Boolean Control Networks 

From genetic regulation to cryptographic protocols, a wide range of complex systems can be represented as Boolean Networks (BNs) and Finite State Machines (FSMs). These models provide a unifying language for discrete dynamics - but they also expose a fundamental computational barrier.

Understanding the long-term behavior of such systems—fixed points, cyclic attractors, basins of attraction, and reachability between states—is an NP-hard problem, intrinsically constrained by the curse of dimensionality. As the number of variables increases, the state space grows exponentially, rendering exhaustive exploration and classical graph-based methods infeasible even for moderately sized networks. 

This work is aims to address the following.

"Can scalable linear algebraic methods unlock new ways to analyze large-scale Boolean and finite-state systems?"

We focus on developing linear algebra–based computational frameworks for targeted classes of BNs and FSMs, aimed at characterizing asymptotic dynamics, reachability, and control-relevant properties without explicit enumeration of the state space. These methods are motivated by applications in systems biology, biochemical reaction networks, communication systems, and cryptography, where understanding emergent behavior is both theoretically challenging and practically essential. 

Research Goals

Broader Perspective

This work aims to bridge Koopman operator theory, discrete dynamical systems, and Boolean network analysis, providing a linear-algebraic lens for studying problems that are traditionally NP-hard. The resulting frameworks have potential implications for systems biology, network control, and formal verification, where understanding attractors and transient behavior is central.

The Koopman operator provides a powerful theoretical framework for lifting nonlinear dynamical systems into an infinite-dimensional linear representation, enabling the use of linear algebraic tools to analyze inherently nonlinear behavior. While Koopman-based methods have been predominantly explored in data-driven and continuous-state settings, this research investigates their theoretical, computational and algorithmic applications to Boolean Networks (BNs).

By exploiting the finite, discrete structure of Boolean dynamics, we seek to develop exact linear representations of BN, enabling scalable analysis without explicit state-space enumeration.

In particular, I am interested addressing the following questions. 

1. Attractor Structure:

Can Koopman-based formulations be used to characterize attractors of Boolean Networks, including the existence and orbit lengths of cyclic attractors?

2. Transient Dynamics:

Can we determine or bound the transient length—the number of iterations required for the system to reach an attractor from an arbitrary initial condition?

3. Spectral Signatures of Dynamics: 

How do spectral properties of invariant Koopman representations reflect asymptotic behavior, attractor multiplicity, and basin structure?

4. Observability:

Can Koopman-invariant observables be used to establish observability conditions for Boolean Networks? How do the spectral properties and invariant subspaces of the Koopman operator relate to state distinguishability and reconstruction of hidden Boolean states? 

5. Control design:

Can Koopman-based linear representations be leveraged to design control strategies that steer trajectories, alter reachability, modify or eliminate specific attractors using linear or bilinear control techniques?

References

1. Ramachandran Anantharaman and Virendra Sule, Koopman operator approach for computing structure of solutions and observability of nonlinear dynamical systems over finite fields, Mathematics of Control, Signals and Systems (MCSS), April 2021.

2. Ramachandran Anantharaman and Virendra Sule,  Observability Attack on Stream Ciphers.  25th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Bayreuth, Germany. September 2022. 

3. Ramachandran Anantharaman and Virendra Sule, On linear representation, complexity and inversion of maps over finite fields, Finite Fields and their Applications (FFA) 2024. 

4. Ramachandran Anantharaman and Virendra Sule, Observability of Nonlinear Dynamical Systems over Finite Fields, March 2024. (Preprint)

Generating Pseudo-random sequences

Pseudo-random number generators (PRNGs) play a central role in communication systems, computer simulation, and cryptography, where the quality of generated sequences directly impacts performance, security, and reliability. In many practical settings—particularly in hardware and embedded implementations—PRNGs are realized using Feedback Shift Register (FSR) configurations due to their simplicity, speed, and deterministic structure.

The generation of pseudo-random sequences with strong statistical properties, long periods, and favorable correlation characteristics remains a fundamental challenge. While classical FSR designs are well understood, their linear structure limits achievable randomness and robustness in advanced applications.

This research explores the integration of classical control theory and digital design methodologies to systematically construct and analyze FSR configurations that generate high-quality pseudo-random sequences. By viewing FSRs as finite-state dynamical systems, we explore design frameworks that enables us to connect state-space structure, spectral properties, and long-term dynamics to measurable randomness criteria.

The broader goal is to develop theory-guided, scalable design of FSR-based PRNGs that bridge control, Boolean dynamics, and hardware-efficient implementation, with applications spanning communications, simulation, and cryptographic systems.

References:

1. Ramachandran Anantharaman and Virendra Sule,  Analysis of Periodic Linear Systems over Finite Fields with and without Floquet Transform,  Mathematics of Control, Signals and Systems (MCSS), September 2021.

2. Ramachandran Anantharaman and Virendra Sule,  Analysis of Periodic Shift Registers. April 2019 (Preprint).