Decrypting Nonlinearity: Koopman Interpretation and Analysis of Cryptosystems

TL;DR

This paper presents a novel approach to analyzing cryptosystems by modeling them as nonlinear dynamical systems and applying Koopman theory to linearize and reconstruct secret keys, offering new insights into their complexity.

Contribution

It introduces a Koopman-based framework for cryptosystem analysis, transforming nonlinear cryptographic processes into linear systems and deriving bounds on the lifting dimension needed for accurate reconstruction.

Findings

Koopman theory can linearize cryptosystems for analysis.

Bound on the lifting dimension aligns with brute-force intractability.

Method extends to data-driven learning of cryptosystem representations.

Abstract

Public-key cryptosystems rely on computationally difficult problems for security, traditionally analyzed using number theory methods. In this paper, we introduce a novel perspective on cryptosystems by viewing the Diffie-Hellman key exchange and the Rivest-Shamir-Adleman cryptosystem as nonlinear dynamical systems. By applying Koopman theory, we transform these dynamical systems into higher-dimensional spaces and analytically derive equivalent purely linear systems. This formulation allows us to reconstruct the secret integers of the cryptosystems through straightforward manipulations, leveraging the tools available for linear systems analysis. Additionally, we establish an upper bound on the minimum lifting dimension required to achieve perfect accuracy. Our results on the required lifting dimension are in line with the intractability of brute-force attacks. To showcase the potential of our approach, we establish connections between our findings and existing results on algorithmic complexity. Furthermore, we extend this methodology to a data-driven context, where the Koopman representation is learned from data samples of the cryptosystems.