SUMS 2024

Math and Cybersecurity

Saturday 16 March 2024

# Plenary Talks

## Jeff Hoffstein (Brown)

Title: Digital Signatures based on lattice problems

Abstract: A digital signature is a strange idea. There is a public key which everyone knows, and a private key that only the person signing a digital document knows. Somehow the digital document gets scrambled with the private key, by the person who knows the private key, and out the other end comes a 128 bit number called a signature. This has to have two properties: Anyone who knows the document and the public key can verify that the person who created the signature knew the private key. However, the signature must leak no information about the private key.

At the root of this lurks a hard mathematical problem, and the problem I'll be talking about is the closest vector problem for high dimensional lattices. A lattice is a vector space, except the only coefficients you are allowed to multiply by are integers. I'll assume no prior knowledge and hopefully get to be able to say some interesting things.

## Anna Lysyanskaya (Brown)

Title: Towards a Privacy-Preserving Digital Dollar

Abstract: When accessing a service, we often need to present authorization credentials or tokens. In this talk, we will give an overview of how to create such tokens in such a way that they cannot be forged, and can be used anonymously but only up to a certain limit, or at a limited rate, with applications to privacy-preserving electronic cash. We will also show how to ensure that, even as the privacy of law-abiding users can be protected, activities that fit well-defined and agreed-upon criteria (such as when a user violates their terms of service) mathematically force the user to reveal the information that de-anonymizes this user. Thus, we can achieve the best of both worlds: privacy for rule followers, and accountability for rule breakers. Along the way, we will also introduce and make use of a few fundamental tools from the cryptographer's toolbox such as digital signature schemes, pseudorandom functions, zero-knowledge proofs, and secure two-party computation.

## Rachel Cummings (Columbia)

Title: Differential Privacy: State of the Art and Challenges

Abstract: Privacy concerns are becoming a major obstacle to using data in the way that we want. It's often unclear how current regulations should translate into technology, and the changing legal landscape surrounding privacy can cause valuable data to go unused. In this talk, we will explore differential privacy as a tool for providing strong privacy guarantees, while still making use of potentially sensitive data. Differential privacy is a parameterized notion of database privacy that gives a mathematically rigorous worst-case bound on the maximum amount of information that can be learned about an individual's data from the output of a computation. In the past decade, the privacy community has developed algorithms that satisfy this privacy guarantee and allow for accurate data analysis in a wide variety of computational settings, including machine learning, optimization, statistics, and economics. This talk will first give an introduction to differential privacy, and then survey recent advances and future challenges in the field of differential privacy.

## Jay Servidea (NSA)

Title: Lessons in Cybersecurity

Abstract: Education is the backbone of building strong cybersecurity professionals and informed citizens. In this talk, I'll discuss some of the biggest cybersecurity challenges the world faces and try to frame them into a historical context.

# Student Talks

### Jake Ginesin (Northeastern)

Title: On Computability and Automated Program Verification

Abstract: Starting from the basic principles of computation as established by Alan Turing, we explore automated program verification. Briefly, we outline the mathematical tools involved in automated reasoning about programs, which includes temporal logics, automata theory, and complexity classes. We then characterize the expressiveness and performance of automated program verification tools in their ability to analyze practical systems such as internet protocols.

### Aidan Hennessey (Brown)

Title: Many-twist Moebius bands

Abstract: Take a strip of paper, add a half-twist, and then tape the ends together. This is a paper Mobius band. With a long strip of paper, this is easy. As your aspect ratio decreases, however, reconnecting the ends becomes harder. Last summer, Prof Rich Schwartz proved that this is impossible if your aspect ratio is less than or equal to sqrt(3). For more than 2 half-twists, determining the minimal aspect ratio is an open problem. I present a construction which allows you to add as many half-twists as you want without the aspect ratio going above 8. This shows that the minimal aspect ratio stays bounded as the number of half twists goes to infinity.

### Stephanie Wang (University of Rochester)

Title: Enhanced Reliability Predictions: An AI-based Framework for Performing Failure Modes, Effects, and Criticality Analysis in Industrial Environments

Abstract: Reliability engineering grapples with the imperative task of predicting and comprehending product failures across an array of sectors. Although traditional approaches like Failure Mode Effects and Analysis (FMEA) and Failure Mode, Effects, and Criticality Analysis (FMECA) have substantially bolstered product safety and quality, their inherent manual intensity and reliance on expert insights introduce limitations. As today's systems surge in complexity, a more holistic approach becomes paramount. In response, we introduce an AI-driven risk assessment tool that guides the user to a host of failure modes and their effects for each component contained in a bigger system. Through a user-friendly graphical interface and a robust statistical modeling backend, the AI-driven tool streamlines the risk assessment process by prompting users to input a system's name and subsequently generating an extensive array of failure modes and associated effects for each constituent component, including Weibull, Rayleigh, and Bathtub distribution curves. By automating this aspect of FMEA/FMECA, the AI-based solution seeks to not only enhance reliability analyses but also optimize development timelines, improve resource allocation, and provide valuable educational avenues for junior across sectors, including chemical, automotive, aerospace, and beyond."

