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International Conference on Computing, Networks and Internet of Things (CNIOT)

A Quantum Steganography Method Based on Directional Transformation of a Perfect Code Generator Matrix

Chaolong Hao, Quangong Ma, Buyu Liu, Qi Chen, Xukui Yang, Dan Qu

Published May 23, 2025• 1 citation
Publisher

Abstract

Current quantum steganography techniques are primarily based on ground state encoding or probability amplitude embedding mechanisms. However, secret information remains confined to classical bit (0/1) formats, which makes it difficult to meet the processing demands of arbitrary superposition states α|0〉 + β |1〉 in fully quantum scenarios. This paper proposes a method that constructs codeword variants through directional transformation of a perfect code generator matrix. These variants exhibit statistical properties that closely resemble those of channel noise parameters under the stabilizer measurement framework, thereby ensuring that eavesdroppers cannot perceive the existence of steganography or decode it effectively. The proposed method offers the following advantages: (1) it achieves the unification of quantum state carriers and secret information, supporting direct steganography of arbitrary superposition states (see Fig. 1); (2) it adopts a Latin square balancing strategy to make the transformed perfect codes match the parameters of depolarizing channel noise, thus reducing the detectability of statistical analysis by an eavesdropper; (3) it uses error-correcting codes as carriers, which possess inherent noise resistance and are suitable for arbitrary Pauli errors. Numerical simulations under Pauli noise demonstrate that with gate noise intensities in (0.01,0.1), eavesdroppers’ quantum state fidelity remains bounded below 0.01 against arbitrary single-qubit Pauli errors, while legitimate users achieve fidelity > 0.93. This protocol proposes a robust quantum information hiding framework for quantum internet and IoT applications.

DOI
10.1109/cniot65435.2025.11070365

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Current quantum steganography techniques are primarily based on ground state encoding or probability amplitude embedding mechanisms. However, secret information remains confined to classical bit (0/1) formats, which makes it difficult to meet the processing demands of arbitrary superposition states α|0〉 + β |1〉 in fully quantum scenarios. This paper proposes a method that constructs codeword variants through directional transformation of a perfect code generator matrix. These variants exhibit statistical properties that closely resemble those of channel noise parameters under the stabilizer measurement framework, thereby ensuring that eavesdroppers cannot perceive the existence of steganography or decode it effectively. The proposed method offers the following advantages: (1) it achieves the unification of quantum state carriers and secret information, supporting direct steganography of arbitrary superposition states (see Fig. 1); (2) it adopts a Latin square balancing strategy to make the transformed perfect codes match the parameters of depolarizing channel noise, thus reducing the detectability of statistical analysis by an eavesdropper; (3) it uses error-correcting codes as carriers, which possess inherent noise resistance and are suitable for arbitrary Pauli errors. Numerical simulations under Pauli noise demonstrate that with gate noise intensities in (0.01,0.1), eavesdroppers’ quantum state fidelity remains bounded below 0.01 against arbitrary single-qubit Pauli errors, while legitimate users achieve fidelity > 0.93. This protocol proposes a robust quantum information hiding framework for quantum internet and IoT applications.
We present and analyze a protocol for quantum steganography where the sender (Alice) encodes her steganographic information into the error syndromes of the perfect (five-qubit) quantum error-correcting code, and sends it to the receiver (Bob) over a depolarizing channel. Alice and Bob share a classical secret key, and hide quantum information in such a way that to an eavesdropper (Eve) without access to the secret key, the quantum message looks like an innocent codeword with a typical sequence of quantum errors. We calculate the average rate of key consumption, and show how the protocol improves in performance as information is spread over multiple codeword blocks. Alice and Bob utilize different encodings to optimize the average number of steganographic bits that they can send to each other while matching the error statistics of the depolarizing channel.
In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender (Alice) and the receiver (Bob) and corrects an arbitrary single-qubit error. In the second half of this thesis we explore the yet uncharted and relatively undiscovered area of quantum steganography. Steganography is the process of hiding secret information by embedding it in an innocent message. We present protocols for hiding quantum information in a codeword of a quantum error-correcting code passing through a channel. Using either a shared classical secret key or shared entanglement Alice disguises her information as errors in the channel. Bob can retrieve the hidden information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for certain protocols.
Steganography is the technique of hiding secret information by embedding it in a seemingly ``innocent'' message. We present protocols for hiding quantum information by disguising it as noise in a codeword of a quantum error-correcting code. The sender (Alice) swaps quantum information into the codeword and applies a random choice of unitary operation, drawing on a secret random key she shares with the receiver (Bob). Using the key, Bob can retrieve the information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We consider two types of protocols: one in which the hidden quantum information is stored locally in the codeword, and another in which it is embedded in the space of error syndromes. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for specific protocols and examples of error channels. We consider both the case where there is no actual noise in the channel (so that all errors in the codeword result from the deliberate actions of Alice), and the case where the channel is noisy and not controlled by Alice and Bob.
Quantum information hiding, as an extension of classical information hiding techniques into the realm of quantum information, currently focuses on embedding classical bits (0/1) within quantum carriers. This includes methods such as disguising classical secret information as channel noise and embedding it within quantum error correction codes. However, the embedding mechanism for arbitrary quantum states $\alpha |0\rangle + \beta |1\rangle $ is still in the exploratory stage. This paper proposes an innovative framework that leverages the redundant space of quantum error correction codes to construct a nonlinear decoding architecture with quantum neural networks. This approach simultaneously achieves both carrier state error correction and secret state embedding and extraction functions. Specifically, the [5,1,3] stabilizer code is used as the carrier, with secret state embedding achieved through single-qubit substitution, and a quantum autoencoder is designed for steganographic state information decoding. The proposed framework features fully quantum-based input/output systems, overcoming the limitations of traditional variational quantum circuits that rely on probabilistic measurements for output generation. By performing full ground-state measurements at the autoencoder bottleneck layer and optimizing the parallel sub-network architecture, the network achieves efficient convergence and effective extraction of single-copy quantum states. Experimental results show that under the conditions of optimized parameters and data size of 20, the training losses for the carrier and secret states are 0.03 and 0.08, respectively, with test fidelities of 0.92 and 0.93. For a data size of 50, the secret states recovery fidelity exceeds 0.87. KS test analysis indicates that the full ground-state measurement and parallel sub-network are key strategies for achieving network performance. Equivalent error analysis shows that this approach successfully utilizes the potential redundant space of quantum error correction codes, providing new research directions for quantum state information hiding.
Quantum steganography is the study of hiding secret quantum information by encoding it into what an eavesdropper would perceive as an innocent-looking message. Here we study an explicit steganographic encoding for Alice to hide her secret message in the syndromes of an error-correcting code, so that the encoding simulates a given noisy quantum channel. We calculate achievable rates of steganographic communication over noiseless quantum channels using this encoding. We give definitions of secrecy and reliability for the communication process, and with these assumptions derive upper bounds on the amount of steganographic communication possible and show that these bounds match the communication rates achieved with our encoding. This gives a steganographic capacity for a noiseless channel emulating a given noisy channel.
Quantum steganography is to send secret quantum information through a quantum channel, such that an unauthorized user will not be aware of the existence of secret data. The depolarizing channel can hide quantum information by disguising it as channel errors of a quantum error-correcting code. We improve the efficiency of quantum steganography with noisy depolarizing channels, by modifying the twirling procedure and adding quantum teleportation. The proposed scheme not only meets the requirements of quantum steganography but also has higher efficiency.