Developing linear algebraic techniques for analysis of non-linear dynamical systems over finite state space.
Observer theory for non-linear systems.
Koopman Operator and its reductions - applications to systems theory and control design.
Application specific interests:
Systems biology - Developing computational framework for analysis of genetric networks, biochemical reactions.
Boolean networks - Analysis, computations and control.
Applications of systems theory to communication networks, cryptanalysis.
My current research focus has been on the intersection of non-linear dynamical systems, finite fields and linear algebra. Dynamical systems evolving over a finite state space (such as a finite set or vector space over a finite field) are highly useful in modeling of communication networks, cryptographic protocols, gene-transcription networks, biochemical reactions, finite state automata to name a few. Current tools for analysis of such non-linear systems do not scale up efficiently with the system dimension and my research focused on developing frameworks for analysis of nonlinear dynamical systems over finite fields using linear algebraic computations. The research outcomes are summarized below.
Computational framework for analysis of non-linear dynamical systems over a finite state space
Developed a framework using Koopman operator for computation of solutions of non-linear systems using linear algebraic computations.
Extended the concept of observability and detectability for non-linear systems over a finite state space.
Designed a Luenberger type observer for non-linear systems using the Koopman operator - established conditions for unique retrieval of the internal states from the output sequence
As an application to cryptanalysis, proposed an attack on stream ciphers called "Observability attack" where the construction of the observer is used as a cryptanalysis tool. Established computational bounds on a class of stream generators called the "filter generators". As a case study, uniquely retrieved the internal state of a 100-bit stream generator by developing a dynamic observer.
Analysis of periodic linear systems over finite fields
Established conditions for Floquet transform for periodic linear systems over finite fields: the existing conditions for Floquet theory for periodic systems over reals was shown to not apply for periodic linear systems over a finite field and hence a new theory was needed.
It is shown that the Floquet transform does not always exist for such systems and I developed an alternative theory for analysis of such systems and characterized points which lie on periodic orbits as a subspace of the state space.
When Floquet transform exists, all possible orbit lengths of such systems are computed explicitly using the transformed linear system constructed through Floquet transform
Linear Representation of non-linear maps and dynamical systems over finite fields
Developed a framework for linear representation of maps over finite fields (or vector spaces over finite fields) using the Koopman Operator.
Linear representation is used for the following:
Translate non-linear operation (like multivariate polynomial compositions) to linear algebraic operations (matrix multiplications)
Solve for the solution of the non-linear difference equations using linear algebraic tools
Deciding invertibility (under composition) of non-linear maps over finite fields and construction of a polynomial inverse for multivariate maps
Extended the concept of linear representation to parametric maps (like Dickson polynomial, parameterized permutation maps) and developed a condition for checking invertibility and constructing the parametric inverse for such maps.
Developed a framework and extended the notion of linear representation to finite groups.