ChatFully Homomorphic Encryption (FHE) allows computing on encrypted data but suffered from noise accumulation that required crippling, compute-heavy bootstrapping; GRAFHEN introduces group-based FHE that executes unlimited computations without generating noise.
Fully Homomorphic Encryption is the holy grail of digital privacy: it allows untrusted cloud servers to process, analyze, and train AI models on encrypted medical and financial data without ever decrypting the underlying information.
All standard lattice-based FHE schemes (such as BGV and CKKS) inject deliberate mathematical noise to guarantee security. As computations progress, this noise accumulates exponentially, requiring computationally crushing 'bootstrapping' operations that slow down processing by factors of ten thousand.
GRAFHEN breaks this thirty-year paradigm by constructing fully homomorphic encryption over non-abelian algebraic group actions rather than noisy polynomial lattices. By operating in noise-free algebraic group structures, computations can proceed indefinitely without ever requiring noise reduction or bootstrapping.
Eliminating the bootstrapping bottleneck transforms FHE from a sluggish cryptographic theoretical proof-of-concept into a lightning-fast reality, enabling privacy-preserving cloud computing, encrypted database queries, and secure federated medical AI.
Introducing GRAFHEN: Group-based Fully Homomorphic Encryption without Noise
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.
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