Joint Data Hiding and Partial Encryption of Compressive Sensed Streams

Cristian Damian; Cristina-Elena Popa; Daniela Coltuc

REVIEW 3 major objections 7 minor 43 references

Joint Data Hiding and Partial Encryption of Compressive Sensed Streams

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a stream of compressive sensing measurements can be partially encrypted and reversibly embed its encrypted parts into the remaining measurements, so that only key holders recover the exact stream and others see a…

desk verdict Genuinely new RDH-on-CS combination undermined by a false zero-mean claim and an unstated centering step in the real-data experiments; worth refereeing but needs major revision. read the letter →

arxiv 2505.23357 v1 pith:UMFEFISW submitted 2025-05-29 cs.CR

Cristina-Elena Popa , Cristian Damian , Daniela Coltuc This is my paper

classification cs.CR

keywords compressivesensingreversibledatahidingpartialencryptionsinglepixelcamerapredictionerrorexpansionon-the-flyembeddingCSPRNGrate-distortioncurve

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Compressive sensing (CS) streams, such as those produced one measurement at a time by a single pixel camera, are normally collected and then encrypted as a block. This paper proposes to instead protect a subset of the measurements with a secret key and reversibly embed those encrypted values into the remaining measurements as they arrive. Because the embedding is reversible, an authorized user recovers the original measurements exactly, while an unauthorized user reconstructs only a visibly degraded image from the modified and shortened stream. The paper derives a capacity formula for this insertion and reports that embedding on 10 levels yields roughly 18 dB distortion for 256x256 images recovered with the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). The practical interest is that the scheme avoids buffering and delivers controlled degradation rather than a noise-like ciphertext.

What carries the argument

The load-bearing object is a modified prediction-error expansion, a reversible data-hiding rule generalized to insert multiple bits per measurement, applied to the CS measurement stream. Instead of predicting each value from correlated neighbors, the predictor is the mean of all measurements, which for scrambled Hadamard sensing is zero; hence $d_i = y_i - \bar{y} = y_i$ and the insertion rule acts directly on the measurements. It expands in-range values by $2^n$ and adds the encrypted chunk, and shifts out-of-range values to avoid overlap, making extraction lossless. The capacity identity $C_r \approx \frac{P n}{1 + $P^{2}$ n/16}$ and the compression-rate consequence $r \approx 2/(1+n/16)$ carry the quantitative argument: they say how many bits can be hidden per measurement and how much the transmitted volume grows, both controlled coarsely by $n$ and finely by the threshold $T$ through $P$.

What would settle it

Take the real S-matrix measurements from Section 7 and compute their mean without any centering. If the mean is not zero, $d_i = y_i - \bar{y} = y_i$ fails and the stated insertion rule is not the rule actually applied to the real data; a reader could then run the published extraction on those measurements and check whether it recovers the original values exactly, which would settle whether the scheme as written supports the reported capacity and distortion figures.

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Extended reading notes

Core claim

The central claim is that the statistics of CS measurements$\textemdash$zero-mean, Gaussian-like, statistically independent$\textemdash$make them ideal carriers for their own encrypted parts. The algorithm takes each measurement, represents it on 16 bits, XORs it with a CSPRNG key, splits the result into $n$-bit chunks, and inserts each chunk into later measurements using a prediction-error expansion rule. Because the predictor is the global mean and the mean is taken to be zero for a scrambled Hadamard matrix, the prediction error of a measurement is the measurement itself, so insertion happens directly on the values. The paper derives the relative capacity $C_r \approx \frac{P n}{1 + $P^{2}$ n/16}$, where $P$ is the Gaussian mass between the expansion thresholds, and shows that with loose thresholds this gives a volume-expansion factor $r \approx 2/(1+n/16)$. With 10 insertion levels and 40% measurements, reconstruction quality drops to about 18 dB for simulated data, leaving content visible but degraded to an unauthorized user while an authorized user extracts the embedded chunks and recovers the exact measurements.

