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REVIEW 5 major objections 6 minor 37 references

An Effective Equivalence Model of Analyzing PLS of Multiple Eavesdroppers Facing Low-altitude Communication Systems

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that a virtual Eve with movable antennas can reproduce the secrecy loss caused by multiple colluding eavesdroppers, and it gives an optimization to make the match quantitative.

desk verdict A virtual-Eve equivalence for multiple colluding eavesdroppers that is mostly an artifact of choosing the distance to match the expected SNR; the math is clean but the claim overreaches. read the letter →

arxiv 2507.05878 v1 pith:L6HT5NZK submitted 2025-07-08 cs.IT eess.SPmath.IT

Yujia Zhao , Zhiyong Feng , Kan Yu , Qixun Zhang , Dong Li This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords physicallayersecuritymovableantennassecrecyrateequivalencemodelcolludingeavesdropperslow-altitudecommunicationsfield-responsechannelalternatingoptimization
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

Multiple colluding eavesdroppers at unknown positions are the main obstacle to closed-form secrecy-rate analysis in low-altitude links, because their random geometry defeats simple expressions. This paper tries to remove that obstacle by showing that the combined effect of $M$ fixed-position eavesdroppers can be represented by a single 'virtual Eve' equipped with an MA array: the antennas' positions model the different eavesdropping channels, and an equivalent distance models the path loss. The authors derive closed-form expected SNRs for both configurations and minimize their difference by alternating between the equivalent distance and the MA positions. If the equivalence holds, secrecy-rate analysis reduces to tuning one surrogate receiver, and the numerical results in the paper put the remaining secrecy-rate error around $1\%$ across the tested settings.

What carries the argument

The carrying object is the virtual Eve, a hypothetical linear array of $M$ movable antennas that replaces $M$ colluding single-FPA eavesdroppers. Each antenna position $r_m$ changes only the phase of each propagation path through factors $\alpha_u(r_m) = e^{j\frac{2\pi}{\lambda}r_m\cos\theta_u}$, while the transmitter-side factors $\gamma_u = \sum_n w_n\beta_u(t_n)$ encode the beamforming and MA positions at the BS; after expectation over path gains, cross terms vanish and the SNR gap $\frac{g_0 d^{-\alpha}}{L_t\sigma^2}\sum_m |\sum_u \alpha_u(r_m)\gamma_u|^2 - \sum_m \frac{g_0 d_m^{-\alpha}}{L_t\sigma^2}\sum_u|\gamma_u|^2$ separates cleanly into $d$ and $R$. Alternating optimization then fixes $R$ to solve $d$ in closed form and fixes $d$ to solve $R$ by a convex slack-variable reformulation, which is what makes the equivalence numerically tunable.

What would settle it

Place two Eve antennas at the same distance but different azimuth angles under the paper's 28 GHz, $\alpha=4$ simulation settings, compute the exact colluding SNR, and optimize a two-MA virtual Eve with the paper's algorithm; if the minimal achievable gap $E[\mathrm{SNR}_{\mathrm{veve}}] - E[\mathrm{SNR}_{\mathrm{col}}]$ stays clearly positive as the angular separation grows, or the secrecy-rate error exceeds the claimed $\sim1\%$, the equivalence fails outside the common-direction case.

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Extended reading notes

Core claim

The central claim is that a virtual Eve with $M$ movable antennas can reproduce the eavesdropping capability of $M$ colluding single-FPA Eves, and that the reproduction can be made quantitatively close by choosing an equivalent distance $d$ and an MA position matrix $R$. Concretely, under a field-response channel model with i.i.d. zero-mean complex-Gaussian path gains, the expected SNR of the virtual Eve is $E[\mathrm{SNR}_{\mathrm{veve}}] = \frac{g_0 d^{-\alpha}}{L_t\sigma^2}\sum_{z=1}^{M}\left|\sum_{u=1}^{L_t}\alpha_u(r_z)\gamma_u\right|^2$ and the expected colluding SNR is $E[\mathrm{SNR}_{\mathrm{col}}] = \sum_{m=1}^{M}\frac{g_0 d_m^{-\alpha}}{L_t\sigma^2}\sum_{u=1}^{L_t}|\gamma_u|^2$; the model keeps $E[\mathrm{SNR}_{\mathrm{veve}}] - E[\mathrm{SNR}_{\mathrm{col}}] \ge 0$, so the virtual Eve is a conservative stand-in. The gap is minimized at the boundary $d = d_{\max}$ for fixed antenna positions, and the position subproblem admits a convex reformulation, yielding a stationary point by alternation. In the paper's simulations the resulting secrecy-rate error is under $1\%$.

