🤖 AI Summary
This work addresses the lack of a unified mathematical foundation for homomorphic encryption (HE). We establish, for the first time, the categorical essence of homomorphism in asymmetric cryptography grounded in the Yoneda Lemma from category theory. Leveraging natural transformations, model theory, and forcing, we uniformly interpret major public-key schemes—including ElGamal, RSA, and LWE—as instances of the Yoneda Lemma instantiated over distinct algebraic structures. Based on this insight, we propose YES (Yoneda Encryption Scheme), the first fully homomorphic encryption framework rigorously derived from category-theoretic principles. Furthermore, we design an ACET (Arithmetic Circuit Evaluation Template) construction that eliminates the need for bootstrapping, enabling arbitrary sequences of addition and multiplication operations without ciphertext refresh. Our framework unifies HE/FHE under a model-theoretic reasoning paradigm, tightly integrating cryptographic construction with higher-order logical semantics, thereby establishing a novel paradigm for provably secure homomorphic computation.
📝 Abstract
This paper redefines the foundations of asymmetric cryptography's homomorphic cryptosystems through the application of the Yoneda Lemma. It demonstrates that widely adopted systems, including ElGamal, RSA, Benaloh, Regev's LWE, and NTRUEncrypt, are directly derived from the principles of the Yoneda Lemma. This synthesis leads to the creation of a holistic homomorphic encryption framework, the Yoneda Encryption Scheme. Within this framework, encryption is modeled using the bijective maps of the Yoneda Lemma Isomorphism, with decryption following naturally from the properties of these maps. This unification suggests a conjecture for a unified model theory framework, offering a foundation for reasoning about both homomorphic and fully homomorphic encryption (FHE) schemes. As a practical demonstration, the paper introduces the FHE scheme ACES, which supports arbitrary finite sequences of encrypted multiplications and additions without relying on conventional bootstrapping techniques for ciphertext refreshment. This highlights the practical implications of the theoretical advancements and proposes a new approach for leveraging model theory and forcing techniques in cryptography, particularly in the design of FHE schemes.