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A Novel Steganography Scheme Using Quantum Hilbert Transform

Nitin Jha, Abhishek Parakh

arXiv:2609.07894v1quant-phcs.CRcs.ITmath.IT

TL;DR

Classical phase-domain information hiding has not been fully explored for quantum systems, and classical Hilbert-transform operations cannot simply be copied into coherent quantum evolution. This paper defines a finite-dimensional unitary QHT and builds a steganography protocol around weak signed phase perturbations of quantum cover states. The protocol preserves exceptional Fourier modes and recovers hidden bits through state discrimination, aggregation, and error-correcting decoding, while its per-carrier detectability result is not a full-message covertness guarantee.

  • Problem

    Phase-domain information hiding has not been fully explored for quantum systems, and classical Hilbert-transform operations may violate the unitarity required for coherent quantum evolution.

    Source: §1 · pp. 1–2

  • Method

    The paper defines a finite-dimensional QHT using QFT-based, frequency-sign-dependent phases and embeds classical bits as weak signed unitary perturbations of quantum cover states.

    Source: §1 · p. 2, §2 · pp. 4–5

  • Results

    The proposed construction preserves DC and Nyquist modes, and Bob recovers hidden bits through binary state discrimination, block aggregation, and classical error-correcting decoding.

    Source: §5 · p. 16

  • Takeaways & Limitations

    The QHT framework supports studying covert leakage risks by varying embedding strength, spreading length, whitening, and cover-state structure.

    Source: §5 · p. 17

    Whitening’s O(ϵ2) trace-distance residue is a per-carrier detectability scaling, not a full-message covertness guarantee, and spreading across L carriers accumulates evidence for the warden.

    Source: §4.2 · p. 14

Abstract

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The main goal of steganography is to transmit hidden messages in legitimate-looking communication messages. Phase-domain information hiding, however, has not been fully explored for quantum systems. This work introduces a finite-dimensional Quantum Hilbert Transform (QHT) as a unitary phase operator based on the Quantum Fourier Transform. Using this construction, we develop a QHT-based quantum steganography scheme that embeds classical bits as weak signed phase perturbations of quantum cover states. Bob recovers the hidden message through binary state discrimination, block aggregation, and classical error-correcting decoding.

1 Introduction

Classical Hilbert-transform methods support signal analysis and information hiding, but lifting their phase behavior to quantum steganography requires a unitary construction. This work introduces a finite-dimensional QHT and proposes it as a coherent phase-domain primitive for hiding information in quantum communication.

1 Introduction

  • Classical Hilbert-transform methods have supported covert audio channels and image or signal steganography, including Hilbert–Huang-based detection of embedded data (Li et al., 2011; Deng, 2006; Huang, 2014; Kandregula, 2009; Wu et al., 2011; Sharma et al., 2022).Wu et al. (2011) used instantaneous amplitude and frequency descriptors with a machine-learning classifier to detect embedded data across image formats.Source: §1 · pp. 1–2
  • Quantum Hilbert-transform steganography must preserve norm and remain unitary, unlike classical operations that may suppress or amplify frequency components.The paper proposes a finite-dimensional QHT and a steganography protocol using weak signed phase perturbations of quantum cover states.Source: §1 · p. 2

1.1 Discrete Hilbert Transform

  • The discrete Hilbert transform applies a phase-only multiplier to Fourier coefficients, whereas the analytic-signal mask is nonunitary because it includes magnitudes 0 and 2.That analytic-signal mask therefore cannot directly represent a closed-system quantum operation.Source: §1.1 · p. 3

1.2 Quantum Fourier Transform

  • The quantum Fourier transform maps a quantum state from the computational basis to the Fourier basis, whose coefficients provide phase information used to define the QHT.The QFT is presented as a core primitive for phase estimation, period finding, and quantum signal processing.Source: §1 · p. 2, §1.2 · p. 4

2 Proposed Quantum Hilbert Transform

The proposed QHT implements a Hilbert-transform phase rule in a finite-dimensional Fourier basis while preserving exceptional modes to remain unitary. A skew-Hermitian generator also describes signed fractional QHT-domain displacements.

2 Proposed Quantum Hilbert Transform

  • The QHT maps states to the Fourier basis, applies Hilbert-transform phases, then returns them to the computational basis.Positive modes receive −π/2 and negative modes +π/2 phase shifts.Source: §2 · p. 5
  • Preserving the DC and Nyquist modes keeps the finite-dimensional QHT unitary, unlike annihilating the zero-frequency component in the classical setting.For even N, the DC and Nyquist modes are preserved while the positive and negative modes receive the Hilbert-phase convention.Source: §2 · pp. 4–6

2.1 Fractional QHT Generator

  • Smaller signed powers of the skew-Hermitian generator GH represent fractional QHT-domain displacements.The embedding unitary can be expressed using the corresponding Hermitian observable KH.Source: §2.1 · p. 7
  • On the nonzero, non-Nyquist Fourier subspace, the squared QHT acts as −I while leaving the exceptional modes unchanged.This difference from the classical Hilbert transform is necessary to maintain unitarity.Source: §2.1 · p. 7

3 Quantum Steganography Protocol

The protocol hides classical information in weak QHT-domain phase perturbations of legitimate quantum cover states. Bob discriminates the perturbation sign, aggregates decisions across carrier blocks, and uses classical error-correcting decoding to recover the message.