Load-bearing premise

The load-bearing premise is that all eavesdroppers share the same set of path directions from the transmitter, so a single beamforming factor multiplies every Eve's channel; if real Eves sit at different angles, one virtual Eve cannot generally reproduce their combined SNR by moving antennas and adjusting distance alone.

Editorial extensions

If this is right

  • If the model is right, the secrecy rate of $M$ colluding Eves has a closed-form proxy, so designers can predict how physical-layer security changes with path-loss exponent, noise power, Eve count, and distance without heavy Monte Carlo simulation.
  • The jointly optimized equivalent distance and MA positions give a conservative approximation: the virtual Eve's expected SNR is kept at least as large as the colluding Eves' total, so the virtual-Eve secrecy rate is a lower bound on the true secrecy rate.
  • Increasing the number of MAs at the virtual Eve narrows the equivalence gap, with diminishing returns beyond roughly seven antennas in the simulated settings.
  • The equivalent distance shrinks as the number of colluding Eves or the path-loss exponent grows, while noise power and MA moving range have little effect on the quality of the match.

Reading between the lines

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

  • A testable extension the paper does not run: keep the same equivalence objective but allow the virtual Eve's antennas to move in two dimensions; the extra angular degrees of freedom should relax the common-direction assumption and widen the regime where the model holds.
  • The closed-form virtual-Eve SNR could be inverted, so a designer could choose $M$, $d$, and $R$ to enforce a target secrecy margin instead of simulating many Eve deployments—an operational use the paper leaves implicit.
  • The same compression idea—many distributed single-antenna nodes replaced by one MA-equipped surrogate—could apply to legitimate receivers or cooperative jammers, not only eavesdroppers.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper considers an uplink MA-assisted secure communication system with one legitimate receiver and M colluding single-FPA eavesdroppers, and proposes to approximate the multi-Eve system by a single virtual Eve equipped with M movable antennas. The authors define a secrecy-rate gap, replace it with an expected-SNR gap, and minimize that gap by alternately optimizing the virtual Eve's equivalent distance d and its antenna positions R. They claim this yields a closed-form approximation of the multi-Eve secrecy rate and validate the model through simulations over path-loss exponent, noise power, number of Eves, and antenna moving range.

Significance. The problem addressed is relevant: a tractable model for the secrecy performance of multiple colluding eavesdroppers in low-altitude wireless systems would be useful for system design. The paper also has a clear problem formulation and a systematic numerical study, which are strengths. However, the central equivalence claim is not established. The derivation of the colluding Eves' expected SNR implicitly assumes that all Eves and the virtual Eve share identical path directions, a restriction not stated in the model or simulations. The distance update in Eq. (18) forces the expected SNR gap to zero by construction, so the numerical agreement is an in-sample artifact of the fitting procedure rather than independent evidence of equivalence. In addition, the secrecy-rate gap has a sign error, and the claimed convexity of the antenna-position subproblem is incorrect. These are load-bearing issues that affect the formulation, the algorithm, and the interpretation of the results. The paper does not provide machine-checked proofs or code, and no closed-form secrecy-rate expression is actually derived.

