Kara Fagerstrom | Searcharxiv

arXiv2026

Cohomological support varieties for monomial ideals

Let $R$ be a local or positively graded ring with a regular presentation $R \cong Q/I$ where $I$ is a monomial ideal generated by $n$ elements on a regular sequence. In Briggs-Grifo-Pollitz (2025), the authors classify the cohomological support varieties $\mathcal{V}_R(R)$ for $n \leqslant 5$. In this paper we extend their results to classify the varieties that can occur as $\mathcal{V}_R(R)$ for $n=6$. Moreover, we provide two families of rings, one realizing cohomological support varieties of unbounded codimension, the other realizing an unbounded number of components. Finally, we answer a question of Gintz (2026) about the varieties that occur as $\mathcal{V}_R(R)$ where $I$ is given by the edge ideal of a cycle.

math.AC↗

arXiv2021

Probability Distributions for Elliptic Curves in the CGL Hash Function

Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to an elliptic curve by traversing an input-determined path through an isogeny graph. The nodes of an isogeny graph are elliptic curves, and the edges are special maps betwixt elliptic curves called isogenies. Knowing which hash values are most likely informs us of potential security weaknesses in the hash function. We use stochastic matrices to compute the expected probability distributions of the hash values. We generalize our experimental data into a theorem that completely describes all possible probability distributions of the CGL hash function. We use this theorem to evaluate the collision resistance of the CGL hash function and compare this to the collision resistance of an "ideal" hash function.

cs.CR↗