Unclonable Secret Sharing
Prabhanjan Ananth, Vipul Goyal, Jiahui Liu, Qipeng Liu
TL;DR
The paper introduces unclonable secret sharing (USS), a quantum secret-sharing primitive that prevents copying of shares by non-communicating adversaries while enabling reconstruction when enough shares meet. It establishes deep connections between USS and Unclonable Encryption (UE), showing USS$_1$ is impossible in IT, but USS$_d$ can be achieved for large $d$ (e.g., $d=oldsymbol{omega}( ext{log}\lambda)$) via BB84-based XOR constructions, and links to nonlocal computation and position verification. In the computational setting, USS$_1$ is infeasible with low-$T$-gate reconstructions, yet a quantum-random-oracle-based model yields USS secure against unbounded entanglement; the work also demonstrates USS implies UE in a precise reduction and that USS under disconnected-graph adversaries can be realized via UE, while revealing barriers in the fully entangled, plain-model setting. Collectively, these results map the fundamental limits and potential pathways for USS, highlighting its conceptual significance for unclonable cryptography and its implications for quantum cryptographic primitives, including UE and quantum position verification.
Abstract
Unclonable cryptography utilizes the principles of quantum mechanics to addresses cryptographic tasks that are impossible classically. We introduce a novel unclonable primitive in the context of secret sharing, called unclonable secret sharing (USS). In a USS scheme, there are $n$ shareholders, each holding a share of a classical secret represented as a quantum state. They can recover the secret once all parties (or at least $t$ parties) come together with their shares. Importantly, it should be infeasible to copy their own shares and send the copies to two non-communicating parties, enabling both of them to recover the secret. Our work initiates a formal investigation into the realm of unclonable secret sharing, shedding light on its implications, constructions, and inherent limitations. ** Connections: We explore the connections between USS and other quantum cryptographic primitives such as unclonable encryption and position verification, showing the difficulties to achieve USS in different scenarios. **Limited Entanglement: In the case where the adversarial shareholders do not share any entanglement or limited entanglement, we demonstrate information-theoretic constructions for USS. **Large Entanglement: If we allow the adversarial shareholders to have unbounded entanglement resources (and unbounded computation), we prove that unclonable secret sharing is impossible. On the other hand, in the quantum random oracle model where the adversary can only make a bounded polynomial number of queries, we show a construction secure even with unbounded entanglement. Furthermore, even when these adversaries possess only a polynomial amount of entanglement resources, we establish that any unclonable secret sharing scheme with a reconstruction function implementable using Cliffords and logarithmically many T-gates is also unattainable.