Backward Iteration in Z-Space and Convergence-Preserving Encryption: A Multi-Layer Framework Based on Implicit Function Nesting

This paper proposes a convergence-preserving multi-layer encryption framework based on backward iteration in Z-space. The framework originates from the Z-U-V triple duality theory established in previous work [1,2]. Its core idea leverages the contrast between the computational ease of forward iteration in Z-space and the difficulty of backward iteration (i.e., solving an implicit function). By nesting implicit functions in multiple layers, the encryption strength increases exponentially with the number of layers, while the global convergence of U-iteration [1] ensures the system never diverges. Unlike traditional cryptography that relies on number-theoretic hardness, the security of this framework stems from structural complexity. Using a function family related to the Riemann zeta function as an example of an implicit function layer [3], we present its corresponding forms in U, V, and Z spaces, and analyze the error sources of backward iteration (approximation error at large scales and numerical error at small scales). Finally, we discuss the framework's potential advantages in applications such as reversible coding [4], trajectory backtracking [5], and data hiding [6].