Invited Talk by Dr. Maria Francis on Polynomial Rings over Commutative Rings - Algorithmic techniques and Applications

Title: Polynomial Rings over Commutative Rings - Algorithmic techniques and Applications
Speaker: Dr. Maria Francis
Host Faculty:  Dr. N R Aravind
Room No: A-220, academic block A.


 Polynomial rings over  commutative rings have applications in several areas like cryptography, control theory, coding theory and algebraic geometry. For example, in lattice based cryptography, most arithmetic operations are over integers, and in control theory parametric equations with polynomials themselves as coefficients are very common.

Algorithmic techniques for polynomial rings over fields are well studied with Groebner bases being one of the fundamental tools. Even though various approaches have been proposed to extend Groebner bases theory to polynomial rings over rings, these techniques have only looked at extending basic definitions and concepts.

In this talk, I will present an algorithmic characterization that allows us to extend certain fundamental results in the theory of Groebner bases like the Macaulay-Buchberger basis theorem to polynomial rings over rings.  I will also present an algorithmic method to compute the Krull dimension of residue class rings of polynomial rings. Finally, I will show how these techniques can be used to extend the notion of ideal lattices to the multivariate case, an important class of lattices that have several applications in lattice based cryptography. I will also show how to build collision resistant hash functions using these ideal lattices. This is a joint work with Prof. Ambedkar Dukkipati and ongoing work with Prof. Manuel Kauers.

Speaker bio:

Maria Francis is a postdoctoral researcher at the Institute for Algebra at the Johannes Kepler University (JKU), Linz, Austria where she works under the supervision of Prof. Manuel Kauers. Previously, she was a doctoral student at the Computer Science and Automation department at the Indian Institute of Science (IISc), Bangalore, India. Her research supervisor was Dr.Ambedkar Dukkipati. Her research interests are in Computer Algebra, Symbolic Computation, Commutative Algebra and Lattice Cryptography.

Friday, January 12, 2018 - 12:00