arXiv subjects

David Thomson

Publications and source records attributed to David Thomson.

4 recordsLinked to original sources

arXiv2019

A note on depth-$b$ normal elements

In this paper we study elements $\beta \in \mathbb{F}_{q^n}$ having normal $\alpha$-depth $b$; that is, elements for which $\beta, \beta - \alpha, \ldots, \beta-(b-1)\alpha$ are simultaneously normal elements of $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$. In [1], the authors present the definition of normal $1$-depth but mistakenly present results for normal $\alpha$-depth for some fixed normal element $\alpha \in \mathbb{F}_{q^n}$. We explain this discrepancy and generalize the given definition of normal $(1-)$depth from [1] as well as answer some open questions presented in [1].

math.NT↗

arXiv2018

Unicyclic Strong Permutations

In this paper, we study some properties of a certain kind of permutation $\sigma$ over $\mathbb{F}_{2}^{n}$, where $n$ is a positive integer. The desired properties for $\sigma$ are: (1) the algebraic degree of each component function is $n-1$; (2) the permutation is unicyclic; (3) the number of terms of the algebraic normal form of each component is at least $2^{n-1}$. We call permutations that satisfy these three properties simultaneously unicyclic strong permutations. We prove that our permutations $\sigma$ always have high algebraic degree and that the average number of terms of each component function tends to $2^{n-1}$. We also give a condition on the cycle structure of $\sigma$. We observe empirically that for $n$ even, our construction does not provide unicylic permutations. For $n$ odd, $n \leq 11$, we conduct an exhaustive search of all $\sigma$ given our construction for specific examples of unicylic strong permutations. We also present some empirical results on the difference tables and linear approximation tables of $\sigma$.

cs.CR↗

arXiv2017

Existence of primitive $1$-normal elements in finite fields

An element $\alpha \in \mathbb F_{q^n}$ is \emph{normal} if $\mathcal{B} = \{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ forms a basis of $\mathbb F_{q^n}$ as a vector space over $\mathbb F_{q}$; in this case, $\mathcal{B}$ is a normal basis of $\mathbb F_{q^n}$ over $\mathbb F_{q}$. The notion of $k$-normal elements was introduced in Huczynska et al (2013). Using the same notation as before, $\alpha$ is $k$-normal if $\mathcal{B}$ spans a co-dimension $k$ subspace of $\mathbb F_{q^n}$. It can be shown that $1$-normal elements always exist in $\mathbb F_{q^n}$, and Huczynska et al (2013) show that elements that are simultaneously primitive and $1$-normal exist for $q \geq 3$ and for large enough $n$ when $\gcd(n,q) = 1$ (we note that primitive $1$-normals cannot exist when $n=2$). In this paper, we complete this theorem and show that primitive, $1$-normal elements of $\mathbb F_{q^n}$ over $\mathbb F_{q}$ exist for all prime powers $q$ and all integers $n \geq 3$, thus solving Problem 6.3 from Huczynska, et al (2013).

math.NT↗

arXiv2005

UV Luminosity Function at z~4, 3, and 2

We use very deep (R_lim=27) UGRI imaging to study the evolution of the faint end of the UV-selected galaxy luminosity function from z~4 to z~2. We find that the number of sub-L* galaxies increases from z~4 to z~3 while the number of bright ones appears to remain constant. We find no evidence for continued evolution to lower redshift, z~2. If real, this differential evolution of the luminosity function suggests that differentially comparing key diagnostics of dust, stellar populations, etc. as a function of z and L may let us isolate the key mechanisms that drive galaxy evolution at high redshift and we describe several such studies currently underway.

astro-ph↗