Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
The aim of the talk is to give a nice visual introduction to the world of knot theory. This talk provides the foundation of a new cryptographic protocol, involving knots. This is joint work with Silvia Sconza.
Join the Zoom Meeting at 3 PM (CST UT-6)
Meeting ID: 958 5298 7437
Passcode: 922447
In this talk, we start providing a friendly introduction to Cryptography, focusing on the (Generalized) Diffie-Hellman Key Exchange. Building on this foundation, we present a novel cryptographic protocol that utilizes the semigroup of oriented knots under the connected sum operation. The shared secret key is derived from a knot invariant, computed from two distinct representations of the same knot. The security of this protocol relies on two computationally hard problems: the Decomposition Problem, which involves determining the prime decomposition of a knot, and the Recognition Problem, which asks whether two knot diagrams represent the same knot. This is a joint work with Arno Wildi.
Join the Zoom Meeting at 3 PM (CST UT-6)
Meeting ID: 958 5298 7437
Passcode: 922447
We will discuss several algebraic public key exchange proposals involving semigroups, highlighting similarities and differences and the best known attacks on each.
Join the Zoom Meeting at 3 PM (CST UT-6)
Meeting ID: 958 5298 7437
Passcode: 922447
Let k ≥ 2 be an even integer. Let q be a prime power such that q ≡ k+1 (mod 2k). We define the k-th power Paley digraph of order q, Gk(q), as the graph with vertex set 𝔽q where a → b is an edge if and only if b−a is a k-th power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of transitive subtournaments of order four contained in G_k(q), 𝒦4(Gk(q)), which holds for all k. We also provide a formula, in terms of Jacobi sums, for the number of transitive subtournaments of order three contained in Gk(q), 𝒦3(G_k(q)). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of 𝒦4(Gk(q)) (resp. 𝒦3(Gk(q))) yield lower bounds for the multicolor directed Ramsey numbers Rk⁄2(4)=R(4,4,...,4) (resp. Rk⁄2(3)). We state explicitly these lower bounds for k ≤ 10 and compare to known bounds, showing improvement for R2(4) and R3(3). Combining with known multiplicative relations we give improved lower bounds for Rt(4), for all t ≥ 2, and for Rt(3), for all t ≥ 3.
We will discuss p-adic numbers, the p-adic absolute value, completions, and additive valuations in order to understand Hensel's Lemma and its applications.
We discuss a new notion of lifting modules over a Noetherian local ring to a regular local ring. We define a module to be Serre liftable if it lifts to a module of correct dimension over the regular ring. We will discuss an interesting property of such Serre liftable modules: a finite length Serre liftable module must have length at least the Hilbert-Samuel multiplicity of the local ring. This project is joint with Nawaj KC.
We will discuss the Rubik's Cube and other similar combination puzzles, as well as God's Number, how it was found, and finally the Devil's Algorithm.