major comments (5)
  1. [Section III-D/E, Eq. (13)] The expression E[SNRcol] = (1/(Ltσ^2)) Σ_m g0 d_m^{-α} Σ_u |γ_u|^2 drops the Eve index from the path directions and path gains. In Eq. (1), the m-th Eve channel is h_e^m(t)=Σ_l σ_l e^{j2π/λ t^T p_l^m}, with Eve-dependent p_l^m and, per the earlier definition of Σ_m, Eve-dependent gains σ_{l,m}. However, γ_u in Eq. (13) is defined with β_u(t_n)=e^{-j2π/λ t_n^T p_u}, independent of m. Unless every Eve and the virtual Eve see exactly the same transmit-side path directions {p_u} and the same path gains, the colluding expected SNR is Σ_m (g0 d_m^{-α}/(Lt σ^2)) Σ_u |Σ_n w_n e^{j2π/λ t_n^T p_u^m}|^2, which is not proportional to a common factor Σ_u |γ_u|^2. The equivalence model is therefore restricted to Eves that differ only in distance, and this restriction is not stated in the system model or in the simulation setup. This is a load-bearing gap because Eq. (15), the objective (16), and Eq. (18) all rely on the factored form.
  2. [Section III-E, Eq. (3)] The sign in the secrecy-rate gap is inconsistent. With R_s(C_e)=[C_bob-C_e]^+, the difference R_s(C_ve)-R_s(C_col) equals C_col-C_ve when C_ve ≥ C_col and the projection is active, not C_ve-C_col as written in Eq. (3). The statement that the virtual Eve should have a better channel quality and hence a lower secrecy rate implies ΔR_sec ≤ 0, whereas the paper imposes ΔR_sec ≥ 0 and minimizes it. This inconsistency propagates into Eq. (4) and the sign of E[ΔSNRdif] in problem (5), so the quantity actually minimized is not the secrecy-rate difference described in the text.
  3. [Section IV-A, Eq. (18)] The distance update sets d = d_max, which by construction makes E[ΔSNRdif] = 0 for the current R. Substituting d_max into the first term of Eq. (15) yields exactly E[SNRcol], so the expected SNR gap is zero after the distance subproblem. The subsequent R-subproblem in problem (25) then becomes a fitting step that tries to maintain this equality. Consequently, the small secrecy-rate gaps reported in Figs. 6-11 are in-sample consequences of the optimization objective, not independent evidence that a single virtual MA Eve reproduces the M-Eve secrecy rate. Validation on configurations not used to fit d and R would be needed to support the equivalence claim.
  4. [Section IV-B, Eqs. (21)-(23)] The claim that constraints (21)-(23) are convex is incorrect. The quantities p_m and q_m are the real and imaginary parts of Σ_u α_u(r_m)γ_u, where α_u(r_m)=e^{j2π/λ r_m cos θ_u} is a sinusoidal function of the antenna position r_m. The inequalities v_m ≥ p_m^2+q_m^2 are second-order cone constraints only if p_m and q_m are treated as free variables, but the defining equalities (21)-(22) are nonlinear and non-convex in r_m. Therefore the statement that constraints (5b), (21), (22), (23), (24), and (25b) are convex is not justified, and the CVX-based solution of problem (25) is not substantiated.
  5. [Abstract and Section I] The abstract and introduction claim that the paper derives a closed-form expression for the secrecy rate and equates the secrecy rates of M Eves with single FPAs to those of a virtual Eve with an MA array. What is actually derived is an expected-SNR gap, Eq. (15), under the symmetry assumption discussed above, and the antenna positions R are obtained numerically by CVX within an alternating-optimization loop. No closed-form secrecy-rate expression appears in the paper. The claims should be revised to describe an SNR-matching optimization rather than a closed-form equivalence.
minor comments (6)
  1. [Abstract and Section I] Please align the wording with the actual contribution: the paper provides an optimization-based SNR-matching procedure, not a closed-form secrecy-rate expression.
  2. [Index Terms] The index term 'Physcal layer security' should read 'Physical layer security'.
  3. [Throughout] The symbol M is used both for the number of Eves and for the number of MAs at the virtual Eve, and Section V varies the two independently (e.g., Figs. 6 and 10); distinct notation would avoid confusion.
  4. [Section IV-C, Fig. 4] The discussion of Fig. 4 states that the first-iteration distance is too large and makes the objective negative, but Eq. (18) sets d=d_max, which should yield E[ΔSNRdif]=0 for the given R; these statements need to be reconciled.
  5. [Table II and Fig. 11] The reported secrecy rates are on the order of 10^-7 bps/Hz; please justify the parameter choices or clarify the scaling of the y-axis in Fig. 11.
  6. [Section V-C-2] There is a duplicated phrase 'of the optimization problem (5)' and a sentence fragment beginning 'The distance d between the BS and the virtual Eve, known as an crucial factor...'; these should be rewritten.

Circularity Check

2 steps flagged · score 8.0 of 10

The equivalence is forced by construction: Eq. (18) selects d so that E[SNRveve] = E[SNRcol], and Eq. (13) drops Eve-dependent path directions, so the virtual-Eve model matches by definition rather than by independent prediction.

  1. self definitional [Section III-G, proof of Eq. (13)]
    "The power strength of received signal is given by |he_m(T)^H w|^2 = | w1 Σ_u σ_u β_u(t1) + ... + wN Σ_u σ_u β_u(tN)|^2 = |Σ_u σ_u γ_u|^2. By using the same method ..., E[SNRcol] = Σ_m g0 d_m^{-α}/(Lt σ^2) (Σ_u |γ_u|^2)."