Load-bearing premise

The scheme assumes every stream of compressive sensing measurements has mean zero, so that a measurement can be used as its own prediction error; if that does not hold, the embedding and extraction formulas as written would need a centering step that the paper does not state.

Editorial extensions

If this is right

An authorized receiver reconstructs the original image exactly, since the embedding is reversible; an unauthorized receiver gets a visibly distorted image rather than noise.
Embedding on 10 levels at 40% sampling gives about 18 dB median distortion, strong enough to obscure content while keeping it discernible.
The data volume grows by about 23% at 10 levels, so the encrypted stream costs more bandwidth but still compresses to about 98% of the original image size overall.
Because insertion happens as measurements arrive, a single pixel camera can transmit securely without buffering measurements for a separate encryption stage.
The number of insertion levels $n$ acts as a coarse distortion and capacity knob, while the threshold $T$ provides a finer control whose range is limited by the 32-bit representation constraint.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

We infer that the same embedding rule could be applied to any sensing matrix whose measurements are centered, not just scrambled Hadamard, by replacing the zero-mean assumption with an explicit centering step; the real S-matrix experiments would then be a direct application rather than a separate case.
The paper's error-concealment results show an attacker can strip the encrypted chunks and improve PSNR by about 16 dB, so we infer the security goal is better described as controlled degradation plus exact recovery for key holders, not cryptographic confidentiality in the strongest sense.
A testable extension is to use the capacity formula $C_r(P,n)$ to choose $n$ and $T$ from a target PSNR, turning the reported rate-distortion curve into a design procedure for other CS acquisition rates and patch sizes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this paper is a real new combination, but the zero-mean simplification is wrong and the real-data experiments depend on an unstated centering step. The core idea—XOR-encrypting a subset of CS measurements and reversibly embedding them into the others on the fly—is genuinely different from [33], which only encrypts signs and hides side information. The capacity derivation from a Gaussian model (Eq. 14) is a real derivation, not a fit, and the rate-distortion analysis separating truncation from embedding distortion is sensible. The authors also honestly report the ECA weakness: without the key, an attacker can discard the embedded chunks and push PSNR to about 31 dB. That is a useful data point for the security framing.

The soft spots are real. Eq. (3) claims the measurement mean is always zero because of the Hadamard construction. That is false. After removing the all-ones row, the remaining rows sum to a vector with first entry N-1 and other entries -1, so the sample mean is a particular weighted sum of the scene, generally nonzero. For the 0/1 S-matrix used in the real-data experiments the claim is even less tenable—measurements are nonnegative, so the [-T,T] insertion window would be empty unless the data are centered or normalized. Nothing in the text says they did that. Eq. (4) is also printed with missing operators in the shifted branches, so the exact reversible rule is ambiguous. There is no code or raw data, which matters because the real-data results are the only validation of the on-the-fly hardware scenario.

The paper is worth a serious referee. It is not a desk reject: the problem is well-motivated, the novelty is clear, and the capacity model is a real contribution. But the current version is conditional. A referee should ask for a corrected Eq. (3)/(6) with an explicit treatment of the DC component, a clarified real-data protocol, a precise statement of Eq. (4), and ideally artifacts. Once those land, the method would be citable for SPC security applications.

Referee Report

3 major / 7 minor

Summary. The paper proposes a reversible data-hiding scheme for compressive sensing (CS) streams: a subset of CS measurements is XOR-encrypted with a CSPRNG keystream and embedded, on-the-fly, into the remaining measurements via a modified prediction-error-expansion algorithm. An authorized receiver recovers the exact measurements and reconstructs the original image; an unauthorized receiver reconstructs only a visibly distorted image from the truncated, modified stream. The paper derives a closed-form capacity relation C_r ≈ P·n/(1+P²·n/16), analyzes the resulting volume expansion r ≈ 2/(1+n/16), and reports experiments on 359 patches of a sky image and on real single-pixel-camera measurements, with a median PSNR of 18.21 dB at 10 insertion levels for the simulated data.

