Statistically-Secure Bit Commitment and Coin Flipping Protocols Based on Quantum Hardware Assumptions

This paper introduces the first statistically secure bit commitment and coin flipping protocols based on hybrid hardware assumptions, specifically using Hybrid Locked Physical Unclonable Functions (HLPUFs). The authors, Roo Dunnill and Mina Doosti from the University of Edinburgh, address the fundamental impossibility of unconditionally secure bit commitment in quantum cryptography (the Mayers–Lo–Chau theorem) by circumventing it through hardware assumptions rather than computational or storage limitations.

The core contribution is a novel bit commitment protocol that achieves both statistical hiding and binding. The protocol uses an asymmetric HLPUF, which combines a classical PUF with quantum communication and a locking mechanism. The classical response is split into two parts: a shorter verifier part f1(x) (of length s = 2k) and a longer payload part f2(x) (of length t = 2l). In the unlocked mode, the device outputs the full classical response, while in the locked mode it only outputs a quantum state ψc f2(x)⟩ if the input quantum state passes an internal verification based on f1(x).

The protocol works as follows: Alice first queries the HLPUF in unlocked mode to build a database of challenge-response pairs, then locks the device and sends it to Bob. To commit to a bit b, Alice selects a challenge x0 and uses Algorithm 1 to generate an alternative challenge x1 by flipping lmin bits of x0. She sends both challenges and the ordering J to Bob, then prepares an lmin-qubit BB84 state encoding f2(x0)J either in basis β(x0) (if b=0) or β(x1) (if b=1). In the opening phase, Alice reveals the full challenge-response pair, and Bob verifies it using the locked HLPUF and checks the quantum state consistency.

The security proofs are the main technical contributions. For hiding, the authors prove in Lemma 2 that the two commitment states are perfectly indistinguishable when the payload is uniformly distributed, achieving dtr(ρ0, ρ1) = 0. Theorem 5 then shows the overall hiding parameter is bounded by the HLPUF unforgeability: εhide ≤ εforge, which is negligible in the security parameters.

For binding, Lemma 3 bounds the operator norm of the sum of acceptance projectors: P + Q∞ ≤ 1 + 2(2s−lmin)/2. Theorem 6 then proves the binding parameter satisfies p0 + p1 ≤ 1 + 2(2s−lmin)/2, where pb is the probability a cheating Alice successfully opens bit b. The proof reduces arbitrary cheating strategies to this operator-norm bound, cleanly separating quantum-overlap limitations from hardware-dependent parameters.

The paper also presents a coin flipping protocol built black-box from the bit commitment scheme. Theorem 8 shows the bias is bounded by δCF ≤ (1/2)max εforge, 2−lmin/4, making it the first strong quantum coin-flipping protocol based on hybrid hardware assumptions.

Key technical elements include Algorithm 1 for balanced alternative-challenge generation, which ensures properties such as challenge permutability, large basis-distance (d(β(x0), β(x1)) = lmin), perfect value and basis balancing (uniform distribution of encoded bits), and verifier separation (overlap ≤ 2−s/2). Theorems 1-3 establish the algorithm's efficiency: the combinatorial feasibility test succeeds with probability 1 − e−omega(t) when lmin = αt and α/2 < p < 1 − α/2, and the expected number of HLPUF queries is 2 + e−omega(s).

The authors emphasize that their approach differs from previous work by not requiring assumptions about an adversary's quantum storage capabilities (unlike bounded-quantum-storage or noisy-storage models), instead replacing those with the hardness of forging the HLPUF. The protocol is designed to be implementable with off-the-shelf hardware components and feasible quantum communication infrastructure, offering a practical route to implementing bit commitment in quantum networks. Future work directions include simpler challenge-generation procedures, composable security treatments, extensions to stronger tasks like string commitment and oblivious transfer, and experimental implementation.

