On a secondary construction of quadratic APN functions

Almost perfect nonlinear functions possess the optimal resistance to the differential cryptanalysis and are widely studied. Most known constructions of APN functions are obtained as functions over finite fields F27 and very little is known about combinatorial constructions in F2n. We consider how to obtain a quadratic APN function in n + 1 variables from a given quadratic APN function in n variables using special restrictions on new terms

Функции на расстоянии один от APN-функций от малого числа переменных

Исследуется вопрос существования APN-функций на расстоянии один друг от друга. Доказано, что гипотеза о том, что таких APN-функций нет, выполнена для большинства известных APN-функций от не более чем восьми переменных

On values of vectorial Boolean functions and related problems in APN functions

In this paper we prove that there are only differential 4-uniform functions which are on distance 1 from an APN function. Also we prove that there are no APN functions of distance 1 from another APN functions up to dimension 5. We determine some properties of the set of values of an arbitrary vectorial Boolean function from F_n^2 to F_n^2 in connection to the set of values of its derivatives. These results are connected to several open question concerning metric properties of APN functions

On relations between CCZ- and EA-equivalences

In the present paper we introduce some sufficient conditions and a procedure for checking whether, for a given function, CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. It is known from Budaghyan et al. (IEEE Trans. Inf. Theory 52.3, 1141–1152 2006; Finite Fields Appl. 15(2), 150–159 2009) that for quadratic APN functions (both monomial and polynomial cases) CCZ-equivalence is more general. We prove hereby that for non-quadratic APN functions CCZ-equivalence can be more general (by studying the only known APN function which is CCZ-inequivalent to both power functions and quadratics). On the contrary, we prove that for power non-Gold APN functions, CCZ equivalence coincides with EA-equivalence and inverse transformation for n ≤ 8. We conjecture that this is true for any n.acceptedVersio

A Note on 5-bit Quadratic Permutations’ Classification

Classification of vectorial Boolean functions up to affine equivalence is used widely to analyze various cryptographic and implementation properties of symmetric-key algorithms. We show that there exist 75 affine equivalence classes of 5-bit quadratic permutations. Furthermore, we explore important cryptographic properties of these classes, such as linear and differential properties and degrees of their inverses, together with multiplicative complexity and existence of uniform threshold realizations

Линейный спектр квадратичных APN-функций

Работа посвящена изучению почти совершенно нелинейных (APN) функций. Введено понятие линейного спектра квадратичной APN-функции; доказана теорема о нулевых значениях линейного спектра при чётном числе переменных; приведены вычислительные данные при малых значениях переменных n = 3, 4, 5, 6. Для известного класса APN-функций Голда F(x) = x2 +1, где (k,n) = 1, доказана теорема о крайнем значении линейного спектра

On Two Fundamental Problems on APN Power Functions

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 xi∘x1/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 xi∘L∘x1/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≤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 ∑k−1i=122ni−1 which generalize the inverse and Dobbertin families. We present data on the Walsh spectrum of the Dobbertin function for n≤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≤21 show that these points cover about 2/3 of the field.acceptedVersio

On a remarkable property of APN Gold functions

In [13] for a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself it was defined an associated Boolean function $\gamma_F(a,b)$ in $2n$ variables that takes value~$1$ iff $a\neq{\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. In this paper we introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. It is an interesting open problem to describe differential equivalence class of a given APN function. We consider the APN Gold function $F(x)=x^{2^k+1}$, where gcd$(k,n)=1$, and prove that there exist exactly $2^{2n+n/2}$ distinct affine functions $A$ such that $F$ and $F+A$ are differentially equivalent if $n=4t$ for some $t$ and $k = n/2 \pm 1$; otherwise the number of such affine functions is equal to $2^{2n}$. This theoretical result and computer calculations obtained show that APN Gold functions for $k=n/2\pm1$ and $n=4t$ are the only functions (except one function in 6 variables) among all known quadratic APN functions in $2,\ldots,8$ variables that have more than $2^{2n}$ trivial affine functions $A^F_{c,d}(x)=F(x)+F(x+c)+d$, where $c,d\in\mathbb{F}_2^n$, preserving the associated Boolean function when adding to $F$