Significance. If the construction is correct, the scheme offers a practical method for protecting CS streams without buffering, with tunable distortion and a compact analytical capacity estimate. The derivation of Eq. (14) from a stated Gaussian measurement model is a useful contribution, and the experimental rate-distortion curve together with the execution-time and memory table provide evidence of feasibility. However, the central embedding rule is built on an incorrect zero-mean claim in Eq. (3), and the real-data results depend on an undocumented centering or normalization step. Until these are corrected, the quantitative claims of the paper are not supported. The paper does not provide code or the real dataset, which would materially help reproducibility.

major comments (3)

[§3.2, Eq. (3)] The premise that 'due to the construction of Hadamard matrix, the mean estimate is always zero' is false. For a Sylvester Hadamard matrix with the first row removed, the sample mean of the measurements is μ = (1/L)Σ_i y_i = (1/L)(Σ_i φ_i)·x, and Σ_i φ_i is not the zero vector. For example, the 4×4 Hadamard rows 2–4 sum elementwise to (3,−1,−1,−1), so for x = (1,0,0,0) the three measurements are all 1 and their mean is 1. Therefore d_i = y_i − μ differs from y_i, and Eqs. (4)–(6) are not the actual embedding rule for the simulated experiments. The real-data experiments are worse: the S-matrix has nonnegative entries, its rows have positive sums, and the measurements are nonnegative, so a zero mean is impossible; under the stated T=500 and Eq. (6), essentially no real measurement falls in [−T,T], yet the paper reports successful real-data insertion (PSNR 10.48 dB at n=10). This indicates an unstated centering or normalization step. This is load-bearing for the embedding rule, the capacity derivation, and all real-data results, and it must be fixed and explicitly stated.
[§3.2, Eqs. (4) and (6)] Equations (4) and (6) are printed with missing operators in the shifted branches: 'd_i + bn_max T+bn_max' and 'd_i − bn_max T' are not well-formed arithmetic expressions. The standard prediction-error-expansion shift requires adding or subtracting (bmax + T) to or from d_i, but the printed forms are ambiguous and cannot be used to verify overlap-freeness or reversibility. The same typesetting problem appears in Eq. (16). The authors must print the correct piecewise definitions, including the exact handling of negative values in the expansion branch, before the method can be reproduced or verified.
[§4, Eqs. (12)–(14), and §7, Fig. 10] The capacity formula Eq. (14) is not experimentally validated. The only capacity plot, Fig. 10, is computed directly from Eq. (15), not measured by running the insertion procedure and counting how many measurements actually carry data. Because Eq. (14) is a central claimed contribution, and because it relies on the approximation q ≈ P² derived only for the case of two carrying measurements per payload measurement (n between 8 and 16), the paper should provide a direct comparison of predicted and achieved capacity, especially for tight thresholds where the q approximation matters. Without such a comparison, the capacity claim, and the statement that T provides fine control of capacity and distortion, remain unverified.

minor comments (7)

[§3.1] The extraction procedure is underspecified: after 'going through y_wm to extract embedded data,' the paper does not state how the location map is used, how carriers are identified, or how remainder bits across consecutive measurements are reassembled. The reference to [37] may cover this, but the paper should be self-contained enough for a reader to implement the method.
[§4, Eq. (12)] The derivation of q contains the phrase 'where m is a the number of terms' and the error analysis for the geometric-series truncation is reported only for m=1. Please clarify how m is chosen and why the 1% and 5% error estimates extend to the actual streams used in the experiments.
[§7, Fig. 11] The plots report only the median PSNR over 359 patches; adding interquartile ranges or box plots would show the spread across patches with different sparsity and would make the distortion-control claims more convincing.
[§7, real-data experiments] For the real S-matrix data, the paper should state explicitly how the raw measurements were scaled, centered, or otherwise preprocessed before applying the embedding rule of Eq. (6). Without this, the reported real-data PSNRs cannot be reproduced.
[Abstract and §7] The abstract says 'approximately 18 dB' while Section 7 reports a median PSNR of 18.21 dB; please align the wording and report the exact measure (median vs. mean) consistently.
[References] References [28] and [32] are identical; one of them should be removed or replaced with the correct distinct source.
[§8, Conclusion] The conclusion states the volume expansion is '40% to 6%' for n∈[7,14], whereas Section 7 and Eq. (22) give approximately 39% to 7%; please correct the inconsistency.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: capacity is an analytic model and PSNR values are empirical; the only self-citations are minor and not load-bearing.

