Introducing GRAFHEN: Group-based Fully Homomorphic Encryption without Noise
Pierre Guillot, Auguste Hoang Duc, Michel Koskas, Florian Méhats
TL;DR
GRAFHEN proposes a noise-free approach to fully homomorphic encryption by encoding messages as elements of a nonabelian group and performing computations through group operations encoded via polynomial maps. Security hinges on hard subgroup membership in a group presented by generators and relations, with rewriting systems (and their pseudo-bounded variants) enabling efficient, bounded representations and word reductions. The authors demonstrate concrete encodings (notably into $\operatorname{SL}_3(\mathbb{K})$ and permutation groups), develop a full private-key protocol with a public-rule counterpart, and analyze a broad spectrum of attacks—providing enhancements via admissible rule sets, semidirect products, and automorphism-based variants. A benchmark against OpenFHE shows orders-of-magnitude speedups for basic operations, illustrating potential practical impact for encrypted computation without the noise growth inherent to lattice-based 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.