Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
individual
Campus
Daytona Beach
Authors' Class Standing
Osasumwen Omobude, Senior
Lead Presenter's Name
Osasumwen Omobude
Lead Presenter's College
DB College of Engineering
Faculty Mentor Name
Dr. Ronald Adams
Abstract
This project investigates determinants and matrix invertibility in finite modular systems, focusing on matrices over Zn. Using the Hill cipher as context, it examines the algebraic conditions under which a matrix is invertible in modular arithmetic. In particular, the project studies how the determinant determines invertibility, showing that a matrix over Zn is invertible if and only if its determinant is coprime with n. The project further compares invertibility over the real numbers with invertibility over modular systems, highlighting the distinction between prime moduli Zp and composite moduli. In the prime case, matrices behave similarly to those over fields, where a nonzero determinant guarantees invertibility. In contrast, when n is composite, additional number-theoretic conditions are required, and not all nonzero determinants yield invertible matrices. The aim is to develop a deeper understanding of how linear algebra concepts extend to modular arithmetic and to explore their significance in matrix-based cryptographic systems.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Algebra Commons, Cybersecurity Commons, Number Theory Commons
Determinants and Invertibility in Finite Modular Systems
This project investigates determinants and matrix invertibility in finite modular systems, focusing on matrices over Zn. Using the Hill cipher as context, it examines the algebraic conditions under which a matrix is invertible in modular arithmetic. In particular, the project studies how the determinant determines invertibility, showing that a matrix over Zn is invertible if and only if its determinant is coprime with n. The project further compares invertibility over the real numbers with invertibility over modular systems, highlighting the distinction between prime moduli Zp and composite moduli. In the prime case, matrices behave similarly to those over fields, where a nonzero determinant guarantees invertibility. In contrast, when n is composite, additional number-theoretic conditions are required, and not all nonzero determinants yield invertible matrices. The aim is to develop a deeper understanding of how linear algebra concepts extend to modular arithmetic and to explore their significance in matrix-based cryptographic systems.