{"id":"aba3b624-e76a-4afd-8e8c-4b4f52837337","arxiv_id":"2502.08967","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A near-field THz wiretap scheme jointly optimizes signal and artificial-noise focusing points via two one-dimensional searches and a closed-form power split, improving secrecy rate over prior low-complexity designs.","lead":"This paper designs a low-complexity artificial noise beam focusing scheme for near-field terahertz wiretap links by jointly choosing where to focus the signal and the noise and how to split power between them. A generalist would read it to see a practical attempt to make 6G terahertz links harder to eavesdrop on without heavy computation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1's sign assertion is unproven; a counterexample sweep over the beam-focusing optima would determine whether the closed-form alpha occasionally misses the true maximizer.","rationale":"The reader's weakest_assumption correctly identifies Lemma 1's sign assertion as the load-bearing analytical step. My independent reading of Eqs. 28-30 confirms that the claim 'F2 and F1 are always smaller than zero' is asserted without derivation and does not follow from the two ratio-maximization objectives alone. This matters because the closed-form alpha in Eq. 27 is the paper's headline contribution over the benchmark; if the sign assertion fails only in a few geometries, the main numerical conclusions could still hold, but the analytical claim is not proven. I agree with CONDITIONAL rather than REJECT because the numerical evidence in Fig. 3 and Fig. 4 plausibly supports the scheme's performance for the tested parameters, and the gap is a proof gap that a numerical sweep could either confirm or refute. The test I propose directly probes the proof's linchpin: evaluate F1 and F2 at the beam-focusing optima over a broad geometry grid and compare the closed-form alpha with an exhaustive search. This is a concrete, finite computation that settles whether the unproven sign assertion is merely a missing justification or an actual counterexample to the claimed optimal power allocation.","tokens_in":9749,"tokens_out":1808,"duration_ms":14260,"concrete_test":"Numerically sweep the Lemma 1 domain: for a grid of N in {129,257,513}, rB in {3,...,8} m, rE in {1,...,10} m, thetaB = 0, thetaE = 0, and use the paper's Sec. III-A search to compute (rS*, rA*) and the resulting rho_i. For each case, evaluate F1 and F2 at these optima and record any instance with F1>=0 or F2>=0; also compare the secrecy rate at the Lemma 1 alpha against a fine one-dimensional grid over alpha in [0,1). If any grid point shows F1,F2>=0 or the closed-form alpha underperforming the grid by more than a small tolerance, Lemma 1's sign claim is falsified and the manuscript must be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central analytical result is the closed-form power split in Lemma 1 (Eq. 27), and its proof rests on the claim that F2<0 and F1<0 always, because 'the beam focusing design maximizes rho1^2/rho2^2 and rho4^2/rho3^2' (Sec. III-B). That implication is not established. F1 in Eq. 29 contains P gB rho1^2 rho2^2 (rho3^2 - rho4^2) + P gE rho3^2 rho4^2 (rho2^2 - rho1^2) + sigma^2 (rho2^2 rho3^2 - rho1^2 rho4^2); F2 in Eq. 30 contains P gB gE rho2^2 rho4^2 (rho2^2 rho3^2 - rho1^2 rho4^2) plus sign-indefinite terms. The two design objectives pin down ratios rho1^2/rho2^2 and rho4^2/rho3^2, but not the signs of mixed products such as rho2^2 rho3^2 - rho1^2 rho4^2, nor the relative weights gB vs gE needed for the bracket terms. Also, Lemma 1 optimizes the exact secrecy-rate expression while the beam-focusing points come from the noise-neglected approximation of Eq. 20; even if F1,F2<0 held at the approximate optima, the chosen alpha would maximize the exact secrecy rate only at two specific beam-focusing points, not over the coupled (ws,wz,alpha) design. The supplied numerical verification (Fig. 3) covers only the displayed geometry, so it does not close the proof gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers a downlink near-field terahertz wiretap system in which a base station with a large-scale uniform linear array transmits to a single-antenna legitimate user while an eavesdropper is present. The authors formulate a secrecy-rate maximization problem over the signal beam focusing vector, the artificial-noise (AN) beam focusing vector, and the power allocation ratio. They decouple the problem into separate signal and AN beam focusing subproblems under a noise-neglect and small-η approximation, solve each by a one-dimensional search, and then derive a closed-form power allocation ratio α in Lemma 1. Numerical results show that the proposed scheme performs close to the optimal joint beam focusing design and outperforms existing low-complexity benchmarks at O(NM) complexity.","tokens_in":10087,"tokens_out":7394,"duration_ms":72667,"significance":"If