Constructing a fully homomorphic encryption scheme with the Yoneda Lemma (2401.13255v4)

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.

• The paper proposes a novel Yoneda Encryption Scheme that unifies popular cryptosystems into a fully homomorphic framework without the need for bootstrapping.
• It leverages category theory and the Yoneda Lemma to streamline encryption processes and preserve functional fidelity during sequential operations.
• The scheme offers practical efficiencies for secure, cloud-based computation and advances post-quantum cryptographic research with its robust algebraic design.

Constructing a Fully Homomorphic Encryption Scheme with the Yoneda Lemma: An Essay

This paper presents a compelling and innovative approach to reinterpreting the foundations of asymmetric cryptography's homomorphic cryptosystems using the Yoneda Lemma. It theorizes a unifying framework named the Yoneda Encryption Scheme that harmonizes widely adopted systems such as ElGamal, RSA, Benaloh, Regev's LWE, and NTRUEncrypt, under a single theoretical foundation. The formulation of a novel fully homomorphic encryption (FHE) scheme, capable of processing sequential encryptions without bootstrapping, is a noteworthy development. This paper has implications not only in advancing theoretical aspects but also in practical engagements with model theory and category theory in cryptographic design.

Theoretical Advancements and Methodological Approach

Central to the paper is the Yoneda Lemma, a fundamental result in category theory, which establishes a correspondence between objects in a category and sets of morphisms into those objects. By leveraging this lemma, the authors claim that various cryptosystems can be unified within a cohesive mathematical structure that elucidates shared properties across these systems—streamlining the conceptual complexity traditionally associated with FHE.

Homomorphic encryption, particularly FHE, is pivotal due to its capability to allow computations on ciphertexts, yielding results that are encrypted representations of operations carried on plaintexts. Previous FHE implementations generally relied on computationally intensive techniques like bootstrapping to manage noise accumulation during operations. In contrast, the Yoneda Encryption Scheme, as introduced, aims to circumvent these challenges through a structured categorical and algebraic approach, which potentially simplifies the construction and analysis of FHE schemes.

Key Results and Cryptographic Implications

The Yoneda Encryption Scheme is explored through detailed constructions of various cryptosystems demonstrating shared characteristics. The application of the Yoneda Lemma ensures that the encryption and decryption processes maintain functional fidelity across operations, thus supporting homomorphic properties such as addition and multiplication without degradation in performance—a crucial requirement for FHE.

Notably, the paper posits that the developed FHE system can securely execute arbitrary sequences of multiplicative and additive operations on encrypted data without necessitating bootstrapping. This provides a substantial leap in efficiency, addressing a significant limitation of current FHE schemes. The adoption of limit sketch theory within the category-theoretic framework promises a formal approach for enforcing desirable properties within cryptosystems.

Practical and Theoretical Implications

The research not only contributes to a more unified theoretical framework for encryption schemes but also proffers practical implications in secure computations, suggesting improvements in cloud computing and collaborative environments where data privacy is paramount. The introduction of arithmetic channels and the explicit use of the underlying algebraic structures could inspire further investigation into post-quantum cryptographic techniques, potentially complicating the task of quantum adversaries when solving underlying hard problems.

From a practical perspective, the paper includes a Python implementation that can be accessed via GitHub, advocating for an interactive exploration and potential application of the introduced concepts in research.

Future Directions and Concluding Thoughts

As the paper consolidates critical aspects of cryptographic theory under the figurative umbrella of the Yoneda Lemma, it leaves open several avenues for future research. One significant direction concerns the further optimization of encryption algorithms based on this framework to cater to real-world deployment scenarios. Moreover, the robustness of this theoretical framework against quantum computing paradigms remains an open question warranting exploration.

In conclusion, this paper presents a rigorous yet flexible framework potentially transforming our approach to handling homomorphic encryption. By revisiting established cryptographic constructs through the lens of category theory, it lays the groundwork for more efficient and generalizable encryption schemes, promising to bridge theoretical advancements with practical, secure applications in the ever-evolving landscape of cryptography.