Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties so that certain subsets can reconstruct the secret, while unauthorized subsets obtain no information. While QSS was introduced over twenty years ago, previous works focused only on existence of perfectly secure schemes, and the share size of the known schemes is exponential even for access structures computed by polynomial size monotone circuits. This stands in contrast to the classical case, where efficient computationally-secure schemes have been long known for all access structures in $\mathsf{monotone~P}$, and one can even obtain shares which are much shorter than the secret which is impossible with perfect security. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time QSS schemes under standard assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS required exponential share size. We also construct QSS schemes for which the size of the shares is significantly smaller than the size of the secret. As in the classical case, this is impossible with perfect security. We also use our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of access structures to $1.5^{n+o(n)}$, improving upon best known schemes and matching the best known result for general access structures in the classical case. Finally, we show construct efficient schemes for all access structures in $\mathsf{P}$ and $\mathsf{NP}$ when the quantum secret sharing scheme is given multiple of copies of the secret.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties so that certain subsets can reconstruct the secret, while unauthorized subsets obtain no information. While QSS was introduced over twenty years ago, previous works focused only on existence of perfectly secure schemes, and the share size of the known schemes is exponential even for access structures computed by polynomial size monotone circuits. This stands in contrast to the classical case, where efficient computationally-secure schemes have been long known for all access structures in monotoneP, and one can even obtain shares which are much shorter than the secret which is impossible with perfect security. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time QSS schemes under standard assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS required exponential share size. We also construct QSS schemes for which the size of the shares is significantly smaller than the size of the secret. As in the classical case, this is impossible with perfect security. We also use our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of access structures to 1.5n+o(n), improving upon best known schemes and matching the best known result for general access structures in the classical case. Finally, we show construct efficient schemes for all access structures in P and NP when the quantum secret sharing scheme is given multiple of copies of the secret.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties in such a way that certain authorized subsets can reconstruct the secret, while unauthorized subsets obtain no information about it. Previous works on QSS for general access structures focused solely on the existence of perfectly secure schemes, and the share size of the known schemes is necessarily exponential even in cases where the access structure is computed by polynomial size monotone circuits. This stands in stark contrast to the classical setting, where polynomial-time computationally-secure secret sharing schemes have been long known for all access structures computed by polynomial-size monotone circuits under standard hardness assumptions, and one can even obtain shares which are much shorter than the secret (which is impossible with perfect security). While QSS was introduced over twenty years ago, previous works only considered information-theoretic privacy. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes, just as in the classical case. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time computationally-secure QSS schemes under standard hardness assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS necessarily required exponential share size. In fact, we can go even further: We construct QSS schemes for which the size of the quantum shares is significantly smaller than the size of the secret. As in the classical setting, this is impossible with perfect security. We also apply our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of n-party access structures to 1.5^{n+o(n)}, improving upon best known schemes and matching the best known result for general access structures in the classical setting. Finally, among other things, we study the class of access structures which can be efficiently implemented when the quantum secret sharing scheme has access to a given number of copies of the secret, including all such functions in 𝖯 and NP.
Quantum secret sharing (QSS), as a quantum extension of classical secret sharing, uses the basic principles of quantum mechanics to share information safely among multiple parties, providing a new paradigm for information security. As a key foundation for secure multiparty quantum communication and distributed quantum computing, QSS has attracted considerable attention since its emergence. Currently, research in this field includes both classical and quantum scenarios, and continuous progress has been made in both theoretical and experimental aspects. This paper first reviews the current development of QSS for classical information. In this regard, significant and parallel progress has been made in both discrete-variable QSS and continuous-variable QSS. The QSS protocols for sharing classical information, from entangled states to single photons and then to coherent light, have been continuously optimized to better utilize available resources and achieve more efficient implementation under current technological conditions. Meanwhile, round-robin, measurement-device-independent, and other protocols have been steadily improving the security of QSS. Next, one will focus on QSS scheme for quantum secrets, which begins with the symmetry of access structures and introduces basic (k, n) threshold protocols, dynamic schemes that support adaptive agent groups, and symmetric quantum information splitting through entanglement. It further introduces hierarchical quantum secret sharing schemes for asymmetric splitting of quantum information. Considering practical laboratory conditions of quantum states as resources, an overall discussion is conducted on quantum secret sharing with graph states. Afterwards, the design of a continuous-variable scheme for quantum secret sharing is outlined, and entanglement state sharing and quantum teleportation between multiple senders and receivers are introduced. Finally, this review discusses and outlines the future development directions of QSS, thereby inspiring readers to further study and explore the relevant subjects.
