⚛️ quantum physics

Traceable Quantum Secret Sharing

This paper introduces a framework for traceable quantum secret sharing that establishes leakage-verifiable and malicious-traceable protocols to attribute leaked shares and identify participants submitting forged shares, respectively, while providing security definitions and demonstrating robustness against noise through numerical simulations.

Original authors: Mengyan Qiao, Jian Long, Yu Li, Zhu Cao

Published 2026-10-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mengyan Qiao, Jian Long, Yu Li, Zhu Cao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of secure communication, there is a long-standing challenge known as secret sharing. Imagine a group of people who need to protect a valuable piece of information, such as a digital key or a confidential message. Instead of giving the whole secret to one person, they split it into many pieces, called shares, and hand one piece to each member of the group. The rule is simple: if enough people come together—say, three out of five—they can combine their pieces to rebuild the original secret. However, if fewer than that number try to put the pieces together, they learn absolutely nothing. This method has been a cornerstone of quantum cryptography, where the laws of physics are used to protect data in ways that classical computers cannot. For decades, the primary goal of these systems was to ensure that the secret remained hidden from unauthorized eyes and could be correctly rebuilt by the right people.

But a new problem has emerged in real-world scenarios. What happens if someone who is supposed to be honest decides to leak their piece of the secret to a stranger? Or what if a dishonest participant tries to sabotage the process by handing over a fake piece during the rebuilding phase? In traditional systems, once the secret is leaked or the reconstruction fails, there is no way to know who was responsible. The system protects the data, but it offers no way to hold anyone accountable. This lack of accountability is a significant gap, especially in distributed networks where trust is limited and the consequences of a leak are severe. Researchers have long wondered if it is possible to build a quantum secret sharing system that not only keeps the secret safe but also leaves a clear trail pointing to the person who caused the trouble.

A team of researchers from East China University of Science and Technology and Tongji University has now answered this question by establishing a framework for what they call traceable quantum secret sharing. They did not just propose a theoretical idea; they designed specific protocols and proved mathematically that they work under two distinct types of threats. The first scenario they addressed involves participants who follow the rules but then leak their valid shares to an outsider. The second scenario deals with participants who actively try to break the system by submitting forged, fake shares during the reconstruction process. By creating systems tailored to these specific dangers, the researchers have moved quantum secret sharing beyond simple confidentiality into a realm where responsibility can be assigned.

To handle the problem of leaking shares, the team developed a method called leakage-verifiable and traceable quantum secret sharing. In this system, every piece of the secret comes with a unique digital tag, much like a serial number, which is hidden inside the quantum data. When a group of people tries to rebuild the secret using a stolen piece and some honest pieces, the system allows a verifier to check the reconstruction. If the secret is successfully rebuilt, the verifier can use a special key to decrypt the hidden tags and see exactly which participants contributed the stolen pieces. The researchers proved that this process is secure: it is impossible for an attacker to hide their identity, and it is equally impossible for the system to falsely accuse an honest person who did not leak anything. The system ensures that if a leak happens, the source can be identified with high certainty.

For the second threat, where malicious actors try to disrupt the process by submitting fake data, the researchers introduced a different protocol called malicious-traceable and accountable quantum secret sharing. Here, the goal is not just to find a leak but to catch the person trying to break the reconstruction. In this setup, when the group attempts to rebuild the secret, the system generates a record of the process. If someone submits a forged share, the system can analyze the record to separate the honest participants from the dishonest ones. The researchers showed that this method can identify the cheaters and provide evidence that others can verify independently. Crucially, the system is designed so that the act of tracing does not compromise the secrecy of the original data; the key used to find the cheater reveals nothing about the secret itself.

