Introducing GRAFHEN: Group-based Fully Homomorphic Encryption without Noise

TL;DR

GRAFHEN is a novel cryptographic scheme that achieves fully homomorphic encryption without noise or bootstrapping by leveraging group encodings, significantly enhancing performance and security.

Contribution

We introduce GRAFHEN, a noise-free fully homomorphic encryption scheme based on group encodings, with improved efficiency and security against various attacks.

Findings

Runs several orders of magnitude faster than existing standards

Provides security by making subgroup membership problem maximally hard

Eliminates the need for bootstrapping in FHE schemes

Abstract

We present GRAFHEN, a new cryptographic scheme which offers Fully Homomorphic Encryption without the need for bootstrapping (or in other words, without noise). Building on the work of Nuida and others, we achieve this using encodings in groups. The groups are represented on a machine using rewriting systems. In this way the subgroup membership problem, which an attacker would have to solve in order to break the scheme, becomes maximally hard, while performance is preserved. In fact we include a simple benchmark demonstrating that our implementation runs several orders of magnitude faster than existing standards. We review many possible attacks against our protocol and explain how to protect the scheme in each case.