    In the physical channel model of Eq. (1), each Eve has its own path directions p_l^m, so the beamforming inner sum should be γ_u^m = Σ_n w_n exp(j2π/λ t_n^T p_u^m). The proof drops the Eve index m and writes the same γ_u for every Eve. The colluding SNR therefore becomes a distance-weighted multiple of one common beamforming gain, which a single virtual Eve can absorb by choosing one equivalent distance. The claimed equivalence between M Eves and the virtual MA Eve is thus built into the definition of E[SNRcol] rather than derived from the channel model.

  2. fitted input called prediction [Section IV-A, Eq. (18)]
    "d ≤ dmax = [ g0 ( |Σ_u α_u(r1)γ_u|^2 + ... + |Σ_u α_u(rM)γ_u|^2 ) / (E[SNRcol] Lt σ^2) ]^{1/α}. ... Thus, when d = dmax, the objective function reaches the minimum value, and dmax is the optimal solution of the subproblem (17a)."

    Because E[SNRveve] = g0 d^{-α}/(Lt σ^2) Σ_z |Σ_u α_u(r_z) γ_u|^2, substituting d = dmax makes E[SNRveve] = E[SNRcol] identically. The equivalent distance d is not an independently estimated physical parameter; it is solved from the very quantity the model is supposed to predict. The later simulations and Table II report small secrecy-rate gaps as validation, but those gaps are a consequence of the fitted d forcing the expected SNR difference to zero, not evidence that a real MA Eve at a physical distance reproduces M colluding FPA Eves.

full rationale

The central claim of the paper is that a single virtual Eve with an MA array can equate the secrecy rate of M colluding single-FPA Eves. The derivation makes this true by construction twice. First, Eq. (13) and its proof replace the Eve-specific path directions p_l^m of Eq. (1) with a common γ_u; this suppresses the angular diversity of real Eves, leaving only distance differences. Second, Eq. (18) chooses the equivalent distance d so that E[SNRveve] equals E[SNRcol], making the objective E[ΔSNR] zero at the optimum. The numerical validation therefore does not test a physical equivalence; it tests a model whose expected SNR difference was zeroed by the fitted distance, and only for Eves that are artificially made to share the same transmit-side path directions. The paper does not rely on a load-bearing self-citation chain: reference [12] is the authors' prior work but is used for context, not to force the equivalence. The circularity is in the equations themselves: the predicted matching quantity is the input used to select the equivalence parameters. Score 8 reflects that the central equivalence result is forced by construction rather than by a genuinely falsifiable physical model.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests on several modeling assumptions that are either stated but idealized (perfect CSI, Rayleigh path-loss) or hidden (identical path directions across Eves), plus artificial degrees of freedom (the virtual distance d and MA positions R) that are fitted to make the equivalence hold. The virtual Eve is an invented analytical entity with no independent falsifiable handle.

free parameters (3)
  • Equivalent distance d between BS and virtual Eve = d = d_max from Eq. (18)
    The distance is an artificial variable chosen to make E[SNRveve] equal E[SNRcol]; it is not a measured physical quantity and is fit to the eavesdroppers' deployment.
  • MA positions R at the virtual Eve = optimized by CVX in Eq. (25)
    Positions of antennas on a non-existent Eve are tuned to minimize the expected SNR gap; they carry no physical constraint from any real receiver.
  • Beamforming vector w at the BS = not specified
    The paper never states how w is chosen; derived expressions and simulation secrecy rates depend on it.
assumptions (5)
  • domain assumption Perfect CSI of Bob and all Eves is available at the BS
    Stated in Section III-C; the authors acknowledge Eves' CSI is hard to obtain in low-altitude systems.
  • ad hoc to paper All Eves and the virtual Eve share the same set of path directions p_u so that gamma_u is identical across links
    Used in Eq. (13) where E[SNRcol] factors as sum_m g0 d_m^{-alpha} sum_u |gamma_u|^2; without this, the m-th Eve has its own p_u^m and the derivation fails.
  • domain assumption Path gains sigma_u are i.i.d. CN(0, g0 d^{-alpha}/L_t)
    Section III-G; standard Rayleigh fading path-loss model, but the variance depends on the distance to each receiver.
  • ad hoc to paper Minimizing the secrecy-rate gap Delta Rsec is equivalent to minimizing the expected SNR gap E[Delta SNRdif]
    Stated in Section III-F (Eqs. 3-4); log(1+SNR) difference is not generally equivalent to SNR difference.
  • domain assumption The virtual Eve can always be placed so that its channel is at least as good as the colluding Eves, giving Delta Rsec = C_ve_MA - C_col
    Section III-E, the '(a)' step; this is an assumption about feasibility of the equivalence model.
invented entities (1)
  • Virtual Eve with M movable antennas
    purpose: Serves as an analytical proxy for M physically separate colluding eavesdroppers with single fixed antennas
    The paper states the virtual Eve 'does not actually exist'; its distance and antenna positions are free variables tuned to match the real Eves, and it makes no falsifiable prediction outside the model.

