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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Quantum Steganography with High Efficiency with Noisy Depolarizing Channels

Xin Liao, Qiaoyan Wen, Tingting Song, Jie Zhang

Published Jan 1, 2013• 12 citations
Publisher

Abstract

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.

DOI
10.1587/transfun.e96.a.2039

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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.
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.
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.
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.
Steganography is the art of concealing secret information by embedding it in an apparently innocent-looking message. Quantum steganography applies the principles of quantum mechanics to traditional steganography and, compared to the latter, offers significant advantages, including heightened security, improved concealment, and increased data-hiding capacity. Traditionally, quantum steganography disguises the covert communication as channel noise, which is corrected using preshared classical randomness. This method requires the steganalytic eavesdropper Eve to overestimate the level of channel noise, so that the bounds on the stego channel capacity depend on this assumed gap in Eve's knowledge of the channel. In this work, we point out that by means of preshared quantum entanglement the secret message can be encoded into nonlocal correlations, obviating the need for such an assumption of Eve's ignorance. Consequently, the capacity bounds on the stego channel can then come from the channel capacity of the quantum communication channel. We introduce three such entanglement-based quantum steganographic protocols that make use of catalytic quantum error-correcting codes (QECCs), degenerate entanglement-assisted QECCs, or the phase bit of preshared entanglement. Here catalytic QECCs enable recycling entanglement, while entanglement assistance allows both sender and receiver to contribute to the protocol's secrecy. We derive upper and lower bounds on the secrecy capacity of each protocol, and demonstrate their practical robustness.
Characterizing secret communication over noisy quantum channels is an interesting problem from both a practical and theoretical perspective. Suppose Alice and Bob wish to communicate secret information so that an eavesdropper Eve will not suspect any type of encoded communication between the two. Classical or quantum cryptography will not suffice since it is always clear secret communication is taking place. Therefore Alice and Bob must execute what is known as a quantum steganographic protocol. Assuming Eve only has partial knowledge of the channel connecting Alice and Bob, we show that for the bit-flip and depolarizing channels Alice can use Eve's lack of knowledge of the channel parameter to encode quantum information steganographically. We give an explicit encoding procedure and calculate the rate at which Alice and Bob can communicate secretly. We also show that our encoding is optimal for nondegenerate quantum codes. We calculate the rate at which a secret key must be consumed. Finally, we discuss the possibility of steganographic communication over more general quantum channels and conjecture a general formula for the steganographic rate.
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.