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Decrypting Nonlinearity: Koopman Interpretation and Analysis of Cryptosystems

Robin Strässer, Sebastian Schlor, Frank Allgöwer

TL;DR

This work reframes DH and RSA as nonlinear dynamical systems and constructs linear Koopman representations in lifted spaces. It derives concrete, finite lifting dimensions for exact linearization in both DH and RSA, links these results to algorithmic complexity and Willems' Lemma, and provides a data-driven EDMD pathway to learn the representations from samples. The approach offers a new lens on cryptographic hardness and enables secret reconstruction within a linear systems framework, while not claiming practical breaks of established schemes. It also outlines future directions toward broader cryptosystems, finite-field extensions, and potential certifications leveraging dynamical-systems insights. Overall, the paper bridges cryptography and systems theory, highlighting how linearization and data-driven methods can illuminate classical hardness assumptions.

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.

Decrypting Nonlinearity: Koopman Interpretation and Analysis of Cryptosystems

TL;DR

This work reframes DH and RSA as nonlinear dynamical systems and constructs linear Koopman representations in lifted spaces. It derives concrete, finite lifting dimensions for exact linearization in both DH and RSA, links these results to algorithmic complexity and Willems' Lemma, and provides a data-driven EDMD pathway to learn the representations from samples. The approach offers a new lens on cryptographic hardness and enables secret reconstruction within a linear systems framework, while not claiming practical breaks of established schemes. It also outlines future directions toward broader cryptosystems, finite-field extensions, and potential certifications leveraging dynamical-systems insights. Overall, the paper bridges cryptography and systems theory, highlighting how linearization and data-driven methods can illuminate classical hardness assumptions.

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.
Paper Structure (23 sections, 63 equations, 4 figures, 2 algorithms)

This paper contains 23 sections, 63 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman operator
  4. Extended Dynamic Mode Decomposition
  5. Considered cryptosystems and their dynamical system interpretation
  6. The Diffie-Hellman cryptosystem
  7. The Rivest–Shamir–Adleman cryptosystem
  8. Reconstruction of secrets
  9. Reconstruction of the secret integer $e$ (Problem \ref{['problem:DH_message']})
  10. Reconstruction of the secret key $d$ (Problem \ref{['problem:RSA_key']})
  11. Koopman representation
  12. Observables on the complex unit circle
  13. Observables as a value list
  14. Minimal number of observables
  15. Minimal number of observables for DH
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Comparison of the angles of the complex numbers occurring in the reconstruction of $e$.
  • Figure 1: Computation times of Algorithm \ref{['alg:DH']}.
  • Figure 2: Observables on complex unit circle exemplary for $p=5$, $m=3$, $x=1$, and $q=3$.
  • Figure 3: Trajectory of \ref{['eq:dynamical-system']} for $p=19$ and $m=2$ corresponding to a DH cryptosystem \ref{['fig:plot:trajectory']}. The gray area highlights the minimal number of values in the state of the linear system representation according to Theorem \ref{['thm:minimal-number-observables-DH']}.