the analytical claims are fully substantiated, the paper would offer a practical low-complexity design for AN-aided near-field THz physical-layer security, which is a relevant problem for extremely large-scale arrays. The paper is commendably explicit about its computational complexity and provides numerical comparisons against several benchmark schemes. The derivations are traceable to stated references, and the numerical evaluation is reproducible in structure. However, the main analytical result, Lemma 1, depends on an unproven sign assertion, and the beam focusing design relies on a high-SNR, small-η approximation whose interaction with the later power allocation is not fully quantified. These issues are load-bearing for the central claim that the closed-form α and the decoupled focusing points achieve the claimed near-optimal secrecy rate.","major_comments":[{"comment":"The proof of Lemma 1 asserts that F2 and F1 are 'always smaller than zero' because the beam focusing design maximizes ρ1^2/ρ2^2 and ρ4^2/ρ3^2. This implication is not established. F1 in Eq. (29) contains mixed-sign terms such as σ^2(ρ2^2ρ3^2 - ρ1^2ρ4^2) and P gB ρ1^2ρ2^2(ρ3^2 - ρ4^2) + P gE ρ3^2ρ4^2(ρ2^2 - ρ1^2), and F2 in Eq. (30) contains P gB gE ρ2^2ρ4^2(ρ2^2ρ3^2 - ρ1^2ρ4^2) plus additional sign-indefinite terms. The two ratio inequalities do not fix the signs of these expressions, nor do they control the relative weights gB versus gE. Additionally, Eq. (27) returns the stationary point of ∂Ω/∂α without checking whether this root lies in the feasible interval [0,1); if the positive root exceeds 1, the maximum over α∈[0,1) is at the boundary. The numerical test in Fig. 3 covers only the displayed geometry and does not close the proof gap. I recommend either proving the sign claim under explicit sufficient conditions (e.g., the specific operating regime where ρ1^2 > ρ2^2 and ρ4^2 > ρ3^2 with known gB/gE bounds) or adding a systematic parameter sweep comparing the α from Eq. (27) with a numerical line search on the exact secrecy rate expression.","section":"Sec. III-B (Lemma 1, Eqs. (27)-(31))"},{"comment":"The beam focusing design is obtained through two successive approximations: noise is neglected in Eq. (20), and then the small-η expansion log((η + A)/(η + B)) ≈ log(A/B) is used to decouple the problem into P2 and P3. This decoupling is optimal only in the limiting regime η→0 with high SNR, but Lemma 1 may return a non-negligible α for which the focusing points are not necessarily optimal. The paper should quantify the error of these approximations, for example by bounding or numerically evaluating the first-order correction η(1/A - 1/B), and should demonstrate that the focusing points obtained from the approximate objective remain near-optimal for the α delivered by Lemma 1 over a broader parameter range than the single geometry in Fig. 3.","section":"Sec. III-A, Eqs. (20)-(23)"},{"comment":"The manuscript states that θS = θB and θA = θE are adopted because 'the received signal power at the legitimate user is maximized when θS = θB, and the received AN power at the eavesdropper is maximized when θA = θE,' citing [10, Proposition 1]. However, the actual subproblems P2 and P3 maximize the ratios ρ1^2/ρ3^2 and ρ4^2/ρ2^2, not the individual received powers. Maximizing the numerator of a ratio does not generally maximize the ratio when the denominator also depends on the same variable. The cited proposition should be checked for whether it indeed applies to the ratio objectives, or a direct proof or numerical validation of the optimality of θS = θB and θA = θE for P2 and P3 should be provided.","section":"Sec. III-A, paragraph after Eq. (23)"}],"minor_comments":[{"comment":"The expression RS = log2((A+B)/(A+C)) omits the positive-part operator; since the text defines RS = (RB - RE)^+, the expression is valid only when RB > RE. Please clarify that the optimization assumes the regime where the secrecy rate is positive and that the design ensures this regime.","section":"Eq. (16)"},{"comment":"The noise-neglect approximation in Eq. (20) is justified by a high-SNR assumption; when this assumption is later used to set the beam focusing points independently of α, the paper should restate the SNR regime in which the subsequent α optimization remains valid.","section":"Sec. III-A, footnote 2"},{"comment":"The definitions of β2 and β3 contain the term sqrt(d |(1-θE^2)/rE - (1-θS^2)/rS|) with an absolute value, but the earlier expression for β1 uses |(1-θB^2)/rB - (1-θS^2)/rS|. Please ensure the notation is consistent and unambiguous, especially for the arguments of the Fresnel integrals.","section":"Eqs. (24)-(26)"},{"comment":"The verification of Lemma 1 in Fig. 3 is limited to a single geometry; adding a second panel with different rB, rE, or θE would make the claim that the closed-form α matches the exact optimum considerably