  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.
