arXiv subjects
Publications and source records attributed to Krystal Maughan.
In this short note, we give a method for computing a non-torsion point of smallest canonical height on a given elliptic curve $E/\mathbb{Q}$ over all number fields of a fixed degree. We then describe data collected using this method, and investigate related conjectures of Lehmer and Lang using these data.
Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree $\ell$, their structure has been investigated graph-theoretically. We generalise the notion of $\ell$-isogeny graphs to $L$-isogeny graphs (studied in the prime field case by Delfs and Galbraith), where $L$ is a set of small primes dictating the allowed isogeny degrees in the graph. We analyse the graph-theoretic structure of $L$-isogeny graphs. Our approaches may be put into two categories: cycles and graph cuts. On the topic of cycles, we provide: a count for the number of cycles in the $L$-isogeny graph with cyclic kernels using traces of Brandt matrices; an efficiently computable estimate based on this approach; and a third ideal-theoretic count for a certain subclass of $L$-isogeny cycles. We provide code to compute each of these three counts. On the topic of graph cuts, we compare several algorithms to compute graph cuts which minimise a measure called the edge expansion, outlining a cryptographic motivation for doing so. Our results show that a greedy neighbour algorithm out-performs standard spectral algorithms for computing optimal graph cuts. We provide code and study explicit examples. Furthermore, we describe several directions of active and future research.
Motivated by cryptographic applications, we investigate two machine learning approaches to modular multiplication: namely circular regression and a sequence-to-sequence transformer model. The limited success of both methods demonstrated in our results gives evidence for the hardness of tasks involving modular multiplication upon which cryptosystems are based.
Surveys are an important tool for many areas of social science research, but privacy concerns can complicate the collection and analysis of survey data. Differentially private analyses of survey data can address these concerns, but at the cost of accuracy - especially for high-dimensional statistics. We present a novel privacy mechanism, the Tabular DDP Mechanism, designed for high-dimensional statistics with incomplete correlation. The Tabular DDP Mechanism satisfies dependent differential privacy, a variant of Pufferfish privacy; it works by building a causal model of the sensitive data, then calibrating noise to the level of correlation between statistics. An empirical evaluation on survey data shows that the Tabular DDP Mechanism can significantly improve accuracy over the Laplace mechanism.
As AI-based systems increasingly impact many areas of our lives, auditing these systems for fairness is an increasingly high-stakes problem. Traditional group fairness metrics can miss discrimination against individuals and are difficult to apply after deployment. Counterfactual fairness describes an individualized notion of fairness but is even more challenging to evaluate after deployment. We present prediction sensitivity, an approach for continual audit of counterfactual fairness in deployed classifiers. Prediction sensitivity helps answer the question: would this prediction have been different, if this individual had belonged to a different demographic group -- for every prediction made by the deployed model. Prediction sensitivity can leverage correlations between protected status and other features and does not require protected status information at prediction time. Our empirical results demonstrate that prediction sensitivity is effective for detecting violations of counterfactual fairness.
Group fairness metrics can detect when a deep learning model behaves differently for advantaged and disadvantaged groups, but even models that score well on these metrics can make blatantly unfair predictions. We present smooth prediction sensitivity, an efficiently computed measure of individual fairness for deep learning models that is inspired by ideas from interpretability in deep learning. smooth prediction sensitivity allows individual predictions to be audited for fairness. We present preliminary experimental results suggesting that smooth prediction sensitivity can help distinguish between fair and unfair predictions, and that it may be helpful in detecting blatantly unfair predictions from "group-fair" models.
Deep learning has produced big advances in artificial intelligence, but trained neural networks often reflect and amplify bias in their training data, and thus produce unfair predictions. We propose a novel measure of individual fairness, called prediction sensitivity, that approximates the extent to which a particular prediction is dependent on a protected attribute. We show how to compute prediction sensitivity using standard automatic differentiation capabilities present in modern deep learning frameworks, and present preliminary empirical results suggesting that prediction sensitivity may be effective for measuring bias in individual predictions.