Homomorphic Computation in Reed-Muller Codes

Abstract

With the ongoing developments in artificial intelligence (AI), big data, and cloud services, fully homomorphic encryption (FHE) is being considered as a solution for preserving the privacy and security in machine learning systems. Currently, the existing FHE schemes are constructed using lattice-based cryptography. In state-of-the-art algorithms, a huge amount of computational resources are required for homomorphic multiplications and the corresponding bootstrapping that is necessary to refresh the ciphertext for a larger number of operations. Therefore, it is necessary to discover a new innovative approach for FHE that can reduce the computational complexity for practical applications. In this paper, we propose a code-based homomorphic operation scheme. Linear codes are closed under the addition, however, achieving multiplicative homomorphic operations with linear codes has been impossible until now. We strive to solve this problem by proposing a fully homomorphic code scheme that can support both addition and multiplication simultaneously using the Reed-Muller (RM) codes. This can be considered as a preceding step for constructing code-based FHE schemes. As the order of RM codes increases after multiplication, a bootstrapping technique is required to reduce the order of intermediate RM codes to accomplish a large number of operations. We propose a bootstrapping technique to preserve the order of RM codes after the addition or multiplication by proposing three consecutive transformations that create a one-to-one relationship between computations on messages and that on the corresponding codewords in RM codes.