Related papers: A few more functions that are not APN infinitely o…
We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular…
The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for integer powers m and n, which are…
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…
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…
In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z_{2^n-1}, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function \de(n),…
We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $g\geq 1$ defined over a finite field $\F_q$. In particular, we prove that there always exists an effective non-special divisor of…
Although Feynman integrals in general cannot be expressed as well-studied special functions, they can be calculated systematically and efficiently using the \texttt{AMFlow} method in combination with differential equations in the kinematic…
In [arXiv 0811.3913] the authors introduced the notion of quasi-polynomial function as being a mapping f: X^n -> X defined and valued on a bounded chain X and which can be factorized as f(x_1,...,x_n)=p(phi(x_1),...,phi(x_n)), where p is a…
In this article, we focus on the concept of locally-APN-ness (``APN" is the abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow larger…
We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…
In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in \mathbb{N}$, $\ell$ odd). We show several important properties about that function and then…
We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are…
We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…
Assume $n=2m\geq 2$ and let $F(x)=x^{d_1}+x^{d_2}$ be a binomial vectorial function over $\F_{2^n}$ possessing the maximal number (i.e. $2^n-2^m$) of bent components. Suppose the $2$-adic Hamming weights $\wt_2(d_1)$ and $\wt_2(d_2)$ are…
Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and…
We find bounds for the maximal length of a sequence of distinct $\bf{\Gamma_{2n+1,m}}$-sets under $AD$ and show there is no sequence of distinct $\bf{\Gamma_{2n+1}}$-sets of length $\bf{\delta^1_{2n+3}}$. As a special case, there is no…
For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…
A natural model of read-once linear branching programs is a branching program where queries are $\mathbb{F}_2$ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we…
The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special…
Given an odd integer polynomial f(x) of a degree k >=3, we construct a non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of f(x) not greater than n, and a small free coefficient…