full rationale

The derivation chain is self-contained. The insertion-capacity formula in Eq. (14) follows from an explicitly stated Gaussian measurement model with a known standard deviation and a user-chosen threshold T; Eq. (10) is a conservation relation for the number of insertable measurements, and q is approximated, not fitted. The experimental PSNR values, including the 18 dB at 10 levels and 10.48 dB on real data, are measured outputs rather than predictions forced by the model, so there is no fitted-input-called-prediction pattern. The compression-rate formula in Eq. (22) is straightforward algebra from the capacity expression. The only self-references are the authors' prior RDH papers [36,37], which supply the reversible prediction-error-expansion building block; the underlying algorithm is independently attributed to [38], and the present paper specifies its modified predictor and thresholds. That self-citation is a foundation for the method but not a circular justification of the paper's central capacity or distortion claims. The zero-mean assumption in Eq. (3), d_i = y_i - mean(y) = y_i, is a substantive factual premise that may fail for the real-data S-matrix and would require an unstated centering step; however, this is a correctness risk about an unverified input, not a circularity, because the reported distortion results are empirical measurements rather than consequences derived from assuming Eq. (3). Overall: no circular derivation; one minor, non-load-bearing self-citation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on a Gaussian, zero-mean measurement model, and the modified RDH algorithm depends on d_i = y_i, which collapses to y_i only for balanced Hadamard rows. Insertion levels n and threshold T are user-chosen control parameters, not fitted constants. The real-data S-matrix violates the zero-mean premise unless an unstated centering step is applied.

free parameters (3)

insertion levels n = 10 recommended; experiments run for n in [7,14]
A user-chosen control parameter that sets the number of bits embedded per carrier and hence the capacity and distortion. It enters Eqs. (14) and (22) and is the main tuning mechanism in the paper.
threshold T = T=500 loose, T=50 tight, Tmax derived
Controls the probability P that a measurement is eligible for insertion. It enters Eqs. (4), (9), and (16)-(18); choosing T=500 makes all measurements eligible, simplifying the capacity formula.
measurement standard deviation sigma
A scene-dependent input to the Gaussian capacity model in Eq. (9), estimated from measurement statistics rather than fitted to the reported PSNR values.

assumptions (4)

standard math Excluding the first row, scrambled Hadamard rows have zero mean, so prediction error equals the measurement itself (Eq. 3).
Used in Section 3.2 to set d_i = y_i. True for balanced +/-1 Hadamard rows, but not for the binary S-matrix used with real data.
domain assumption CS measurements follow a zero-mean Gaussian distribution with standard deviation sigma.
The capacity derivation in Eqs. (9)-(14) integrates a Gaussian PDF; the paper states in Section 4 that scrambled Hadamard matrices behave like random Gaussian matrices.
domain assumption The scene is sparse in some domain and the sensing columns are incoherent with the sparsity basis.
Standard CS reconstruction premise stated in Section 1; needed for FISTA reconstruction to succeed.
domain assumption An unauthorized user can reconstruct a lower-quality image from the modified measurements and the truncated measurement matrix.
Since some measurements are removed for embedding, the remaining carriers are used with a reduced matrix; this assumption underlies the claimed visible-distortion behavior.