### Aaron Lin (Brown)

Title: Radial Symmetry in PDE's

Abstract: We can't solve PDE's explicitly in most cases. However, we can often deduce some amazing properties of solutions a priori. In this talk, I'll demonstrate how classical techniques like the moving plane method and the narrow region principle reveal that solutions to certain elliptic PDEs are radially symmetric.

### Mattie Ji (Brown)

Title: On the Injectivity of Euler Integral Transforms with Hyperplanes and Quadric Hypersurfaces

Abstract: The Euler characteristic transform (ECT) is an integral transform used widely in topological data analysis. Previous efforts by Curry et al. and Ghrist et al. have independently shown that the ECT is injective on all compact definable sets. In this work, we study the injectivity of the ECT on definable sets that aren't necessarily compact. We then introduce the quadric Euler characteristic transform (QECT) as a natural generalization of the ECT by detecting definable shapes with quadric hypersurfaces rather than hyperplanes. We also discuss some criteria for the invertibility of QECT.

### Ian Benway (Brown)

Title: Heat Maps and Voronoi Diagrams on Generalized Knight Metrics

Abstract: Arising from a failed effort to get better at chess, I look at ways of visualizing knight movements on a chess board. A knight moves one square in one direction and two squares in another, which I call a (1,2)-move. We casually explore the visually stunning results of using more general (n,m)-moves, and even (f(t), g(t))-moves.

### Cassie Ding (Brown)

Title: Bananas, Eggs, and Chips, a recipe for graph gonality

Abstract: In 2008, Baker pioneered the study of the gonality of algebraic curves in the context of discrete finite graphs. This concept, known as graph gonality, has since been extensively investigated, especially in relation to chip-firing games on graphs. The significance of this study lies in the fact that the gonality on metric graphs provides a lower bound for the associated algebraic curves, making the computation of graph gonality a valuable tool for examining algebraic curves. In our research, we delve into the gonality of specific families of knight's, king's, and bishop's graphs, including their toroidal variations. Recent studies have demonstrated that graph gonality can vary based on graph operations like subgraphs, minors, and uniform subdivisions (Josse van Dobben de Bruyn et al 2021). Our exploration of banana graphs, which are path graphs with possibly multiple edges, led to a new discovery about a certain property of graph gonality. Namely, we establish that graph gonality can increase or decrease with the addition or deletion of a single edge by any integer through the use of Dhar’s burning algorithm and modular arithmetic. We also present some related results concerning the computational complexity of establishing lower bounds for graph gonality.

### Annette Belleman (Wellesley)

Title: Knot too Bad: On the Symmetries of Chain Links and their FAL-Equivalent Counterparts

Abstract: The talk will be on knot theory, specifically on symmetry groups of links. The focus is on research I did at an REU on chain links and a class of links related to chain links. We will first begin with a brief introduction to the necessary knot theory to understand the research question and the focus on chain links. Then we will explain several key lemmas with motivating examples and finally end with a theorem that gives us all the possible symmetry groups for the links in question.

### Ignas Karvelis (Brown)

Title: Dynamic One Time Passwords

Abstract: The talk would essentially talk about password security and the approach of generating random numbers. It is inspired by the unique pin code generators in my home country Lithuania for banking. A thing that is not prevalent in the United States. The talk would explore the benefits of Dynamic One Time Passwords (OTPs). As well as, many other topics that are not as mainstream in the United States.

### Abi Bowering (Smith)

Title: System identification with neural networks in a geothermal borehole

Abstract: Artificial neural networks are a type of machine learning. A neural network has an input and an output layer, often with numerous hidden layers - leading to the term “deep learning”. Each layer performs a basic function (Wa+b), adding and rescaling the network’s weights and biases, otherwise known as the network parameters. To train a network, we optimize a loss function of the parameters so that it reaches a minimum.

System identification is an application of neural networks to approximate a system’s governing equations only using observed data. This project utilized the inverse problem, in which we have solutions at various time steps and want to recover the system of ordinary differential equations which produced them. In particular, we attempt to recreate an equation which describes the change in temperature over time in a borehole for the Smith Geothermal Project. Borehole temperature data were collected each foot for 1000 feet of depth, every hour for over a year. In Python, we coded a feedforward network to learn the data. The observation arrays were our training data inputs, and their spline-smoothed derivatives were the target data. Differential equation reconstruction was used to validate our results. Future work will include using recurrent network structures and regularization terms.

### Devin Brown (Northeastern)

Title: Combinatorial Cactus Actions, Crystals, and Toggles

Abstract: Kashiwara crystals, visualized as directed graphs with colored edges, arise in the study of representations of quantum groups. They have numerous applications across mathematics as well as in statistical mechanics and particle physics. Using tools from algebraic combinatorics like heaps and reverse plane partitions, we compare two different actions on crystals to explore their deeper symmetries. On one hand, the cactus group acts by Lusztig’s involutions which generalize the Schützenberger involution in type A. On the other hand, one can define the action of so-called combinatorial toggles on crystals, which can be visualized as adding or removing beads from an abacus. We show that the cactus action factors through the toggle action for a certain class of crystals. At SUMS we will give an overview of crystals and their related combinatorics and illustrate our main results through examples.