QMA Complete Quantum-Enhanced Kyber: Provable Security Through CHSH Nonlocality

TL;DR

This work introduces CHSH-certified Kyber, a hybrid post-quantum key exchange that embeds CHSH quantum nonlocality tests into Kyber’s lattice-based protocol. By linking the CHSH verification to a QMA-complete 2-local Hamiltonian problem and preserving Module-LWE hardness via a Markov-state evolution, the approach offers verifiable quantum-assisted security alongside classical lattice security. The design remains compatible with the Fujisaki–Okamoto transform, preserving CCA security and practical efficiencies, with feasible quantum resources (polynomial in $m$). Empirically, CHSH enhancement yields a meaningful security uplift (roughly 30% gain in effective attack complexity) across Kyber parameter sets, demonstrating a viable path to integrated quantum–classical cryptographic defenses suitable for near-term deployment.

Abstract

Post-quantum cryptography (PQC) must secure large-scale communication systems against quantum adversaries where classical hardness alone is insufficient and purely quantum schemes remain impractical. Lattice-based key encapsulation mechanisms (KEMs) such as CRYSTALS-Kyber provide efficient quantum-resistant primitives but rely solely on computational hardness assumptions that are susceptible to hybrid classical-quantum attacks. To overcome this limitation, we introduce the first Clauser-Horne-Shimony-Holt (CHSH)-certified Kyber protocol, which embeds quantum non-locality verification directly within the key exchange phase. The proposed design integrates CHSH entanglement tests using Einstein-Podolsky-Rosen (EPR) pairs to yield measurable quantum advantage values exceeding classical correlation limits, thereby coupling information--theoretic quantum guarantees with lattice-based computational security. Formal reductions demonstrate that any polynomial-time adversary breaking the proposed KEM must either solve the Module Learning With Errors (Module-LWE) problem or a Quantum Merlin-Arthur (QMA)-complete instance of the 2-local Hamiltonian problem, under the standard complexity assumption QMA $\subset$ NP. The construction remains fully compatible with the Fujisaki-Okamoto (FO) transform, preserving chosen-ciphertext attack (CCA) security and Kyber's efficiency profile. The resulting CHSH-augmented Kyber scheme therefore establishes a mathematically rigorous, hybrid post-quantum framework that unifies lattice cryptography and quantum non-locality to achieve verifiable, composable, and forward-secure key agreement.

Figures (3)

• Figure 1: Kyber 512 security under three lattice attack vectors including Central Reduction, BKZ, and Enumeration, comparing the Standard Kyber implementation with its quantum-enhanced variants (QCS and CHSH). Security levels are expressed in bits as $\log_{2}(T)$ where $T$ is the attack time complexity.
• Figure 2: Security evaluation of Kyber 768 under Central Reduction, BKZ, and Enumeration attacks comparing Standard, QCS-enhanced, and CHSH-enhanced variants. Results show a consistent quantum advantage, with security increasing from 185.2 bits (Standard) to 241.1 bits (CHSH), demonstrating significant cryptographic strengthening.
• Figure 3: Security evaluation of Kyber 1024 under Central Reduction, BKZ, and Enumeration attacks comparing Standard, QCS-enhanced, and CHSH-enhanced variants. The CHSH-enhanced configuration achieves 325.3 bits of security, exceeding the Standard (250 bits) and QCS (300.8 bits) versions, with a consistent 8.2% advantage confirming its superior quantum resilience.

Theorems & Definitions (12)

• Definition 2.1: CHSH Expectation Parameter
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Definition 2.4: Primitivity and Spectral Gap
• Theorem 2.5
• proof
• Theorem 2.6
• proof