To ensure these new protocols are practical, the researchers ran extensive computer simulations to see how they hold up against the noise and errors that are common in real quantum devices. They simulated a scenario where each piece of the secret is subject to a type of random interference known as depolarizing noise, which can corrupt the data. The results showed that the systems remain robust, maintaining high accuracy in both reconstructing the secret and tracing the source of the problem, as long as the noise levels are not too high. They found that the ability to verify the evidence was the most sensitive part of the process, meaning that while the system is strong, the final check requires the cleanest data to work perfectly. Even with these challenges, the simulations confirmed that the protocols can successfully identify leakers and cheaters in a noisy environment.

This work represents a significant shift in how we think about quantum security. For a long time, the focus was almost entirely on keeping secrets safe from prying eyes. This new research demonstrates that in a distributed world, keeping the secret safe is only half the battle; knowing who might have broken the trust is equally important. By integrating the ability to trace and hold participants accountable directly into the quantum protocols, the researchers have provided a foundation for more resilient and trustworthy quantum networks. Their findings suggest that it is possible to have a system that is both secret and fair, where the laws of physics protect the data and the design of the protocol ensures that no one can misbehave without being caught. This approach opens the door for more complex and secure applications of quantum technology in the future, where accountability is built into the very fabric of the communication.

Technical Summary: Traceable Quantum Secret Sharing

Problem Statement

Quantum Secret Sharing (QSS) is a fundamental primitive for protecting distributed quantum information, ensuring that only authorized subsets of participants can reconstruct a secret while unauthorized subsets learn nothing. However, existing QSS schemes focus exclusively on confidentiality and correctness. They lack intrinsic mechanisms for accountability in realistic distributed environments. Specifically, current protocols cannot address two critical threat models:

Leakage of Valid Shares: Compromised participants may leak or resell their valid shares to third parties. Existing schemes cannot identify the source of the leakage.
Malicious Forgery: Malicious participants may submit forged shares during the reconstruction phase to disrupt the process or manipulate the outcome. Existing schemes cannot attribute responsibility for these deviations.

The absence of traceability means that QSS remains a confidentiality-oriented primitive, failing to provide attribution when valid shares are exposed or when reconstruction is corrupted. While Traceable Secret Sharing (TSS) has been formalized in the classical setting, extending these concepts to the quantum domain is nontrivial due to the nature of quantum states, quantum operations, and the requirements of distributed quantum information processing.

Methodology

The authors establish a framework for Traceable Quantum Secret Sharing under two distinct threat models, introducing two new protocols: Leakage-Verifiable and Traceable QSS (LVT-QSS) and Malicious-Traceable and Accountable QSS (MTA-QSS).

1. Honest-but-Leaky Model (LVT-QSS)

In this model, participants follow the protocol but may leak valid shares. The goal is to trace leaked shares back to the specific coalition responsible.

Mechanism: The dealer encodes the secret quantum state $|\psi\rangle$ into $m$ shares. Each share is associated with a unique tag stored in an auxiliary register. These tags are encrypted using a post-quantum symmetric-key encryption scheme and a quantum-secure hash function.
Tracing Process: If a reconstruction box is built using $f$ leaked shares and $k-f$ additional shares, the verifier uses a tracing key to decrypt the auxiliary registers of the reconstructed output. This recovers the tags associated with the leaked shares.
Verification: A verification algorithm checks if the recovered tags correspond to valid verification keys, ensuring the evidence is not forged.
Security Definitions: The scheme defines secrecy (unauthorized sets learn nothing), tracer secrecy (access to the tracing key does not compromise the secret), traceability (leaked shares can be attributed to the coalition), and non-imputability (honest parties cannot be falsely accused).

2. Malicious-but-Accountable Model (MTA-QSS)

In this model, malicious participants actively deviate by submitting forged shares during reconstruction. The goal is to identify the malicious actors and provide publicly verifiable evidence of their misconduct.

