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This paper redefines the foundations of asymmetric cryptography's homomorphiccryptosystems through the application of the Yoneda Lemma. It explicitlyillustrates that widely adopted systems, including ElGamal, RSA, Benaloh,Regev's LWE, and NTRUEncrypt, directly derive from the principles of the YonedaLemma. This synthesis gives rise to a holistic homomorphic encryption frameworknamed the Yoneda Encryption Scheme. Within this scheme, encryption iselucidated through the bijective maps of the Yoneda Lemma Isomorphism, anddecryption seamlessly follows from the naturality of these maps. Thisunification suggests a conjecture for a unified model theory framework,providing a basis for reasoning about both homomorphic and fully homomorphicencryption (FHE) schemes. As a practical demonstration, the paper introduces anFHE scheme capable of processing arbitrary finite sequences of encryptedmultiplications and additions without the need for additional tweakingtechniques, such as squashing or bootstrapping. This not only underscores thepractical implications of the proposed theoretical advancements but alsointroduces new possibilities for leveraging model theory and forcing techniquesin cryptography to facilitate the design of FHE schemes.

Constructing a fully homomorphic encryption scheme with the Yoneda Lemma