On the symmetric properties of APN functions

We study the symmetric properties of APN functions as well as the structure and properties of the range of an arbitrary APN function. We prove that there is no permutation of variables that preserves the values of an APN function. Upper bounds for the number of symmetric coordinate Boolean functions in an APN function and its coordinate functions invariant under a cyclic shift are obtained. For n ≤ 6, some upper bounds for the maximal number of identical values of an APN function are given and a lower bound is found for different values of an arbitrary APN function of n variables.     

On quadratic APN functions and dimensional dual hyperovals

In this paper we characterize the d-dimensional dual hyperovals in PG(2d + 1, 2) that can be obtained by Yoshiara’s construction (Innov Incid Geom 8:147–169, 2008) from quadratic APN functions and state a one-to-one correspondence between the extended affine equivalence classes of quadratic APN functions and the isomorphism classes of these dual hyperovals.     

On the equivalence of quadratic APN functions

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function x^2r+1{x^{2^r+1}} if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2 ⁿ where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.     

On weakly starlike multivalent functions

Journal d'Analyse Mathématique -     

More constructions of APN and differentially 4-uniform functions by concatenation

We study further the method of concatenating the outputs of two functions for designing an APN or a differentially 4-uniform (n, n)-function for every even n. We deduce several specific constructions of APN or differentially 4-uniform (n, n)-functions from APN and differentially 4-uniform (n/2, n/2)-functions. We also give a construction of quadratic APN functions which includes as particular cases a previous construction by the author and a more recent construction by Pott and Zhou.     

On the conjecture on APN functions and absolute irreducibility of polynomials

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field \(\mathbb {F}\) is called exceptional APN, if it is also APN on infinitely many extensions of \(\mathbb {F}\). In this article we consider the most studied case of \(\mathbb {F}=\mathbb {F}_{2^n}\). A conjecture of Janwa–Wilson and McGuire–Janwa–Wilson (1993/1996), settled in 2011, was that the only monomial exceptional APN functions are the monomials \(x^n\), where \(n=2^k+1\) or \(n={2^{2k}-2^k+1} \) (the Gold or the Kasami exponents, respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our results is that all functions of the form \(f(x)=x^{2^k+1}+h(x)\) (for any odd degree h(x), with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture. We also show absolute irreducibility of a class of multivariate polynomials over finite fields (by repeated hyperplane sections, linear transformations, and reductions) and discuss their applications.     

Equivalences among plateaued APN functions

Some general results are obtained about pairs of CCZ-equivalent plateaued APN functions, exploiting character theory of finite groups. It is proven that if a plateaued power APN function f over \({\mathbb {F}}_{2^{2m}}\) is CCZ-equivalent to a plateaued function g, then f and g are EA-equivalent.     

Equivalences of quadratic APN functions

The following conjecture due to Y. Edel is affirmatively solved: two quadratic APN (almost perfect nonlinear) functions are CCZ-equivalent if and only if they are extended affine equivalent.     

Plateaudness of Kasami APN functions

It is shown that the Kasami function defined on ${\mathbb{F}}_{2^n}$ with n even is plateaued. This generalizes a result [3, Theorem 11], where the restriction $(n,3)=1$ is assumed. The result is used to establish the CCZ-inequivalence of the Kasami function defined on ${\mathbb{F}}_{2^n}$ with n even to the other known monomial APN functions [4].     

On the classification of APN functions up to dimension five

We classify the almost perfect nonlinear (APN) functions in dimensions 4 and 5 up to affine and CCZ equivalence using backtrack programming and give a partial model for the complexity of such a search. In particular, we demonstrate that up to dimension 5 any APN function is CCZ equivalent to a power function, while it is well known that in dimensions 4 and 5 there exist APN functions which are not extended affine (EA) equivalent to any power function. We further calculate the total number of APN functions up to dimension 5 and present a new CCZ equivalence class of APN functions in dimension 6.     

Trims and extensions of quadratic APN functions

Designs, Codes and Cryptography - In this work, we study functions that can be obtained by restricting a vectorial Boolean function $$F :\mathbb {F}_{2}^n \rightarrow \mathbb {F}_{2}^n$$ to an...     

New semifields, PN and APN functions

We describe a method of proving that certain functions ${f:F\longrightarrow F}$ defined on a finite field F are either PN-functions (in odd characteristic) or APN-functions (in characteristic 2). This method is illustrated by giving short proofs of the APN-respectively the PN-property for various families of functions. The main new contribution is the construction of a family of PN-functions and their corresponding commutative semifields of dimension 4s in arbitrary odd characteristic. It is shown that a subfamily of order p ^4s for odd s > 1 is not isotopic to previously known examples.     

On the approximation of weakly plurifinely plurisubharmonic functions

In this note, we study the approximation of singular plurifine plurisubharmonic function $u$ defined on a plurifine domain $\Omega$. Under some condition we prove that $u$ can be approximated by an increasing sequence of plurisubharmonic functions defined on Euclidean neighborhoods of $\Omega$.     

Scalar characterizations of weakly cone-convex and weakly cone-quasiconvex functions

The principal aim of this paper is to show that weakly cone-convex vector-valued functions, as well as weakly cone-quasiconvex vector-valued functions, can be characterized in terms of usual weakly convexity and weakly quasiconvexity of certain real-valued functions, defined by means of the extreme directions of the polar cone or by Gerstewitz’s scalarization functions.     

Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions

We survey the properties of two parameters introduced by C. Ding and the author for quantifying the balancedness of vectorial functions and of their derivatives. We give new results on the distribution of the values of the first parameter when applied to F + L, where F is a fixed function and L ranges over the set of linear functions: we show an upper bound on the nonlinearity of F by means of these values, we determine then the mean of these values and we show that their maximum is a nonlinearity parameter as well, we prove that the variance of these values is directly related to the second parameter. We briefly recall the known constructions of bent vectorial functions and introduce two new classes obtained with Gregor Leander. We show that bent functions can be used to build APN functions by concatenating the outputs of a bent (n, n/2)-function and of some other (n, n/2)-function. We obtain this way a general infinite class of quadratic APN functions. We show that this class contains the APN trinomials and hexanomials introduced in 2008 by L. Budaghyan and the author, and a class of APN functions introduced, in 2008 also, by Bracken et al.; this gives an explanation of the APNness of these functions and allows generalizing them. We also obtain this way the recently found Edel?CPott cubic function. We exhibit a large number of other sub-classes of APN functions. We eventually design with this same method classes of quadratic and non-quadratic differentially 4-uniform functions.     

Dimensional dual hyperovals and APN functions with translation groups

In this paper we develop a theory of translation groups for dimensional dual hyperovals and APN functions. It will be seen that both theories can be treated, to a large degree, simultaneously. For small ambient spaces it will be shown that the translation groups are normal in the automorphism group of the respective geometric object. For large ambient spaces there may be more than one translation group. We will determine the structure of the normal closure of the translation groups in the automorphism group and we will exhibit examples which in fact do admit more than one translation group.