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Christof Beierle

Publications and source records attributed to Christof Beierle.

10 recordsLinked to original sources
arXiv2025

Millions of inequivalent quadratic APN functions in eight variables

The only known example of an almost perfect nonlinear (APN) permutation in even dimension was obtained by applying CCZ-equivalence to a specific quadratic APN function in dimension six. Motivated by this result, there have been numerous attempts to construct new quadratic APN functions. Prior to this work, $32\,892$ quadratic APN functions in dimension eight are known and two recent conjectures address their possible total number. The first, proposed by Y. Yu and L. Perrin, suggests that there are more than $50\,000$ such functions. The second, by A. Polujan and A. Pott, argues that their number exceeds that of inequivalent quadratic $(8,4)$-bent functions, which is $92\,515$. We computationally construct $3\,775\,599$ inequivalent quadratic APN functions in dimension eight and estimate the total number to be about six million.

math.CO↗
arXiv2024

A degree bound for planar functions

Using Stickelberger's theorem on Gauss sums, we show that if $F$ is a planar function on a finite field $\mathbb{F}_q$, then for all non-zero functions $G : \mathbb{F}_q \to \mathbb{F}_q$, we have \begin{equation*} d_{\mathsf{alg}}(G \circ F) - d_{\mathsf{alg}}(G) \le \frac{n(p-1)}{2}, \end{equation*} where $q = p^n$ with $p$ a prime and $n$ a positive integer, and $d_{\mathsf{alg}}(F)$ is the algebraic degree of $F$, i.e., the maximum degree of the corresponding system of $n$ lowest-degree interpolating polynomials for $F$ considered as a function on $\mathbb{F}_p^n$. This bound implies the (known) classification of planar polynomials over $\mathbb{F}_p$ and planar monomials over $\mathbb{F}_{p^2}$. As a new result, using the same degree bound, we complete the classification of planar monomials for all $n = \smash{2^k}$ with $p>5$ and $k$ a non-negative integer. Finally, we state a conjecture on the sum of the base-$p$ digits of integers modulo $q-1$ that implies the complete classification of planar monomials over finite fields of characteristic $p>5$.

math.CO↗
arXiv2022

On Matrix Algebras Isomorphic to Finite Fields and Planar Dembowski-Ostrom Monomials

Let $p$ be a prime and $n$ a positive integer. As the first main result, we present a deterministic algorithm for deciding whether the matrix algebra $\mathbb{F}_p[A_1,\dots,A_t]$ with $A_1,\dots,A_t \in \mathrm{GL}(n,\mathbb{F}_p)$ is a finite field, performing at most $\mathcal{O}(tn^6\log(p))$ elementary operations in $\mathbb{F}_p$. In the affirmative case, the algorithm returns a defining element $a$ so that $\mathbb{F}_p[A_1,\dots,A_t] = \mathbb{F}_p[a]$. We then study an invariant for the extended-affine equivalence of Dembowski-Ostrom (DO) polynomials. More precisely, for a DO polynomial $g \in \mathbb{F}_{p^n}[x]$, we associate to $g$ a set of $n \times n$ matrices with coefficients in $\mathbb{F}_p$, denoted $\mathrm{Quot}(\mathcal{D}_g)$, that stays invariant up to matrix similarity when applying extended-affine equivalence transformations to $g$. In the case where $g$ is a planar DO polynomial, $\mathrm{Quot}(\mathcal{D}_g)$ is the set of quotients $XY^{-1}$ with $Y \neq 0,X$ being elements from the spread set of the corresponding commutative presemifield, and $\mathrm{Quot}(\mathcal{D}_g)$ forms a field of order $p^n$ if and only if $g$ is equivalent to the planar monomial $x^2$, i.e., if and only if the commutative presemifield associated to $g$ is isotopic to a finite field. As the second main result, we analyze the structure of $\mathrm{Quot}(\mathcal{D}_g)$ for all planar DO monomials, i.e., for commutative presemifields of odd order being isotopic to a finite field or a commutative twisted field. More precisely, for $g$ being equivalent to a planar DO monomial, we show that every non-zero element $X \in \mathrm{Quot}(\mathcal{D}_g)$ generates a field $\mathbb{F}_p[X] \subseteq \mathrm{Quot}(\mathcal{D}_g)$ and $\mathrm{Quot}(\mathcal{D}_g)$ contains the field $\mathbb{F}_{p^n}$.

math.RA↗
arXiv2022

Generalized Almost Perfect Nonlinear Binomials and Trinomials Over Fields of Prime-Square Order

Let $p>3$ be a prime. We show that, for each integer $d$ with $p \leq d \leq 2(p-1)$, there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over $\mathbb{F}_{p^2}$ of algebraic degree $d$. We start by deriving sufficient conditions for the function $G \colon \mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}, X \mapsto X^{d_1} + u X^{d_2}$ to be GAPN in the case where one of the terms of $G$ is GAPN. We then give explicit constructions of GAPN binomials over $\mathbb{F}_{p^2}$ of any odd algebraic degree between $p$ and $2(p-1)$ and, in the case where $p$ is not a Mersenne prime, also of any even algebraic degree in this range. To obtain GAPN functions of even algebraic degree also in the general case, we finally show how to construct GAPN trinomials over $\mathbb{F}_{p^2}$ of any even algebraic degree between $p$ and $2(p-1)$ by applying a characterization of a special form of GAPN binomials by \"{O}zbudak and S\u{a}l\u{a}gean. Our constructed functions are the first GAPN functions of even algebraic degree over extension fields of odd characteristic reported so far.

