The security of public-key cryptosystems relies on computationally hard problems, that are classically analyzed by number theoretic methods. In this paper, we introduce a new perspective on cryptosystems by interpreting the Diffie-Hellman key exchange as a nonlinear dynamical system. Employing Koopman theory, we transfer this dynamical system into a higher-dimensional space to analytically derive a purely linear system that equivalently describes the underlying cryptosystem. In this form, analytic tools for linear systems allow us to reconstruct the secret integers of the key exchange by simple manipulations. Moreover, we provide an upper bound on the minimal required lifting dimension to obtain perfect accuracy. To demonstrate the potential of our method, we relate our findings to existing results on algorithmic complexity. Finally, we transfer this approach to a data-driven setting where the Koopman representation is learned from data samples of the cryptosystem.

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