APN Functions Research Articles

Correction to: Algebraic Construction of Near-Bent and APN Functions

Remark 4.1 and Remark 4.5 in Section 4 will be true only if a is a power of two. Hence, the proofs of Theorem 4.2 and Theorem 4.6 hold in the case of a being a power of 2. More generally, both Theorems 4.2 and 4.6 are true by observing the change of variable x → y + 1 in the Walsh-Hadamard Transform, as it is a permutation on the finite field, i.e., we get exactly opposite values of the function f(x) = Tr( x)d, where d is the Niho exponent. Since the function f(x) = Tr(x)d is near-bent function, the function f(x) = Tr(x+1)d is also a near bent function

Partially APN functions with APN-like polynomial representations

In this paper we investigate several families of monomial functions with APN-like exponents that are not APN, but are partially 0-APN for infinitely many extensions of the binary field $$\mathbb {F}_2$$. We also investigate the differential uniformity of some binomial partial APN functions. Furthermore, the partial APN-ness for some classes of multinomial functions is investigated. We show also that the size of the pAPN spectrum is preserved under CCZ-equivalence.

On Two-to-One Mappings Over Finite Fields

Two-to-one ($2$-to-$1$) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of $x^rh(x^{(q-1)/d})$, those from permutation polynomials, from linear translators and from APN functions. Then we present $2$-to-$1$ polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of $2$-to-$1$ mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.

Infinite families of 3‐designs from APN functions

AbstractCombinatorial ‐designs have nice applications in coding theory, finite geometries, and several engineering areas. A classical method for constructing ‐designs is by the action of a permutation group that is ‐transitive or ‐homogeneous on a point set. This approach produces ‐designs, but may not yield ‐designs. The objective of this paper is to study how to obtain 3‐designs with 2‐transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are 2‐transitive, is considered. A characterization of such incidence structure to be a 3‐design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of 3‐designs are constructed by employing almost perfect nonlinear functions. Some 3‐designs presented in this paper give rise to self‐dual binary codes or linear codes with optimal or best parameters known. Several conjectures on 3‐designs and binary codes are also presented.

Свойства объединенных булевых функций квадратичных функций APN

Свойства объединенных булевых функций квадратичных функций APN

Determining the Walsh spectra of Taniguchi's and related APN-functions

We introduce a method based on Bezout's theorem on intersection points of two projective plane curves, for determining the nonlinearity of some classes of quadratic functions on F22m. Among those are the functions of Taniguchi (2019) [20], Carlet (2011) [7], and Zhou and Pott (2013) [22], all of which are APN under certain conditions. This approach helps to understand why the majority of the functions in those classes have solely bent and semibent components, which in the case of APN functions is called the classical spectrum. More precisely, we show that all Taniguchi functions have the classical spectrum independent from being APN. We determine the nonlinearity of all functions belonging to Carlet's class and to the class of Zhou and Pott, which also confirms with comparatively simple proofs earlier results on the Walsh spectrum of APN-functions in these classes. Using the Hasse-Weil bound, we show that some simple sufficient conditions for the APN-ness of the Zhou-Pott functions, which are given in the original paper, are also necessary.

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.

Further study on the maximum number of bent components of vectorial functions

In 2018, Pott et al. have studied in (IEEE Trans Inf Theory 64(1):403-411, 2018) the maximum number of bent components of vectorial functions. They have presented many nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondly, we prove the non-existence of APN plateaued functions having the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. Finally, we present some sufficient conditions that the vectorial function defined from $$\mathbb {F}_{2^{2k}}$$ to $$\mathbb {F}_{2^{2k}}$$ by its univariate representation: $$\begin{aligned} \alpha x^{2^i}\left( x+x^{2^k}+\sum \limits _{j=1}^{\rho }\gamma ^{(j)}x^{2^{t_j}} +\sum \limits _{j=1}^{\rho }\gamma ^{(j)}x^{2^{t_j+k}}\right) \end{aligned}$$ has the maximum number of bent components, where $$\rho \le k$$ . Further, we show that the differential spectrum of the function $$ x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})$$ (where $$i,t_1,t_2$$ satisfy some conditions) is different from the binomial function $$F^i(x)= x^{2^i}(x+x^{2^k})$$ presented in the article of Pott et al.

Changing APN functions at two points

We investigate a construction in which a vectorial Boolean function G is obtained from a given function F over $\mathbb {F}_{2^{n}}$ by changing the values of F at two points of the underlying field. In particular, we examine the possibility of obtaining one APN function from another in this way. We characterize the APN-ness of G in terms of the derivatives and in terms of the Walsh coefficients of F. We establish that changing two points of a function F over $\mathbb {F}_{2^{n}}$ which is plateaued (and, in particular, AB) or of algebraic degree deg(F) < n − 1 can never give a plateaued (and AB, in particular) function for any n ≥ 5. We also examine a particular case in which we swap the values of F at two points of $\mathbb {F}_{2^{n}}$. This is motivated by the fact that such a construction allows us to obtain one permutation from another. We obtain a necessary and sufficient condition for the APN-ness of G which we then use to show that swapping two points of any power function over a field $\mathbb {F}_{2^{n}}$ with n ≥ 5 can never produce an APN function. We also list some experimental results indicating that the same is true for the switching classes from Edel and Pott (Adv. Math. Commun. 3(1):59–81, 2009), and conjecture that the Hamming distance between two APN functions cannot be equal to two for n ≥ 5.

