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.

翻译：公钥密码系统的安全性依赖于计算困难问题，传统上通过数论方法进行分析。本文引入一种全新视角，将迪菲-赫尔曼密钥交换和RSA密码系统视为非线性动力系统。通过应用库普曼理论，我们将这些动力系统变换到高维空间，并解析推导出等价的纯线性系统。该表述允许我们利用线性系统分析工具，通过简单运算重构密码系统的秘密整数。此外，我们确立了为实现完美精度所需的最小提升维度的上界。关于所需提升维度的结果与暴力破解的难解性相一致。为展示该方法的潜力，我们将研究成果与算法复杂性的现有结论建立关联。最后，我们将该方法推广至数据驱动场景，即从密码系统的数据样本中学习库普曼表示。