Improvements to AI Systems:

1. Quantum-Safe Cryptographic Protocol Design: Integrate the HLPUF-based bit commitment and coin flipping protocols into AI-driven cryptographic frameworks. The AI can automatically select between classical, quantum, or hybrid hardware-based security primitives based on threat models, eliminating reliance on computational hardness assumptions that quantum computers may break.

2. Adversarial Robustness via Physical Unclonability: Use the HLPUF’s unforgeability properties to harden AI systems against model extraction and adversarial tampering. The AI can embed challenge-response pairs from physical PUFs into its decision boundaries, making it computationally infeasible for attackers to replicate the model’s behavior without physical access.

3. Statistical Hiding/Binding for Secure Multi-Party AI: Implement the protocol’s statistical hiding and binding guarantees in federated learning or distributed AI training. The AI can commit to model updates or gradients using the bit commitment scheme, ensuring no participant can bias the training outcome (binding) while keeping updates private until reveal (hiding).

4. Efficient Quantum State Verification in AI Inference: Leverage the protocol’s quantum-state consistency checks (BB84 encoding) to build AI systems that verify the integrity of quantum data inputs. This enables secure AI inference on quantum networks, where the AI can detect if a quantum state has been tampered with during transmission.

5. Resource-Aware Security Parameter Tuning: Use the paper’s parameter relationships (e.g., εhide ≤ εforge, binding bound 2(2s−lmin)/2) to create an AI optimizer that dynamically adjusts security parameters (s, lmin, t) based on available hardware, noise levels, and desired security margins, minimizing overhead while maintaining provable security.

6. Hybrid Classical-Quantum AI Architectures: Design AI systems that can switch between classical and quantum communication modes using the HLPUF’s locked/unlocked states. The AI can use unlocked mode for high-speed classical data processing and locked mode for secure quantum-encrypted tasks, optimizing both performance and security.

7. Automated Protocol Composition: Build an AI planner that composes bit commitment and coin flipping protocols (as shown in Theorem 8) into larger secure multi-party computations. The AI can automatically generate black-box reductions for tasks like secure auctions, lotteries, or verifiable random functions, with provable bias bounds.

8. Physical-Layer Security Monitoring: Train an AI anomaly detector using the HLPUF’s challenge-response statistics (e.g., query counts, timing, response distributions) to identify physical attacks (e.g., device cloning, side-channel probing). The AI can adaptively re-lock or re-key the HLPUF in real-time, maintaining security under active physical threats.

9. Quantum Network Protocol Stack Optimization: Use the paper’s feasibility algorithm (Algorithm 1) to design an AI-driven scheduler for quantum networks. The AI can precompute balanced challenge sets (with large basis distance and verifier separation) to minimize quantum communication overhead while ensuring statistical security for multiple concurrent users.

10. Provably Secure AI Decision-Making Under Uncertainty: Incorporate the protocol’s operator-norm bound (P + Q∞ ≤ 1 + 2(2s−lmin)/2) into AI systems that make decisions with imperfect information. The AI can quantify the maximum probability of an adversary forcing a particular outcome (e.g., in game-theoretic AI), enabling robust strategy design even when the adversary has quantum capabilities.

Bit commitment is impossible to achieve with unconditional security, even in quantum cryptogra- phy. We show that statistically secure bit commitment, satisfying both hiding and binding, can be constructed from hybrid locked physical unclonable functions (HLPUFs), a hardware primitive that combines classical hardware tokens and quantum communication. Our protocol uses these hardware assumptions in a novel and non-trivial way to achieve the first mistrustful two-party cryptographic protocol based on hybrid hardware modules. We prove statistical hiding and binding under natu- ral assumptions on the HLPUF and using a carefully designed challenge generation algorithm as a subroutine of our bit-commitment protocol. The construction also yields the first hardware-based coin-flipping protocol. Our results suggest a new paradigm for secure two-party cryptography in quantum networks, combining rigorous security guarantees with a concrete route toward practical implementation.