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The six infinite families of power APN functions are among the oldest known instances of APN functions, and it has been conjectured in 2000 that they exhaust all possible power APN functions. Another long-standing open problem is that of the Walsh spectrum of the Dobbertin power family, which is still unknown. Those of Kasami, Niho and Welch functions are known, but not the precise values of their Walsh transform, with rare exceptions. One promising approach that could lead to the resolution of these problems is to consider alternative representations of the functions in questions. We derive alternative representations for the infinite APN monomial families. We show how the Niho, Welch, and Dobbertin functions can be represented as the composition $x^{i} \circ x^{1/j}$ of two power functions, and prove that our representations are optimal, i.e. no two power functions of lesser algebraic degree can be used to represent the functions in this way. We investigate compositions $x^{i} \circ L \circ x^{1/j}$ for a linear polynomial $L$ , show how the Kasami functions in odd dimension can be expressed in this way with $i=j$ being a Gold exponent and compute all APN functions of this form for $n \le 9$ and for $L$ with binary coefficients, thereby showing that our theoretical constructions exhaust all possible cases. We present observations and data on power functions with exponent $\sum _{i = 1}^{k-1} 2^{2ni} - 1$ which generalize the inverse and Dobbertin families. We present data on the Walsh spectrum of the Dobbertin function for $n \le 35$ , and conjecture its exact form. As an application of our results, we determine the exact values of the Walsh transform of the Kasami function at all points of a special form. Computations performed for $n\leq 21$ show that these points cover about 2/3 of the field. [ABSTRACT FROM AUTHOR]

We investigate the differential properties of a vectorial Boolean function $G$ obtained by modifying an APN function $F$. This generalizes previous constructions where a function is modified at a few points. We characterize the APN-ness of $G$ via the derivatives of $F$ , and deduce an algorithm for searching for APN functions whose values differ from those of $F$ only on a given set $U \subseteq \mathbb {F}_{2^{n}}$. We introduce a value $\Pi _{F}$ associated with any $F$ , which is invariant under CCZ-equivalence. We express a lower bound on the distance between a given APN function $F$ and the closest APN function in terms of $\Pi _{F}$. We show how $\Pi _{F}$ can be computed efficiently for $F$ quadratic. We compute $\Pi _{F}$ for all known APN functions over $\mathbb {F}_{2^{n}}$ up to $n \le 8$. This is the first new CCZ-invariant for APN functions to be introduced within the last ten years. We derive a mathematical formula for this lower bound for the Gold function $F(x) = x^{3}$ , and observe that it tends to infinity with $n$. Finally, we describe how to efficiently find all sets $U$ such that, taking $G(x) = F(x) + v$ for $x \in U$ and $G(x) = F(x)$ for $x \notin U$ , $G(x)$ is APN. [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear (APN) functions over fields of characteristic 2 play an important role in cryptography, coding theory and, more generally, mathematics and information theory. In this paper we deduce a new method for constructing APN functions by studying the isotopic equivalence, concept defined for quadratic planar functions in fields of odd characteristic. In particular, we construct a family of quadratic APN functions which provides a new example of an APN mapping over ${\mathbb F}_{2^{9}}$ and includes an example of another APN function $x^{9}+ \mathop {\mathrm {Tr}}\nolimits (x^{3})$ over ${\mathbb F}_{2^{8}}$ , known since 2006 and not classified up to now. We conjecture that the conditions for this family are satisfied by infinitely many APN functions. [ABSTRACT FROM AUTHOR]

We study the problem of existence of APN functions of algebraic degree n over {\mathbb F}_{2^{n}} . We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce several non-existence results which imply, in particular, that for most of the known APN functions F over \mathbb F2^{n} the function x^{2^{n}-1}+F(x) is not APN, and changing a value of $F$ in a single point then results in non-APN functions. This leads us to conjectures that an APN function modified in one point cannot remain APN and that there exists no APN function of algebraic degree $n$ . [ABSTRACT FROM AUTHOR]