Mechanism: Similar to LVT-QSS, shares contain auxiliary classical information (tags and tracing keys). However, the reconstruction process is modeled explicitly rather than via a black-box abstraction.
Tracing Process: During reconstruction, the algorithm measures ancillary registers to obtain classical outcomes (auxiliary information), which are then used to generate a tracing transcript. The tracing algorithm partitions this information into sets of tags corresponding to honest participants and those corresponding to malicious participants (based on consistency with the tracing keys).
Verification: A verification algorithm checks the consistency of the traced evidence against verification keys, confirming which participants contributed valid shares and which contributed forged ones.
Security Definitions: The scheme adapts the definitions of secrecy, tracer secrecy, traceability (identifying honest contributors to infer malicious ones), and non-imputability (preventing false attribution of forged shares to honest parties).

Technical Foundations

Both protocols rely on:

Quantum Error-Correcting Codes (QECCs): Specifically, a coherent threshold encoding based on polynomial interpolation over finite fields.
Post-Quantum Cryptography: A symmetric-key encryption scheme ( $\Pi_{SKE}$ ) satisfying the sparse-hit property and a quantum-secure collision-resistant hash function ( $H$ ).
Quantum Operations: Inverse Vandermonde transforms for reconstruction and linear transformations to remove residual dependencies.

Key Contributions

The paper makes three primary contributions:

LVT-QSS Protocol: Introduces a scheme that attributes leaked shares through an explicit leakage-verification mechanism, enabling the identification of participants who leaked valid shares.
MTA-QSS Protocol: Proposes a scheme that identifies participants submitting forged shares during reconstruction, supporting publicly verifiable accountability.
Security Framework and Analysis:
- Formulates rigorous security definitions for both settings (secrecy, traceability, non-imputability, and tracer secrecy).
- Proves that the proposed protocols satisfy these guarantees against relevant adversaries (Quantum Polynomial-Time adversaries).
- Performs numerical simulations under per-share depolarizing noise to evaluate robustness.

Results

Theoretical Guarantees: The authors prove that both LVT-QSS and MTA-QSS satisfy correctness, secrecy, tracer secrecy, traceability, and non-imputability. The proofs rely on the properties of the underlying polynomial encoding and the security of the cryptographic primitives (collision resistance and sparse-hit property).
Noise Resilience:
- LVT-QSS: Simulations show that reconstruction fidelity and tracing accuracy decrease monotonically with noise probability $p$ . The verification success rate decays more rapidly than the other metrics, as it requires a stricter joint success condition (correct tracing and consistency check). Consequently, verification is identified as the dominant noise-sensitive stage for LVT-QSS. Smaller access structures ( $k, m$ ) exhibit stronger robustness.
- MTA-QSS: Demonstrates higher robustness compared to LVT-QSS. Because the auxiliary key is retained by the dealer and tracing/verification operate on the classical information obtained from the reconstruction (derived from quantum measurements), noise does not accumulate along the verification pathway. Consequently, tracing accuracy and verification success rates nearly coincide and remain high in low-noise regimes, showing no single dominant noise-sensitive stage in the same manner as LVT-QSS.
Dominant Noise Sensitivity: In LVT-QSS, verification is the dominant noise-sensitive stage, whereas in MTA-QSS, tracing and verification exhibit nearly identical sensitivity profiles.

Significance and Claims

The paper claims to provide the first systematic framework for traceable quantum secret sharing that incorporates publicly verifiable accountability.

Beyond Confidentiality: The work extends QSS beyond the traditional goals of secrecy and correct reconstruction, establishing accountability as a core security objective.
Scenario-Dependent Traceability: The authors argue that traceability in quantum settings must be characterized based on the specific adversarial behavior (passive leakage vs. active forgery), leading to the distinct LVT and MTA frameworks.
Foundation for Future Work: The results provide a theoretical starting point for traceable distributed quantum cryptography. The authors note that while their work addresses specific threat models separately, a unified framework capturing both leakage and forgery simultaneously remains an open question for future research. They also highlight that implementing traceability incurs costs (communication, storage, complexity) and that further work is needed to explore stronger guarantees (e.g., reconstruction despite malicious participants) and lower-overhead variants.

The paper concludes that incorporating traceability into the security characterization of distributed quantum cryptographic protocols is both natural and meaningful for realistic multiparty settings.