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Dr. Pamela E. Harris (University of Wisconsin-Milwaukee)
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What is mathematical research? How does a mathematician find problems to work on? How does one build mathematical collaborations? In this talk, I will share my journey to research mathematics, what it entails, how I have developed new research ideas, and how I have found my place within the mathematical community. Mathematical topics of discussion will include lattice point visibility, parking functions, and a connection between vector partition functions and juggling. No prior mathematical background on these topics is expected nor assumed as we will introduce all the needed concepts from first principles. All that is needed is the willingness to wonder and ask the question: “what happens if…?
A useful feature of elliptic curve cryptography is the ability to "hash into" the set of public keys: a user can easily compute a uniformly random public key in a way so that no one (not even the user!) knows the corresponding private key. It is not known whether isogeny-based cryptography enjoys this functionality. In isogeny-based cryptosystems, a private key is an isogeny from a fixed public supersingular elliptic curve, and the corresponding public key is the codomain of that isogeny, another supersingular elliptic curve. Public keys can be thought of as vertices in the supersingular isogeny graph, and private keys are random walks (beginning at the fixed public curve) in that graph. Whether one can "hash into" the supersingular isogeny graph is an open question: there is no known method for quickly computing a random supersingular elliptic curve that does not also provide, as a byproduct, enough information to easily find a path from that curve to the fixed public base curve. This byproduct is the ring of endomorphisms of the supersingular elliptic curve: there is no known efficient algorithm for computing the endomorphism ring, and computing endomorphism rings is exactly as hard as path-finding in isogeny graphs. There are isogeny-based cryptosystems that can only be securely instantiated given a single supersingular elliptic curve with unknown endomorphism ring - a supersingular curve you can trust.
In this talk, I will discuss a practical, distributed protocol for computing a supersingular elliptic curve with unknown endomorphism ring. The main ingredient in this protocol is another protocol, a proof of knowledge, that proves knowledge of a secret isogeny. We show this proof of knowledge is statistically zero-knowledge (the protocol does not reveal anything other than the fact that the prover knows the secret information). To prove that our proof of knowledge is statistically zero-knowledge, we introduce isogeny graphs with level structure and prove that these graphs are Ramanujan graphs: random walks mix as rapidly as possible. In this talk, I will discuss isogeny-based cryptography at a high level and explain our distributed protocol and proof of knowledge. I will then explain why the Ramanujan-Peterrson conjecture implies that our isogeny graphs with level structure have the Ramanujan property, and hence why our proof of knowledge is statistically zero-knowledge. This is based on joint work with Basso, Codogni, Connolly, De Feo, Fouotsa, Lido, Panny, Patranabis, and Wesolowski.
(NOTE SPECIAL TIME) Supersingular curves you can trust
Tue, Oct. 22 12:10pm (MATH …
Daniel Martin (PhD '20) (Clemson)
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“Lattice problems” drew significant interest from cryptographers in the late 90s. The belief was (and still is) that they offer an easy way to encrypt information that comes with a hard-to-hack guarantee. The popularity of lattice problems continued to rise in subsequent decades, and today they are the darlings of so-called post-quantum cryptography. This talk will show you what the fuss is all about. We will explore how lattice-based encryption works, what advantages it has over other systems, and how it invites the algebraic number theorist to the world of cryptography. No prior understanding of cryptography or number theory is necessary.
Networks are everywhere, from social media platforms to intricate systems in nature. But how do we study and understand them? Different fields like computer science, math, and statistics offer their own tools and views. In this talk, we'll explore how these areas approach network analysis and how mathematicians formalize intuition and results from other areas.
The talk will cover joint works with Rémy Sanchis (UFMG), Roberto I. Oliveira (IMPA) and Caio Alves (Oak Ridge National Lab).
Interdisciplinary Perspectives on Network Analysis: A Confluence of Computer Science, Mathematics, and Statistics
How to choose your own mathematical adventures