确定了修正逆函数对应的差分平方，评估了其APN缺陷对$n$的依赖关系，并讨论了相关影响。在本系列研究的后续第二部分中，我们将进一步考察逆函数的各类修正形式，并系统研究低均匀度函数类在$\mathbb{F}_{2^n}$无穷扩张域上的修正变体。我们将给出关于这些函数差分特性的量化结果，特别是其与APN缺陷的关联性分析。