Searcharxiv⌕ Search

arXiv subjects

Gohar M. Kyureghyan

Publications and source records attributed to Gohar M. Kyureghyan.

8 recordsLinked to original sources
arXiv2024

Factorization and irreducibility of composed products

Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees $m$ and $n$ is again irreducible if and only if $m$ and $n$ are coprime and the involved diamond product satisfies a special cancellation property, the so-called conjugate cancellation. This completes the characterization of irreducible composed products, considered in several previous papers. More generally, we give precise criteria when a diamond product satisfies conjugate cancellation. For diamond products defined via bivariate polynomials, we prove simple criteria that characterize when conjugate cancellation holds. We also provide efficient algorithms to check these criteria. We achieve stronger results as well as more efficient algorithms in the case that the polynomials are bilinear. Lastly, we consider possible constructions of normal elements using composed products and the methods we developed.

math.NT↗
arXiv2023

On the size of integer programs with bounded non-vanishing subdeterminants

Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the column number question for integer matrices whose every $r \times r$ minor is non-zero and bounded by a fixed constant $Δ$ in absolute value. Approaching the problem in two different ways, one that uses results from coding theory, and the other from the geometry of numbers, we obtain linear and asymptotically sublinear upper bounds on the maximal number of columns of such matrices, respectively. We complement these results by lower bound constructions, matching the linear upper bound for $r=2$, and a discussion of a computational approach to determine the maximal number of columns for small parameters $Δ$ and $r$.

math.CO↗
arXiv2023

Constructing irreducible polynomials recursively with a reverse composition method

We suggest a construction of the minimal polynomial $m_{β^k}$ of $β^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_β$ for all positive integers $k$ whose prime factors divide $q-1$. The computations of our construction are carried out in $\mathbb F_q$. The key observation leading to our construction is that for $k \mid q-1$ holds $$m_{β^k}(X^k) = \prod_{j=1}^{\frac kt} ζ_k^{-jn} f (ζ_k^j X),$$ where $t= \max \{m\mid \gcd(n,k): f (X) = g (X^m), g \in \mathbb F_q[X]\}$ and $ζ_{k}$ is a primitive $k$-th root of unity in $\mathbb F_q$. The construction allows to construct a large number of irreducible polynomials over $\mathbb F_q$ of the same degree. Since different applications require different properties, this large number allows the selection of the candidates with the desired properties.

math.NT↗
arXiv2021

Image sets of perfectly nonlinear maps

We consider image sets of differentially $d$-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution, by extending methods used for planar maps. We apply the results to study $d$-uniform Dembowski-Ostrom polynomials. Further, we focus on a particularly interesting case of APN maps on binary fields. We show that APN maps with the minimal image size must have a very special preimage distribution. We prove that for an even $n$ the image sets of several well-studied families of APN maps are minimal. We present results connecting the image sets of special maps with their Walsh spectrum. Especially, we show that the fact that several large classes of APN maps have the classical Walsh spectrum is explained by the minimality of their image sets. Finally, we present upper bounds on the image size of APN maps.

math.CO↗
arXiv2020

A recurrent construction of irreducible polynomials of fixed degree over finite fields

In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an irreducible polynomial of degree n and order 2^rt with t odd, the construction produces ord_t(2)/d irreducible polynomials of degree n and order t for a certain divisor d of n. The construction can be used, for example, to search irreducible polynomials with specific requirements on its coefficients.

math.NT↗
arXiv2013

On Inversion in Z_{2^n-1}

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), which for a fixed positive integer d maps integers n\geq 1 to the least positive residue of the inverse of d modulo 2^n-1, if it exists. In particular, we showed that the function \de is completely determined by its values for 1 \leq n \leq \ordb, where \ordb is the order of 2 modulo the largest odd divisor of d.

math.NT↗