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Pith. "Pith review of Joint Data Hiding and Partial Encryption of Compressive Sensed Streams." pith.science (2026). https://pith.science/paper/UMFEFISW

@misc{pith2026250523357,
  author       = {Pith},
  title        = {Pith review of: Joint Data Hiding and Partial Encryption of Compressive Sensed Streams},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMFEFISW}},
  note         = {Machine review of arXiv:2505.23357}
}

abstract

The paper proposes a method to secure the Compressive Sensing (CS) streams. It consists in protecting part of the measurements by a secret key and inserting the code into the rest. The secret key is generated via a cryptographically secure pseudo-random number generator (CSPRNG) and XORed with the measurements to be inserted. For insertion, we use a reversible data hiding (RDH) scheme, which is a prediction error expansion algorithm, modified to match the statistics of CS measurements. The reconstruction from the embedded stream conducts to visibly distorted images. The image distortion is controlled by the number of embedded levels. In our tests, the embedding on 10 levels results in $\approx 18 dB $ distortion for images of 256x256 pixels reconstructed with the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). A particularity of the presented method is on-the-fly insertion that makes it appropriate for the sequential acquisition of measurements by a Single Pixel Camera. On-the-fly insertion avoids the buffering of CS measurements for a subsequent standard encryption and generation of a thumbnail image.

Figures

Figures reproduced from arXiv: 2505.23357 by the authors.

**Figure 1.** Schema of the proposed scenario. (up) Data insertion: if information is available for embedding and the measurement is eligible, the modified measurement is computed and transmitted. If not, the measurement is shifted and transmitted. If information is no longer available for embedding, the current measurement is processed to be used for insertion. (down) Data extraction: if the user is authorised, the embedded info… view at source ↗

**Figure 2.** Embedding process example on 7 bits. With these steps, the following data is transmitted: the marked measurements ywm, the seed value used to construct the measurement matrix and a location map representing the positions in y that were extracted and then embedded in following eligible measurements. An authorised user has access to extra information that allows the original measurements to be computed: the threshold … view at source ↗

**Figure 3.** An example of measurement distribution for scrambled Hadamard matrix. The blue area delimited by the thresholds −T and T is the probability of insertable measurements. Knowing the distribution and the thresholds [−T, T], the number of insertable measurements M can be estimated by multiplying the total number of measurements L by the probability P to be between thresholds: M = L ∗ PT,σ = L ∗ Z T −T 1 σ √ 2π e −x 2 … view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: The theoretical relative capacity as a function of measurements standard deviation for Tmax and parametrized by n. With dashed line, the capacity calculated for no restrictive thresholds [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 6.** Figure 6: The distortion as a function of insertion levels in three instances: reconstruction from truncated measurements with no embedding (blue), marked measurements extended to contain the measurements to be embedded too (red), and including both embedding and truncation (bla…

**Figure 7.** Figure 7: Example of ECA attack: (a,c) Patch reconstructed with insertion on 10 levels, (b,d) The same patch recontructed with clean measurements and reduced Hadamard matrice (after EDA attack). on the less significant bits of the carriers. If loose thresholds are used, by sever…

**Figure 8.** Figure 8: The image of the W51 nebula obtained by NASA’s Spitzer Space Telescope. The image was used to generate the simulated data [40] [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Two examples of patches used for tests: a dense one (left) and a sparse one (right). patches with different sparsities. We simulated the acquisition by calculating random projections on a Hadamard matrix generated using Sylvester’s construction. The matrix only contain…

**Figure 10.** Figure 10: The relative capacity Cr vs. the number of insertion levels n, for loose thresholds (T = 500) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: The median of PSNR experimental values against n for T = 500 and T = 50. The vertical dotted lines marks the range of interest for the application [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 13.** Figure 13: Reconstruction examples for a sparse and a dense patch: (a,e) originals obtained with the 40% genuine measurements; (b,f) images after performing insertion on 7 levels; (c,g) images after insertion on 10 levels; (d,h) images after insertion on 14 levels. Examples of i…

**Figure 14.** Figure 14: Relative Capacity Cr as a function of PSNRs median. The PSNR median at 10 levels is 18.21 dB for the whole set ( [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

**Figure 15.** Figure 15: The compression rate vs. n for insertion with loose thresholds [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]

**Figure 17.** Figure 17: Images reconstructed from real measurements: (a) Original obtained with 40% measurements; (b) image after data insertion on 7 levels; (c) image after insertion on 10 levels; (d) image after insertion on 14 levels. The PSNR is calculated by respect to the Original. 8. …

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