On some quadratic APN functions

A construction of APN functions using the bent function $$B(x,y)=xy$$ is proposed in Carlet (Des Codes Cryptogr 59:89–109, 2011). At this time, two families of APN functions using this construction are known, that is, the family of Carlet (2011) and the family of Zhou and Pott (Adv Math 234:43–60, 2013). In this note, we propose another family of APN functions with this construction, which are not CCZ equivalent to the former two families on $${{\mathbb {F}}}_{2^8}$$ . We also propose a family of presemifields and determined the middle, left, right nuclei and the center of the associated semifields.

Componentwise APNness, Walsh uniformity of APN functions, and cyclic-additive difference sets

In [Characterizations of the differential uniformity of vectorial functions by the Walsh transform, IEEE Transactions on Information Theory 2017], the author has characterized differentially δ-uniform functions by equalities satisfied by their Walsh transforms. This generalizes the characterization of APN functions by the fourth moment of the Walsh transform. We study two notions which are related: (1) the componentwise APNness (CAPNness) of (n,n)-functions, which is a stronger version of APNness, related to the characterization by the fourth moment, in which the arithmetic mean of WF4(u,v) when u ranges over F2n and v is fixed nonzero in F2n equals 22n+1 (2) the componentwise Walsh uniformity (CWU) of (n,m)-functions (m=n, resp. m=n−1), which is a stronger version of APNness (resp. of differential 4-uniformity) related to one of the new characterizations, in which the arithmetic mean of WF2(u1,v1)WF2(u2,v2)WF2(u1+u2,v1+v2) when u1,u2 range independently over F2n and v1,v2 are fixed nonzero and distinct in F2m, equals 23n. Concerning the first notion, it is known from Berger, Canteaut, Charpin and Laigle-Chapuy that any plateaued function is CAPN if and only if it is AB and that APN power permutations are CAPN. We prove that CAPN functions can exist only if n is odd; this solves an eleven year old open problem by these authors. Concerning the second notion, we show that any crooked function (and in particular any quadratic APN function) is CWU, but we observe also that other APN functions like Kasami functions and the inverse of one of the Gold APN permutations are CWU for n≤11. We show that the CWUness of APN power permutations is equivalent to a property of ΔF={F(x)+F(x+1)+1;x∈F2n}. This new property, that we call cyclic-additive difference set property, is more complex than the cyclic difference set property (proved in the case of Kasami APN functions by Dillon and Dobbertin). We prove it in the case of the inverse of Gold function. In the case of Kasami functions, we observe that the cyclic-additive property is also true for n≤10 even and we leave the proof of the CWUness and the cyclic-additive property as open problems.

Decomposition of permutations in a finite field

We describe a method to decompose any power permutation, as a sequence of power permutations of lower algebraic degree. As a result we obtain decompositions of the inversion in GF(2n) for small n from 3 up to 16, as well as for the APN functions, when n = 5. More precisely, we find decompositions into quadratic power permutations for any n not multiple of 4 and decompositions into cubic power permutations for n multiple of 4. Finally, we use the Theorem of Carlitz to prove that for 3 ≤ n ≤ 16 any n-bit permutation can be decomposed in quadratic and cubic permutations.

On Upper Bounds for Algebraic Degrees of APN Functions

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 ${\mathbb F}_{2^{n}}$ 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 $n$ .

On APN functions L1(x3) + L2(x9) with linear L1 and L2

In a recent paper by L. Budaghyan, C. Carlet, and G. Leander (2009) it is shown that functions of the form L1(x3) + L2(x9), where L1 and L2 are linear, are a good source for construction of new infinite families of APN functions. In the present work we study necessary and sufficient conditions for such functions to be APN.

English

English

О построении APN-функций специального вида и их связи с взаимно однозначными APN-функциями

О построении APN-функций специального вида и их связи с взаимно однозначными APN-функциями

On the irreducibility of the hyperplane sections of Fermat varieties in ℙ 3 $\mathbb {P}^{3}$ in characteristic 2. II

Let t be an integer ≥ 3 such that t ≡ 1 mod 4. The absolute irreducibility of the polynomial $\phi _{t}(x, y) = \frac {x^{t} + y^{t} + 1 + (x + y + 1)^{t}}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb {F}_{2}$ ) plays an important role in the study of APN functions. We prove that this polynomial is absolutely irreducible under the assumptions that the largest odd integer which divides t − 1 is large enough and can not be written in a specific form.

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

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

О дифференциальной эквивалентности квадратичных APN-функций

О дифференциальной эквивалентности квадратичных APN-функций

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.