A Note On Square-free Commuting Probabilities of Finite Rings
It is shown that the commuting probability of a finite ring cannot be a fraction with square-free denominator, resolving a conjecture of Buckley and MacHale.
math.RA↗
arXiv subjects
Andrew Mendelsohn
Publications and source records attributed to Andrew Mendelsohn.
Sum-Product Bounds & an Inequality for the Kissing Number in Dimension 16
We obtain an inequality for the kissing number in 16 dimensions. We do this by generalising a sum-product bound of Solymosi and Wong for quaternions to a semialgebra in dimension 16. In particular, we obtain the inequality $$k_{16}\geq \frac{\sum_{x \in \mathcal{R}}\left|\mathcal{S}_{x}\right|}{\left|\bigcup_{x \in \mathcal{R}} \mathcal{S}_{x}\right|}-1,$$ where $k_{16}$ is the 16-dimensional kissing number, and $S_x$ and $\mathcal{R}$ are sets defined below. Along the way we also obtain a sum-product bound for subsets of the octonions which are closed under taking inverses, using a similar strategy to that used for the quaternions. We use the fact that the kissing number in eight dimensions is 240 to achieve the result. Namely, we obtain the bound $$\operatorname{max}(|\mathcal{A}+\mathcal{A}|,|\mathcal{A}\mathcal{A}|)\geq \frac{|\mathcal{A}|^{4/3}}{(1928\cdot\lceil \operatorname{log}|\mathcal{A}|\rceil)^{1/3}},$$ where $\mathcal{A}$ is a finite set of octonions such that if $x\in\mathcal{A}$, $x^{-1}\in\mathcal{A}$ also.
math.CO↗
An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields
Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with discriminant and regulator $Δ_K, R_K$ respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by $O( Δ_K \exp(2n \log(n)+f(n,R_K)))$, where $f(n,R_K)$ is a finite value, satisfying $f(n,R_K)=\frac{\sqrt{n-1}}{2}R_K^{\frac{1}{n-1}}+\frac{4}{n-1}\log(\sqrt{|Δ_K|})^2$ if $n \leq 11$. Moreover, if $K$ is a unit reducible field, the number of classes of perfect unary forms is bound above by $O( Δ_K \exp(2n \log(n)))$.
math.NT↗
Subfield Algorithms for Ideal- and Module-SVP Based on the Decomposition Group
Whilst lattice-based cryptosystems are believed to be resistant to quantum attack, they are often forced to pay for that security with inefficiencies in implementation. This problem is overcome by ring- and module-based schemes such as Ring-LWE or Module-LWE, whose keysize can be reduced by exploiting its algebraic structure, allowing for faster computations. Many rings may be chosen to define such cryptoschemes, but cyclotomic rings, due to their cyclic nature allowing for easy multiplication, are the community standard. However, there is still much uncertainty as to whether this structure may be exploited to an adversary's benefit. In this paper, we show that the decomposition group of a cyclotomic ring of arbitrary conductor can be utilised to significantly decrease the dimension of the ideal (or module) lattice required to solve a given instance of SVP. Moreover, we show that there exist a large number of rational primes for which, if the prime ideal factors of an ideal lie over primes of this form, give rise to an "easy" instance of SVP. It is important to note that the work on ideal SVP does not break Ring-LWE, since its security reduction is from worst case ideal SVP to average case Ring-LWE, and is one way.
cs.CR↗