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.

翻译：不可克隆密码学利用量子力学原理来解决经典密码学无法完成的任务。我们在秘密共享的背景下引入了一种新型不可克隆原语，称为不可克隆秘密共享（USS）。在一个USS方案中，存在$n$个共享者，每个共享者持有一个以量子态表示的经典秘密的份额。当所有参与方（或至少$t$个参与方）带着他们的份额聚集在一起时，他们可以恢复秘密。重要的是，复制他们自己的份额并将副本发送给两个不通信的参与方，使双方都能恢复秘密，这应该是不可行的。我们的工作正式开启了对不可克隆秘密共享领域的系统研究，揭示了其内涵、构造方法和内在局限性。**关联性：** 我们探讨了USS与其他量子密码原语（如不可克隆加密和位置验证）之间的联系，展示了在不同场景下实现USS的困难性。**有限纠缠：** 在对抗性共享者之间不共享任何纠缠或仅共享有限纠缠的情况下，我们展示了USS的信息论构造。**大量纠缠：** 如果我们允许对抗性共享者拥有无限制的纠缠资源（以及无限制的计算能力），我们证明了不可克隆秘密共享是不可能的。另一方面，在量子随机预言机模型中，当对手只能进行有限多项式次数的查询时，我们展示了一种即使在无限制纠缠下也安全的构造。此外，即使这些对手仅拥有多项式数量的纠缠资源，我们证明了任何重构函数可通过Clifford门和对数数量T门实现的不可克隆秘密共享方案同样无法实现。