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Cite this review

Pith. "Pith review of An Effective Equivalence Model of Analyzing PLS of Multiple Eavesdroppers Facing Low-altitude Communication Systems." pith.science (2026). https://pith.science/paper/L6HT5NZK

@misc{pith2026250705878,
  author       = {Pith},
  title        = {Pith review of: An Effective Equivalence Model of Analyzing PLS of Multiple Eavesdroppers Facing Low-altitude Communication Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6HT5NZK}},
  note         = {Machine review of arXiv:2507.05878}
}
read the original abstract

In low-altitude wireless communications, the increased complexity of wireless channels and the uncertainty of eavesdroppers (Eves)--caused by diverse altitudes, speeds, and obstacles--pose significant challenges to physical layer security (PLS) technologies based on fixed-position antennas (FPAs), particularly in terms of beamforming capabilities and spatial efficiency. In contrast, movable antennas (MAs) offer a flexible solution by enabling channel reconstruction through antenna movement, effectively compensating for the limitations of FPAs. In this paper, we aim to derive a closed-form expression for the secrecy rate, a key metric in PLS, which is often unattainable in current studies due to the uncertainty of Eves. We construct an equivalent model that leverages the reconfigurable nature of MAs, equating the secrecy rates obtained by multiple Eves with single FPAs to those achieved by a single virtual Eve equipped with an MA array. To minimize the gap between these two types of secrecy rates, we formulate and solve an optimization problem by jointly designing the equivalent distance between the transmitter and the virtual Eve} and the antenna positions of MAs at the virtual Eve. Numerical simulations validate the effectiveness of the proposed equivalent model, offering a new perspective for PLS strategies. This work provides significant insights for network designers on how system parameters affect PLS performance.

Figures

Figures reproduced from arXiv: 2507.05878 by the authors.

Figure 2
Figure 2. equivalent model between M Eves and a virtual Eve with M MAs which means that the antenna only moves in one dimension. Let dbob-bs/dveve-bs be the communication distance between the BS and the Bob/virtual Eve. The distance parameter dveve-bs and antenna locations of the MAs at the virtual Eve are crucial factors of the equivalent model deviation minimization, since they can affect the signal attenuation via diverse … view at source ↗
Figure 3
Figure 3. spatial angles of the j-th transmitting path Therefore, the received transmission field response vector of the n-th transmitting MA at the Bob can be expressed as gn(tn) = h e j 2π λ t T n p 1 , ej 2π λ t T n p 2 , . . . , ej 2π λ t T n p Lt iT , 1 ≤ n ≤ N. The corresponding field response matrix can be represented as G(T) = [g1(t1), g2(t2), . . . , gN (tN )] ∈ C L t×N . Next, we model the eavesdropping channels of … view at source ↗
Figure 4
Figure 4. , the value of d at this point is relatively large, which means that the virtual Eve is far from the BS, resulting in the objective function being negative. 2) given the value of d, the antenna position matrix R can be optimized, which yields the optimal positions of MAs R∗ at this stage. Since R has a smaller influence on the objective function, even when using R∗ computes the objective function value, it cannot of… view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: antenna positions of MAs before and after moving [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: relationship between equivalent effect and the number of Eves further prove the findings, which describe the percentage error achieved by (E[SNRcol] − E[SNRveve]) /E[SNRcol], and show no clear trend with variable changes. In particular, correspond￾ing percentage error …
Figure 8
Figure 8. Figure 8: relationship between equivalent distance and path-loss exponent, noise power equivalent distance, corresponding reasons are similar to these of [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 10
Figure 10. Figure 10: the impact of the number of MAs at the virtual Eve on equivalence difference minimization of secrecy rate 2 4 6 8 10 12 14 16 18 20 9.4 9.6 9.8 10 10.2 10.4 secrecy rate (bps/Hz) 10-7 equivalent model realistic deployment of Eves incremental distance of Eve(m) [PITH_…
Figure 11
Figure 11. Figure 11: the secrecy rate of two models 48.5m, and 51m. By gradually increasing the distance between the BS and all Eves but without changing the relative position of all Eves, on the one hand, We can find that the secrecy rate obtained by the virtual Eve with M MAs is very cl…

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Reviewed August 6, 2026 · model on record in the stance chip above.