3 Quantum Steganography Protocol

  • The protocol hides information in the phase structure of legitimate quantum states using weak QHT-domain perturbations.The payload is classical, while the embedding operation is a signed phase displacement generated by the QHT.Source: §3 · p. 7, §3.1 · p. 7

3.2 Hidden Message Encoding

  • Alice maps each encoded bit to a sign and embeds it as a weak signed QHT-domain phase displacement.Classical error-correcting coding precedes embedding; a shared key may provide whitening, while the keyless version omits it.Source: §3.2 · pp. 8–9

3.3 Decoding and Extraction

  • Bob treats each carrier as a binary state-discrimination problem, aggregates noisy sign estimates over a block, and majority-decodes each encoded bit.The recovered encoded sequence then passes through classical error-correcting decoding.Source: §3.3 · p. 10

3.1 Cover-State Model

  • The cover-state model has Alice send legitimate quantum states while a passive warden, Wendy, monitors for a payload.Decoding assumes Bob knows the expected cover-state description; the protocol is scoped to structured traffic generated from a known schedule, public metadata, or a pre-established codebook.Source: §3.1 · p. 7, §3.3 · p. 10

3.4 Working Example

  • In the microgrid example, compromised controllers hide a low-rate payload in routine reports while visible traffic patterns and message timing remain unchanged.The example describes weakly structured perturbations of transmitted cover states.Source: §3.4 · p. 11

4 Results

The simulations assess decoding reliability, cover disturbance, and warden detectability while varying embedding strength, spreading, whitening, and depolarizing noise. They show tradeoffs among these measures, with whitening suppressing first-order leakage and error correction helping only when raw quantum errors are within the code’s capacity.

4.1 Simulation Setup

  • Simulations use full N-dimensional matrices, Haar-random covers as a generic stress test, and an expected-cover-state Helstrom receiver with majority voting over L carriers.Reliability is measured by BER, disturbance by fidelity loss, and detectability by trace distance.Source: §4.1 · pp. 11, 13

4.2 Numerical Results

  • Bob’s BER decreases as embedding strength ϵ and spreading length L increase, with majority aggregation across independent carriers improving reliability.The simulation compares L = 1, 3, 5, and 9.Source: Figure 2 · p. 13, §4.2 · p. 13
  • Whitening suppresses first-order signed leakage to an O(ϵ2) trace-distance residue, while mean per-carrier disturbance follows a slope-2 prediction.O(ϵ4) disturbance corrections remain negligible for ϵ ∈ [10−2, 0.2].Source: Figure 3 · p. 14, §4.2 · p. 14
  • The Hamming(7, 4) layer reduces final error when embedding strength is sufficient, but can miscorrect when raw quantum errors exceed the code’s capacity.Figure 4 compares non-encoded transmission, block spreading alone, and the combined spreading-plus-coding scheme.Source: Figure 4 · p. 15, §4.2 · p. 15
  • With depolarizing noise, Bob’s decoding reliability degrades as Wendy’s ability to distinguish stego from cover states also decreases; ϵ = 0.20 retains lower BER than ϵ = 0.05 at small and moderate noise.At high noise, all BER curves approach random guessing.Source: §4.2 · p. 15, Figure 5 · p. 16

5 Discussion and Conclusions

The paper presents a finite-dimensional QHT that preserves the classical transform’s phase-shifting structure while remaining unitary, then uses it to embed hidden bits in quantum cover states. It also frames the protocol as a way to study covert leakage risks in quantum-augmented networks.

5 Discussion and Conclusions

  • The finite-dimensional QHT preserves the DC and Nyquist modes, keeping the operator unitary while retaining the classical Hilbert transform’s phase-shifting structure.It maps states into the Fourier basis, applies a frequency-sign-dependent phase rule, and maps them back.Source: §5 · p. 16
  • The protocol embeds classical bits as weakly signed QHT-domain phase perturbations of legitimate quantum cover states.Bob decodes the sign using binary state discrimination, block-level aggregation, and classical error-correcting decoding.Source: §5 · p. 16
  • In quantum-augmented networks, compromised nodes could use weak QHT-domain perturbations to leak low-rate information while normal protocol execution appears unchanged.The protocol provides a controlled way to define and evaluate such phase-domain covert-channel attacks.Source: §5 · pp. 16–17
  • Embedding strength, spreading length, whitening sequence, and cover-state structure can be varied to study how much information remains hidden before network monitors detect perturbations.The proposed framework is intended to support systematic evaluation of covert leakage risks in quantum-augmented network architectures.Source: §5 · p. 17
  • Future work includes studying channel-noise models beyond depolarization and developing a hardware-level implementation framework for practical use cases.The authors also identify construction of a formal QuANet covert-channel attack model as future work.Source: §5 · p. 17

Funding

The authors report that this work received no specific funding.

Funding

A Proof of Unitarity of the QHT

The proof establishes that the quantum Hilbert transform operator is unitary by relying on the unitarity of the QFT and the unit-magnitude diagonal entries of the phase operator.

A Proof of Unitarity of the QHT

B Helstrom Measurement for Decoding

Bob decodes the hidden sign by distinguishing the two candidate stego states. Under equal priors, the optimal minimum-error binary measurement is Helstrom measurement, and the ideal error depends on the overlap of the QHT-perturbed states.

B Helstrom Measurement for Decoding

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