Research Agenda
Current Work: Rotation Symmetric Boolean Functions
My work focuses on the interaction between two areas: Cryptography and Representation Theory. In cryptography, digital signatures are used for authentication and nonrepudiation of messages. When it comes to large messages (and with it, large signatures), a hash function is used to create a "fingerprint" that can be used for verification, and thus saves time!
If we focus on the efficiency of these hash functions, we find one useful property to have is that they be rotationally symmetric. This means the result we get when we pass a message through this function is the same if we pass the same message but all the characters are shifted to the left or right.
The most-used family of hashing algorithms (MD4 or MD5 family) has a structure which contains a function, which can be thought of as multiple boolean functions (functions whose output is either a 0 or 1). It turns the algorithm would save significant time if these boolean functions were rotationally symmetric.
My current research explores a matrix with comes from studying the Walsh spectrum of rotationally symmetric boolean function. This matrix is used in the calculations of properties of boolean functions, such as nonlinearity and algebraic immunity. My goal was to answer the question posed on my past advisor's paper: "Determine the distinct eigenvalues in exact quantities of this matrix".
At the same time, this matrix has certain mathematical values and properties similar to character tables that you see in representation of finite groups. In fact, we can think of this rotation action as a semi-direct product of two groups, one is the group of 0's and 1's with length similar to that of the message and one being the group of left rotations. The character table from the product coincides with the matrix in all non-complex parts. This is another direction I am exploring this matrix with to better understand its construction and perhaps consider other symmetries.
Past Work: Double-Gyre Flow
We considered a time-dependent, wind-driven, stochastic double-gyre flow, and investigated the interaction between the flow and coupled particles. It is known that noise can cause individual particles to escape from one gyre to another gyre. By computing the Lagrangian coherent structures (LCS) of the system, one can determine regions of erratic behavior (high and low probability regions of particle escape). If we couple two particles together, it is possible to study the effects on particle escape for each scenario. In particular, for intermediate coupling, one can determine if the coupling force decreases or increases the escape time of the particles.
Presentations
Matrix of Rotation Symmetric Boolean Function and its Eigenvalue, The Ohio State University Denison Conference, May 2022
Teaching Mathematics to Deaf Students, University of Iowa GAUSS, April 2022
Expository on Representation of Finite Groups and Characters, University of Iowa Algebra Seminar, February 2022
Rotation Symmetric Boolean Functions and its Matrix, University of Iowa GAUSS, September 2021
Dynamics of Coupled Particles In a Double-Gyre Flow , University of Iowa GAUSS, September 2019
Dynamics of Coupled Particles In a Double-Gyre Flow , Student Research Symposium, March 2018
