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：本文通过应用Yoneda引理重新定义了非对称密码学中同态加密系统的基础。研究明确论证了包括ElGamal、RSA、Benaloh、Regev的LWE以及NTRUEncrypt在内的广泛采用系统，均直接源于Yoneda引理原理。这种综合催生了一个名为Yoneda加密方案的整体同态加密框架。在该方案中，加密过程通过Yoneda引理同构的双射映射得到阐明，而解密过程则自然地从这些映射的自然性中推导得出。这种统一性提出了一个统一模型理论框架的猜想，为同态加密与全同态加密（FHE）方案的推理提供了基础。作为实际验证，本文提出了一种能够处理任意有限序列加密乘法与加法运算的FHE方案，且无需借助压缩或自举等额外调整技术。这不仅凸显了所提理论进展的实际意义，更为密码学中利用模型理论与力迫技术来促进FHE方案设计开辟了新的可能性。