This paper redefines the foundations of asymmetric cryptography's homomorphic cryptosystems through the application of the Yoneda Lemma. It explicitly illustrates that widely adopted systems, including ElGamal, RSA, Benaloh, Regev's LWE, and NTRUEncrypt, directly derive from the principles of the Yoneda Lemma. This synthesis gives rise to a holistic homomorphic encryption framework named the Yoneda Encryption Scheme. Within this scheme, encryption is elucidated through the bijective maps of the Yoneda Lemma Isomorphism, and decryption seamlessly follows from the naturality of these maps. This unification suggests a conjecture for a unified model theory framework, providing a basis for reasoning about both homomorphic and fully homomorphic encryption (FHE) schemes. As a practical demonstration, the paper introduces an FHE scheme capable of processing arbitrary finite sequences of encrypted multiplications and additions without the need for additional tweaking techniques, such as squashing or bootstrapping. This not only underscores the practical implications of the proposed theoretical advancements but also introduces new possibilities for leveraging model theory and forcing techniques in cryptography to facilitate the design of FHE schemes.

翻译：本文通过运用Yoneda引理，重新定义了非对称密码学中同态密码系统的基础。研究明确阐明了包括ElGamal、RSA、Benaloh、Regev的LWE以及NTRUEncrypt在内的广泛采用的系统，均直接源自Yoneda引理原理。这一综合推导出一个名为Yoneda加密方案的整体同态加密框架。在该方案中，加密通过Yoneda引理同构的双射映射得以阐明，而解密则直接遵循这些映射的自然性。这一统一性为统一的模型论框架提出了一个猜想，为推理同态加密和全同态加密方案提供了基础。作为实际演示，本文引入了一种无需额外调整技术（如压缩或自举）即可处理任意有限序列的加密乘法和加法的全同态加密方案。这不仅凸显了所提出的理论进展的实际意义，还引入了利用模型论和力迫技术促进密码学中全同态加密方案设计的新可能性。