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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]

The binomial $B(x) = x^{3} + \beta x^{36}$ (where $\beta $ is primitive in $\mathbb {F}_{2^{2}}$) over $\mathbb {F}_{2^{10}}$ is the first known example of an Almost Perfect Nonlinear (APN) function that is not CCZ-equivalent to a power function, and has remained unclassified into any infinite family of APN functions since its discovery in 2006. We generalize this binomial to an infinite family of APN quadrinomials of the form $x^{3} + a (x^{2^{i}+1})^{2^{k}} + b x^{3 \cdot 2^{m}} + c (x^{2^{i+m}+2^{m}})^{2^{k}}$ from which $B(x)$ can be obtained by setting $a = \beta $ , $b = c = 0$ , $i = 3$ , $k = 2$. We show that for any dimension $n = 2m$ with $m$ odd and $3 \nmid m$ , setting $(a,b,c) = (\beta, \beta ^{2}, 1)$ and $i = m-2$ or $i = (m-2)^{-1} \mod n$ yields an APN function, and verify that for $n = 10$ the quadrinomials obtained in this way for $i = m-2$ and $i = (m-2)^{-1} \mod n$ are CCZ-inequivalent to each other, to $B(x)$ , and to any other known APN function over $\mathbb {F}_{2^{10}}$. [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]

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]

This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are not Carlet—Charpin—Zinoviev (CCZ)-equivalent to power functions (at least for some values of the number of variables). These are two classes of APN binomials from F2n to F2n (for n divisible by 3, resp., 4). We prove that these functions are extended affine (EA)-inequivalent to any power function and that they are CCZ-inequivalent to the Gold, Kasami, inverse, and Dobbertin functions when n ≥ 12. This means that for n even they are CCZ-inequivalent to any known APN function. In particular, for n = 12, 20, 24, they are therefore CCZ-inequivalent to any power function. [ABSTRACT FROM AUTHOR]

A method for constructing differentially 4-uniform quadratic hexanomials has been recently introduced by J. Dillon. We give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from F22m to F22m. We check for m = 3 that some of these functions are CCZ-inequivalent to power functions. [ABSTRACT FROM AUTHOR]

New infinite classes of almost bent and almost perfect nonlinear polynomials are constructed. It is shown that they are affine inequivalent to any sum of a power function and an affine function. [ABSTRACT FROM AUTHOR]