Octonion Algebra and Noise-Free Fully Homomorphic Encryption (FHE) Schemes

TL;DR

This paper introduces the first noise-free fully homomorphic encryption schemes based on advanced algebraic structures, providing security proofs in a weak ciphertext-only model and demonstrating their potential for privacy-preserving computations.

Contribution

It designs noise-free FHE schemes using quaternion/octonion and Jordan algebras, with security based on hard algebraic problems, and shows their applicability in privacy-preserving computations.

Findings

First noise-free FHE scheme with security proof

Secure in weak ciphertext-only model based on hard problems

Potential for efficient privacy-preserving applications

Abstract

Brakerski showed that linearly decryptable fully homomorphic encryption (FHE) schemes cannot be secure in the chosen plaintext attack (CPA) model. In this paper, we show that linearly decryptable FHE schemes cannot be secure even in the ciphertext only security model. Then we consider the maximum security that a linearly decryptable FHE scheme could achieve. This paper designs fully homomorphic symmetric key encryption (FHE) schemes without bootstrapping (that is, noise-free FHE schemes). The proposed FHE schemes are based on quaternion/octonion algebra and Jordan algebra over finite rings Z_n and are secure in the weak ciphertext-only security model assuming the hardness of solving multivariate quadratic equation systems and solving univariate high degree polynomial equation systems in Z_n. It is up to our knowledge that this is the first noise-free FHE scheme that has ever been designed with a security proof (even in the weak ciphertext-only security model). It is argued that the weak ciphertext-only security model is sufficient for various applications such as privacy preserving computation in cloud. As an example, the proposed FHE schemes are used to construct obfuscated programs. This example could be further used to show that the scheme presented in this paper could be combined with existing FHE schemes with bootstrapping to obtain more efficient FHE schemes with bootstrapping in the fully CPA model. At the end of the paper, we point out the insecurity of several recently proposed noise-free FHE schemes.