On the Cryptographic Reduction of Rational Elliptic Curve Systems to Modular Form Structures

This paper reinterprets Wiles' Modularity Theorem as a cryptographic framework exploiting the correspondence between elliptic curves E/ℚ and modular forms f ∈ S₂(Γ₀(N)). The asymmetric nature of this correspondence, where the forward mapping (Curve → Form) is tractable but the inverse (Form → Curve) is computationally hard, functions as a cryptographic trapdoor mechanism.We introduce two hardness assumptions: Modular Correspondence Hardness (MCH) and Form-to-Curve Trapdoor (FCT), capturing the intractability of reconstructing modular forms from partial data. Under Sato-Tate distribution, Fourier coefficients {aₚ} provide pseudorandomness with entropy bounds H({aₚᵢ}) ≥ k · log₂(√N). We present concrete constructions including Modular Form-based Key Exchange (MFKE), q-expansion key derivation, and Hecke transform signatures, demonstrating potential post-quantum advantages as these problems resist known quantum algorithms.