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.

翻译：公钥密码系统的安全性依赖于计算困难问题，经典分析方法采用数论方法。本文通过将Diffie-Hellman密钥交换解释为非线性动力系统，引入密码系统分析的新视角。借助Koopman理论，我们将该动力系统映射到更高维空间，解析地推导出等价描述底层密码系统的纯线性系统。在此形式下，线性系统的解析工具使我们能够通过简单操作重建密钥交换中的秘密整数。此外，我们给出了实现完美精度所需最小提升维度的上界。为展示该方法潜力，我们将研究结果与已有算法复杂度结论相关联。最后，我们将该方法推广至数据驱动场景，从密码系统的数据样本中学习Koopman表示。