Hardness of Modular Form Reconstruction from Partial q-Expansions: A Cryptographic Trapdoor Candidate

We study the computational hardness of reconstructing modular forms from partial q-expansion data, focusing on Hecke eigenforms in S₂(Γ₀(N)). Motivated by the Modularity Theorem and its correspondence between rational elliptic curves and modular forms, we propose the Modular Form Reconstruction Problem (MFRP): recovering a normalized newform from only a few initial Fourier coefficients aₙ. While computing aₙ from a known form is easy, the inverse appears intractable—this asymmetry underlies our proposed cryptographic trapdoor structure. We formalize this through the Modular Trapdoor Assumption (MTA) and its quantum variant (QMTA), and analyze their hardness using arithmetic properties of Hecke operators, the exponential growth of modular form spaces, and the lack of known algorithms for modular form reconstruction. Our analysis surveys classical and quantum attack vectors and finds no efficient techniques that break MFRP. Using this foundation, we introduce cryptographic primitives—including trapdoor functions, key exchange, signatures, and encryption—built on modular form structures rather than group-based assumptions. Our constructions show promising efficiency (100–200 Byte key sizes) and strong post-quantum resilience. We outline implementation challenges and optimization strategies, highlighting this direction as a novel and promising path for post-quantum cryptography.