Secret sharing is ubiquitous throughout cryptography. All possible access structures are classically feasible, and in the case of threshold access structures, the protocols are even very efficient. However, when moving to the quantum setting, the no-cloning theorem shows that many access structures are impossible. In fact, no-cloning exactly characterizes feasibility: for thresholds, quantum secret sharing is possible if and only if the threshold t is strictly more than n/2.
In this work, we propose a variant of quantum secret sharing (QSS) where two or more identical copies of the input state are provided, but the output is only required to recover one copy. This notion circumvents the simple one-copy no-cloning obstruction, though the natural k-copy generalization still gives much milder obstructions. For thresholds using k copies, no-cloning implies that QSS is impossible whenever t≤n/(k+1).
It is tempting to hypothesize that no-cloning continues to exactly characterize the many-copy case. However, we show that this is not the case. We give positive results showing that multiple copies allow for going slightly beyond the single-copy obstruction: for thresholds, we construct QSS whenever t>(n−k+1)/2. On the other hand, we give a novel obstruction we call the Clique Path obstruction, which applies to arbitrary access structures. For thresholds, it shows that QSS is impossible whenever t≤(n−1)/k. Our upper and lower bounds exactly match for k=2. Both our results leverage connections to the k-colorability of certain graphs derived from the access structure. We leave closing the gap for k≥3 copies as a fascinating direction for future work.
Secret sharing is closely related to error correction, which can also be considered in the many-copy setting. Our results imply similar obstructions for quantum error correction for erasure channels.
Quantum secret sharing (QSS) is a cryptographic protocol that leverages quantum mechanics to distribute a secret among multiple parties. With respect to the classical counterpart, in QSS, the secret is encoded into quantum states and shared by a dealer such that only an authorized subsets of participants, i.e., the players, can reconstruct it. Several state-of-the-art studies aim to transpose classical secret sharing into the quantum realm, while maintaining their reliance on traditional network topologies (e.g., star, ring, and fully connected), and require that all thenplayers calculate the secret. These studies exploit the Greenberger–Horne–Zeilinger state, which is a type of maximally entangled quantum state involving three or more qubits. However, none of these works account for redundancy, enhanced security/privacy features, or authentication mechanisms able to fingerprint players. To address these gaps, in this article, we introduce a new concept of QSS, which leans on a generic distributed quantum network, based on a threshold scheme, where all the players collaborate also to the routing of quantum information among them. The dealer, by exploiting a custom flexible weighting system, takes advantage of a newly defined quantum Dijkstra algorithm to select the most suitable subset oftplayers, out of the entire set onnplayers, to involve in the computation. To fingerprint and authenticate users, CRYSTAL-Kyber primitives are adopted, while also protecting each player’s privacy by hiding their identities. We show the effectiveness and performance of the proposed protocol by testing it against the main classical and quantum attacks, thereby improving the state-of-the-art security measures.
Quantum secret sharing (QSS) is a process in which the state of a quantum system is partitioned into multiple shares, allowing only specific subsets of shareholders to reconstruct the state, while others gain no information about its identity. In this work, we propose a variant of QSS, called ``helper QSS,'' which designates a special shareholder whose participation with any other group of parties is sufficient for recovering the state. We present different families of helper codes that function for an arbitrary number of parties and system sizes. As an application, we introduce the idea of a programmable access structure in QSS, which allows for a third-party programmer to independently choose the access structure of the code even after the shares are delivered to all the shareholders. This choice is made blindly, meaning that the programmer has no information about the secret being encoded, and it is implemented using the nonlocal process of quantum steering.
Quantum secret sharing (QSS) is a cryptographic protocol in which a quantum secret is distributed among a number of parties where some subsets of the parties are able to recover the secret while some subsets are unable to recover the secret. In the standard ((k, n)) quantum threshold secret sharing scheme, any subset of k or more parties out of the total n parties can recover the secret while other subsets have no information about the secret. But recovery of the secret incurs a communication cost of at least k qudits for every qudit in the secret. Recently, a class of communication efficient QSS schemes were proposed which can improve this communication cost to d−k+1d by contacting d ≥ k parties where d is fixed prior to the distribution of shares. In this paper, we propose a more general class of ((k, n)) quantum secret sharing schemes with low communication complexity. In these schemes the combiner can contact any d parties at the time of recovery where k ≤ d ≤ n. This is the first such class of universal communication efficient quantum threshold schemes.