math.CO↗
arXiv2022

Gold Functions and Switched Cube Functions Are Not 0-Extendable in Dimension $n > 5$

In the independent works by Kalgin and Idrisova and by Beierle, Leander and Perrin, it was observed that the Gold APN functions over $\mathbb{F}_{2^5}$ give rise to a quadratic APN function in dimension 6 having maximum possible linearity of $2^5$ (that is, minimum possible nonlinearity $2^4$). In this article, we show that the case of $n \leq 5$ is quite special in the sense that Gold APN functions in dimension $n>5$ cannot be extended to quadratic APN functions in dimension $n+1$ having maximum possible linearity. In the second part of this work, we show that this is also the case for APN functions of the form $x \mapsto x^3 + \mu(x)$ with $\mu$ being a quadratic Boolean function.

cs.IT↗
arXiv2021

New Instances of Quadratic APN Functions

In a recent work, Beierle, Brinkmann and Leander presented a recursive tree search for finding APN permutations with linear self-equivalences in small dimensions. In this paper, we describe how this search can be adapted to find many new instances of quadratic APN functions. In particular, we found 12,921 new quadratic APN functions in dimension eight, 35 new quadratic APN functions in dimension nine and five new quadratic APN functions in dimension ten up to CCZ-equivalence. Remarkably, two of the 35 new APN functions in dimension nine are APN permutations. Among the 8-bit APN functions, there are three extended Walsh spectra that do not correspond to any of the previously-known quadratic 8-bit APN functions and, surprisingly, there exist at least four CCZ-inequivalent 8-bit APN functions with linearity $2^7$, i.e., the highest possible non-trivial linearity for quadratic functions in dimension eight.

cs.IT↗
arXiv2021

Trims and Extensions of Quadratic APN Functions

In this work, we study functions that can be obtained by restricting a vectorial Boolean function $F \colon \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n$ to an affine hyperplane of dimension $n-1$ and then projecting the output to an $n-1$-dimensional space. We show that a multiset of $2 \cdot (2^n-1)^2$ EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on $\mathbb{F}_2^n$. Further, for all of the known quadratic APN functions in dimension $n < 10$, we determine the restrictions that are also APN. Moreover, we construct 6,368 new quadratic APN functions in dimension eight up to EA-equivalence by extending a quadratic APN function in dimension seven. A special focus of this work is on quadratic APN functions with maximum linearity. In particular, we characterize a quadratic APN function $F \colon \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n$ with linearity of $2^{n-1}$ by a property of the ortho-derivative of its restriction to a linear hyperplane. Using the fact that all quadratic APN functions in dimension seven are classified, we are able to obtain a classification of all quadratic 8-bit APN functions with linearity $2^7$ up to EA-equivalence.

cs.IT↗
arXiv2021

A Further Study of Quadratic APN Permutations in Dimension Nine

Recently, Beierle and Leander found two new sporadic quadratic APN permutations in dimension 9. Up to EA-equivalence, we present a single trivariate representation of those two permutations as $C_u \colon (\mathbb{F}_{2^m})^3 \rightarrow (\mathbb{F}_{2^m})^3, (x,y,z) \mapsto (x^3+uy^2z, y^3+uxz^2,z^3+ux^2y)$, where $m=3$ and $u \in \mathbb{F}_{2^3}\setminus\{0,1\}$ such that the two permutations correspond to different choices of $u$. We then analyze the differential uniformity and the nonlinearity of $C_u$ in a more general case. In particular, for $m \geq 3$ being a multiple of 3 and $u \in \mathbb{F}_{2^m}$ not being a 7-th power, we show that the differential uniformity of $C_u$ is bounded above by 8, and that the linearity of $C_u$ is bounded above by $8^{1+\lfloor \frac{m}{2} \rfloor}$. Based on numerical experiments, we conjecture that $C_u$ is not APN if $m$ is greater than $3$. We also analyze the CCZ-equivalence classes of the quadratic APN permutations in dimension 9 known so far and derive a lower bound on the number of their EA-equivalence classes. We further show that the two sporadic APN permutations share an interesting similarity with Gold APN permutations in odd dimension divisible by 3, namely that a permutation EA-inequivalent to those sporadic APN permutations and their inverses can be obtained by just applying EA transformations and inversion to the original permutations.

cs.IT↗
arXiv2021

Linearly Self-Equivalent APN Permutations in Small Dimension

All almost perfect nonlinear (APN) permutations that we know to date admit a special kind of linear self-equivalence, i.e., there exists a permutation $G$ in their CCZ-equivalence class and two linear permutations $A$ and $B$, such that $G \circ A = B \circ G$. After providing a survey on the known APN functions with a focus on the existence of self-equivalences, we search for APN permutations in dimension 6, 7, and 8 that admit such a linear self-equivalence. In dimension six, we were able to conduct an exhaustive search and obtain that there is only one such APN permutation up to CCZ-equivalence. In dimensions 7 and 8, we performed an exhaustive search for all but a few classes of linear self-equivalences and we did not find any new APN permutation. As one interesting result in dimension 7, we obtain that all APN permutation polynomials with coefficients in $\mathbb{F}_2$ must be (up to CCZ-equivalence) monomial functions.

cs.IT↗