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Constructing a fully homomorphic encryption scheme with the Yoneda Lemma

Rémy Tuyéras

TL;DR

This work seeks a unified foundation for asymmetric homomorphic encryption by applying the Yoneda Lemma and limit-sketch model theory to reinterpret classical schemes as instances of a Yoneda Encryption Scheme. It develops ACES, an arithmetic-channel–based leveled FHE that does not rely on bootstrapping, and shows how ElGamal, RSA, Benaloh, NTRU, and LWE-based cryptosystems arise as Yoneda constructions within corresponding limit sketches and module structures over $\mathbb{Z}[X]$. The approach integrates model theory, category theory, and forcing techniques to motivate a broader framework for reasoning about homomorphic properties and noise management, including a new refresh mechanism that yields a proper FHE. Practically, ACES provides a path to non-bootstrapped FHE with controllable noise via affine decompositions and a Python implementation, signaling potential for more efficient and flexible FHE designs rooted in deep mathematical structure.

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.

Constructing a fully homomorphic encryption scheme with the Yoneda Lemma

TL;DR

This work seeks a unified foundation for asymmetric homomorphic encryption by applying the Yoneda Lemma and limit-sketch model theory to reinterpret classical schemes as instances of a Yoneda Encryption Scheme. It develops ACES, an arithmetic-channel–based leveled FHE that does not rely on bootstrapping, and shows how ElGamal, RSA, Benaloh, NTRU, and LWE-based cryptosystems arise as Yoneda constructions within corresponding limit sketches and module structures over . The approach integrates model theory, category theory, and forcing techniques to motivate a broader framework for reasoning about homomorphic properties and noise management, including a new refresh mechanism that yields a proper FHE. Practically, ACES provides a path to non-bootstrapped FHE with controllable noise via affine decompositions and a Python implementation, signaling potential for more efficient and flexible FHE designs rooted in deep mathematical structure.

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.
Paper Structure (26 sections, 20 theorems, 198 equations)

This paper contains 26 sections, 20 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Short presentation
  3. Background
  4. Motivation and roadmap
  5. Acknowledgment
  6. Preparation
  7. Conventions
  8. Yoneda lemma for functors
  9. Yoneda lemma for models of theories
  10. Limit sketches
  11. Modules
  12. Cryptosystems and the Yoneda Lemma
  13. Yoneda encryption
  14. ElGamal encryption
  15. RSA encryption
  16. ...and 11 more sections

Key Result

Proposition 3.9

Let $A$ be a small category and $\mathcal{C}$ be a category. If $\mathcal{C}$ has limits over $A$, then there is a natural isomorphism (in $X$ and $F:A \to \mathcal{C}$), where the rightmost limit is defined on the functorial mapping $a \mapsto \mathcal{C}(X,F(a))$ Similarly, if $\mathcal{C}$ has colimits over $A$, then there is a natural isomorphism (in $X$ and $F:A \to \mathcal{C}$), where the r

Theorems & Definitions (106)

  • Definition 3.6: Constant functor
  • Definition 3.7: Limits and colimits
  • Example 3.8: Sets
  • Proposition 3.9
  • proof
  • Definition 3.10: Full and faithful
  • Theorem 3.11: Yoneda Lemma
  • proof
  • Theorem 3.12: Yoneda Lemma
  • proof
  • ...and 96 more