On the Structural Reduction of Discrete Logarithms under Height-Constrained Arithmetic Embeddings

This paper investigates the structural properties of the discrete logarithm problem (DLP)over finite fields when subjected to strict arithmetic and topological constraints. We intro-duce a logarithmic embedding, denoted as the P1→3 mapping, which projects cryptographicinstances into a metrized arithmetic manifold. By imposing admissibility conditions analo-gous to the strong form of the abc conjecture, we demonstrate a dimension-folding effect onthe parameter space. Our analysis establishes that within this height-controlled region, theeffective search domain for the discrete logarithm undergoes a polynomial compression rela-tive to its radical weight. Consequently, we conclude that cryptographic systems relying on asingle arithmetic separation mechanism exhibit structural instability under such embeddings,resulting in a theoretical transition from exponential to polynomial search complexity.