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We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…
We present new invariants, APN-extendibility criterion and a backtracking approach to identify several numerical facts supporting the conjecture that the set of 6-bit \APN functions is limited to 14 CCZ-classes.
In this paper we compute the Fourier spectra of some recently discovered binomial APN functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis.…
The set of linear structures of most known balanced Boolean functions is nontrivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any APN function in even dimension…
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…
In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with $2^{2m}$ elements, where $m$ is even. We obtain this result by proving that the APN functions introduced by Pott and the…
In this article, we study algebraic decompositions and secondary constructions of almost perfect nonlinear (APN) functions. In many cases, we establish precise criteria which characterize when certain modifications of a given APN function…
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…
In this paper, we propose a new construction of quadratic bent functions in polynomial forms. Right Euclid algorithm in skew-polynomial rings over finite fields of characteristic 2 is applied in the proof.
Planar functions are functions over a finite field that have optimal combinatorial properties and they have applications in several branches of mathematics, including algebra, projective geometry and cryptography. There are two relevant…
We examine combinatorial counting functions with two parameters, $n$ and $q$. For fixed $q$, these functions are (quasi-)polynomial in $n$. As $q$ varies, the degree of this polynomial is itself polynomial in $q$, as are the leading…
We study combinatorial properties of plateaued functions $F \colon \mathbb{F}_p^n \rightarrow \mathbb{F}_p^m$. All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on…
Many underlying structural and functional factors that determine the fault behavior of a combinational network, are not yet fully understood. In this paper, we show that there exists a large class of Boolean functions, called root…
Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low $c$-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of…
We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the…
Permutation polynomials with coefficients 1 over finite fields attract researchers' interests due to their simple algebraic form. In this paper, we first construct four classes of fractional permutation polynomials over the cyclic subgroup…
It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields…
In this paper, we show that there is no vectorial Boolean function of degree 4e, with e satisfaying certain conditions, which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture…
Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the…
Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between…