Quantum Anonymous Secret Sharing with Permutation Invariant Codes

Abstract: Quantum secret sharing schemes are a family of quantum cryptographic protocols which provide secure quantum encodings, mapping one secret to multiple shares of information such that the original secret cannot be accessed without an authorized set of shares present for decoding. In this work, we describe a protocol that enables sender-anonymity during the secret decoding process. By using permutation-invariant QEC codes along with a set of anonymous quantum transmission algorithms, we construct a quantum anonymous secret sharing scheme that achieves sender-anonymity. We quantify information leakage in ramp quantum secret sharing schemes via the quantum conditional min-entropy, justifying it as a valid measure of leaked information by relating it to the Knill-Laflamme quantum error correction conditions. Finally, we evaluate several permutation-invariant codes using this measure to make observations on the information leakage of intermediate shares for each quantum anonymous secret sharing scheme.

• The paper introduces a novel quantum secret sharing protocol that integrates permutation-invariant codes to ensure sender anonymity and robust error correction.
• It employs quantum conditional min-entropy to quantify information leakage in a single-shot setting, enhancing security for threshold access.
• Numerical evaluations confirm the protocol’s resilience, with consistent conditional min-entropy profiles across various qubit PI codes.

Quantum Anonymous Secret Sharing Enabled by Permutation-Invariant Codes

Secret Sharing in the Quantum Paradigm

Quantum secret sharing (QSS) augments classical secret sharing by leveraging entanglement and quantum error correction to distribute sensitive information among multiple participants. The primary objective is to ensure that the reconstruction of the original secret requires cooperation amongst an authorized subset of participants, rendering the secret inaccessible to unauthorized parties. Historically, QSS has focused on securing secrets without necessarily safeguarding participant identities—anonymity was not intrinsic to quantum secret sharing protocols.

The necessity for anonymity, particularly sender anonymity, emerges in contexts such as anonymous voting or organizational decision-making, where participant confidentiality is essential even from the recoverer. Classical anonymous secret sharing schemes have addressed this challenge, but quantum analogues are constrained by the structure of entangled states (notably GHZ and W states) and requirements for full-shareholder participation, limiting their practicality for threshold-based access structures.

Permutation-Invariant Codes and Anonymous Transmission

Permutation-invariant (PI) quantum error-correcting codes constitute a pivotal component of the proposed quantum anonymous secret sharing (QASS) protocol. PI codes, whose codespaces remain invariant under arbitrary reorderings of physical qubits, support erasure correction based solely on the count of erasures, rather than the identity of erased qubits. This feature is vital for anonymous protocols, since it enables decoding without revealing the identities of participants providing shares.

The QASS protocol integrates PI codes with anonymous quantum transmission subprotocols sourced from Christandl and Wehner. These subprotocols employ shared n-qubit GHZ states to enable anonymous entanglement creation, anonymous classical bit broadcast, and anonymous quantum teleportation (ANONQ), thereby obscuring sender identity throughout the secret recovery process both from external adversaries and other protocol participants, including the decoder.

Quantification of Information Leakage

A significant innovation in the protocol is the adoption of quantum conditional min-entropy as a measure for information leakage in ramp QSS schemes. While quantum mutual information and coherent information have traditionally been employed for assessing recoverable information, they aggregate over multiple instances and are not suited for single-shot adversarial scenarios. Conditional min-entropy, by maximizing achievable overlap between a reference and post-error experimental state over all possible recovery channels, provides a single-shot metric for quantifying the adversary's best-case attempt at extracting secret information with an intermediate set of shares.

For stabilizer codes—common QEC codes in QSS—the conditional min-entropy is efficiently calculable via the cleaning lemma and properties of maximum likelihood decoding under erasure channels. The operational meaning is explicit: min-entropy is inversely proportional to the number of logical operators that can be "cleaned" off the shares held by the adversary, thus quantifying information loss.

Protocol Design and Hybrid Schemes

The QASS protocol proceeds as follows: each participating shareholder uses ANONQ to transmit their share anonymously to the decoder, who then reconstructs the secret using the PI code’s erasure correction capability. Sender anonymity is robust against an unbounded adversary capable of corrupting up to $n-2$ participants. Furthermore, the protocol is traceless—adversaries with access to all protocol randomness cannot deduce sender identities.

A hybrid extension of QASS leverages classical secret sharing schemes alongside quantum encoding, “lifting” the security of quantum secrets to match that of classical ones. By encoding auxiliary classical secrets that specify Pauli randomization applied to the quantum secret, the adversary must first successfully decode the classical secrets before any attempt at quantum secret reconstruction, thereby enhancing security.

Numerical Results and Code Comparisons

Extensive numerical evaluations using semidefinite programming and explicit calculations for PI and stabilizer codes demonstrate the non-linear decay of information remaining as shares are erased. Across the tested 4-, 7-, 9-, 11-, and 13-qubit PI codes, the conditional min-entropy profiles sharply differentiate between fully authorized sets, intermediate share sets, and vanishing sets. Strong results indicate identical loss profiles across multiple 4-qubit codes under the same erasure weights, affirming fundamental robustness in PI code construction. The hybrid QASS schemes exhibit strict threshold behavior, with no intermediate sets leaking information, thus resembling perfect secret sharing in classical terms.

Practical and Theoretical Implications

This research establishes a general framework for quantum anonymous secret sharing, applicable to any PI quantum code and capable of supporting thresholds with missing shareholders. The protocol fundamentally addresses sender anonymity, which is essential for a wide range of privacy-sensitive applications including anonymous voting and confidential decision-making. By demonstrating the operational validity and tractability of quantum conditional min-entropy as a metric for information leakage in ramp QSS schemes, the paper underpins a rigorous quantitative approach to evaluating protocol security.

Future directions include substituting GHZ-state-based anonymous transmission protocols with more noise-resilient alternatives (e.g., W-state protocols), incorporating verification stages for shared entangled resources, and formalizing connections between ramp QSS and approximate QSS schemes. Optimization of PI code constructions for gradual ramp leakage, as well as improving computational efficiency for min-entropy evaluations via quantum weight enumerators or specialized bounds, represent substantial technical avenues.

Conclusion

Quantum anonymous secret sharing protocols based on permutation-invariant codes advance quantum cryptography by enabling sender anonymity and flexible threshold access structures, while leveraging quantum conditional min-entropy to precisely quantify information leakage. The architecture offers improved resilience for practical quantum information management in contexts where participant confidentiality is critical, and it foresees theoretical progressions in anonymous communication, quantum code design, and information-theoretic security assessment for quantum secret sharing (2604.27284).