Modular Forms and Hecke Operators for Post-Quantum Cryptography

Trey Li

Paper 2025/1681

Modular Forms and Hecke Operators for Post-Quantum Cryptography

Trey Li, The University of Manchester

Abstract

We introduce modular forms and Hecke operators to cryptography and propose the Hecke problem as a new foundation for post-quantum cryptography. Given two modular forms, the Hecke problem asks to recover the Hecke operator that maps one to the other. While there is a deep relation to isogeny problems through the modularity theorem, this problem is rooted in arithmetic geometry and differs fundamentally in structure and mechanism. We prove NP-hardness of this problem and use it to construct a non-interactive key exchange scheme that achieves higher efficiency than isogeny-based schemes and smaller key sizes than lattice-based and code-based schemes.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint.
Keywords: Modular Forms Hecke Operators Hecke Problem Post-Quantum Cryptography Non-Interactive Key Exchange
Contact author(s): trey li @ manchester ac uk
History: 2025-09-18: approved; 2025-09-16: received; See all versions
Short URL: https://ia.cr/2025/1681
License: CC BY

BibTeX

@misc{cryptoeprint:2025/1681,
      author = {Trey Li},
      title = {Modular Forms and Hecke Operators for Post-Quantum Cryptography},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/1681},
      year = {2025},
      url = {https://eprint.iacr.org/2025/1681}
}