more convincing.","section":"Fig. 3"},{"comment":"The statement that the proposed scheme has complexity O(NM) would benefit from a more explicit description of how M scales with the desired accuracy of the one-dimensional search, and from a statement of the complexity per beam focusing subproblem rather than only the total order.","section":"Sec. IV, complexity discussion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a competent low-complexity design paper that builds directly on the framework of [10]. The main gap is a proof issue in Lemma 1 rather than a demonstrably false result; the numerical curves are consistent with the claims. I believe the paper can become acceptable if the authors either provide a rigorous sign proof for F1 and F2 under explicit conditions or replace the proof claim with a systematic numerical validation sweep over the relevant parameter space, and if they address the small-η approximation issue. The scope is appropriate for a correspondence/letter venue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is an honest, subfield-scoped correspondence that extends [10] by jointly optimizing the signal and AN focusing points plus a closed-form power split. The complexity story is real (O(NM)), and the numerical figures show the proposed scheme tracking the 'optimal' benchmark and beating [10] in the tested geometries. I believe the design insight that AN helps most when the eavesdropper is closer to the BS.\n\nThe genuinely new pieces are the decoupled subproblems P2/P3, the Fresnel-based one-dimensional searches for rS and rA, and Lemma 1's alpha formula. The decoupling is reasonable because with MRT analog beamforming, the signal and AN focusing points affect separate correlations, so splitting the problem is not crazy. The Fresnel approximations are inherited from [16] and are cited properly.\n\nThe soft spots are about what is proved versus what is asserted. First, Lemma 1's proof says F2 and F1 are 'easily verified' to be negative because the beam focusing maximizes rho1^2/rho2^2 and rho4^2/rho3^2. Looking at Eqs. (29)-(30), the expressions contain mixed products like rho2^2 rho3^2 - rho1^2 rho4^2 and weighted differences that depend on gB vs gE and sigma^2. Those signs are not pinned down by the two design objectives. So the closed-form alpha in Eq. (27) is not proven to be the global maximizer of the exact secrecy rate. The stress-test suggestion is right: a sweep over the beam-focusing optima, varying rB, rE, thetaE, and P/sigma^2, would settle whether the formula misses the true maximizer. Fig. 3 only checks one geometry.\n\nSecond, the beam focusing design is based on the noise-neglected approximation in Eq. (20). Footnote 2 justifies this via moderate/high SNR, which is plausible, but Lemma 1 then optimizes the exact secrecy rate at those approximate focusing points. That mismatch is fixable by stating the theorem as an approximation, but it should be stated.\n\nThird, the 'optimal signal and AN beam focusing' benchmark in Fig. 4 is not fully specified. They say it solves P1 with O((NM)^3), but no algorithm is described, so the near-optimal claim cannot be reproduced. Minor, but worth fixing.\n\nI do not see circularity. The correlation ratios are optimized, not fitted; alpha is checked against the exact expression numerically. The citation pattern is normal, with [10] credited as the direct ancestor.\n\nNet: the paper deserves a serious referee. It is competent, the new design is meaningful within near-field PLS, and the gap is concrete and likely fixable. I would ask the authors to either prove the sign claims in Lemma 1 or replace them with a checked condition, and to report at least one off-design geometry in Fig. 3. If that comes back firm, publish as a correspondence.","headline":"A competent, subfield-scoped extension of [10] with a real closed-form power split and credible simulations, but Lemma 1's sign assertion is unproven and needs a numerical sweep before the alpha formula is trustworthy.","tokens_in":10649,"tokens_out":2396,"would_cite":false,"duration_ms":22194,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Near-field terahertz links can be secured with a low-complexity beam focusing scheme that nearly matches optimal secrecy rates.","keywords":["near-field terahertz communications","physical layer security","artificial noise","beam focusing","secrecy rate","low-complexity design","power allocation"],"falsifier":"Evaluate the correlation values $\\rho_i$ for a geometry with the user at $(5\\text{ m}, 0)$ and the eavesdropper at $(3.5\\text{ m}, 0.01\\text{ rad})$, substitute them into $F_1$ and $F_2$ in Eqs. (29)-(30), and check whether either coefficient is nonnegative. If either is, the derivative sign argument behind Lemma 1 fails, and the closed-form $\\alpha$ is not shown to maximize the exact secrecy rate; a full grid search over $\\alpha$ would then settle whether the claimed optimality still holds.","tokens_in":1694,"feed_emoji":"🔐","tokens_out":2706,"duration_ms":86886,"temperature":0.7,"pith_summary":"This paper claims that the secrecy rate of a near-field terahertz wiretap link can be pushed close to the true optimal with a low-complexity design. Instead of jointly optimizing the signal and artificial-noise beams over the large array, the authors split the problem into two independent one-dimensional searches: where to aim the signal beam to maximize the legitimate-to-eavesdropper signal ratio, and where to aim the artificial noise to maximize the eavesdropper-to-legitimate noise ratio. They then give a closed-form expression for the power split between signal and noise. If correct, practical THz base stations with hundreds of antennas can secure links against nearby eavesdroppers at a fraction of the computational cost of exhaustive optimization. The gain is largest when the eavesdropper is closer to the base station than the legitimate user.","feed_headline":"Two quick searches nearly match optimal THz secrecy rates","feed_subtitle":"Decoupling secrecy design into two radial searches plus a closed-form power split lowers complexity from cubic to near-linear.","key_machinery":"The machinery is the ratio-symmetric decomposition of the secrecy rate. With MRT-type analog beam focusing, the secrecy rate becomes $\\log_2\\left(\\frac{\\eta + \\rho_1^2/\\rho_2^2}{\\eta + \\rho_3^2/\\rho_4^2}\\right)$ after dropping noise, where $\\eta = \\alpha/(1-\\alpha)$; for small $\\eta$ this separates into a signal subproblem depending only on $\\rho_1/\\rho_3$ and an AN subproblem depending only on $\\rho_4/\\rho_2$. The correlations $\\rho_i$ are defined through near-field steering-vector inner products, and the Fresnel-integral approximations of $\\rho_1$ and $\\rho_3$ (from [16]) convert each subproblem into a one-dimensional search over the focusing radius. Lemma 1 then supplies a closed-form $\\alpha$ by setting the derivative of $\\Omega=(A+B)/(A+C)$ to zero, treating the numerator as a quadratic in $\\alpha$.","core_discovery":"On the paper's own terms, the central discovery is that the near-field secrecy-rate maximization P1, which is non-convex and costly, can be approximated by two decoupled designs: choose the signal focusing point $Q_S$ to maximize $\\log_2(\\rho_1^2/\\rho_3^2)$ and the AN focusing point $Q_A$ to maximize $\\log_2(\\rho_4^2/\\rho_2^2)$, with both angular coordinates fixed to the user's and eavesdropper's angles, respectively. The correlations $\\rho_i$ are the magnitudes of inner products between near-field steering vectors, and after a Fresnel-integral approximation (from [16]) each subproblem becomes a one-dimensional search over the radial distance. The paper further derives a closed-form power allocation ratio $\\alpha$ (Lemma 1) that maximizes the full noise-included secrecy-rate expression once the two focusing points are fixed. The numerical message is that this $O(NM)$ scheme achieves a secrecy rate very close to the $O((NM)^3)$ optimal joint design and consistently higher than the benchmark schemes in [7], [9], and [10].","pith_inferences":["A natural next test is multi-user/multi-eavesdropper operation: the paper's footnote pairs each user with its most correlated eavesdropper, but joint power allocation across pairs and the effect of shared eavesdroppers on the one-dimensional search remain to be verified by simulation.","The sign claims for $F_1$ and $F_2$ in Lemma 1 could be replaced by explicit conditions on the correlations; if those conditions fail in some geometry, the closed-form $\\alpha$ would need to be replaced by a one-dimensional search over $\\alpha$, preserving low complexity.","The high-SNR noise-neglect approximation suggests the scheme's secrecy-rate advantage may shrink at low SNR; a direct low-SNR extension would likely need to couple the two focusing subproblems again.","The same decoupling idea could be applied to secure beam focusing in other near-field bands or with planar arrays, where the angular dimension adds a second search degree."],"forward_implications":["The proposed scheme achieves a secrecy rate close to the optimal joint signal-and-AN beam focusing design while cutting computational complexity from $O((NM)^3)$ to $O(NM)$.","The optimal signal and AN focusing points are not the user and the eavesdropper themselves; they move to balance higher received power against higher leakage, and move closer to the user as the eavesdropper approaches the legitimate user.","Artificial noise is only beneficial when the eavesdropper is closer to the base station than the legitimate user; otherwise the closed-form power allocation sets $\\alpha = 0$.","The performance advantage over existing signal beam focusing, null-space AN, and AN beam focusing schemes is most pronounced in the $r_E < r_B$ regime."],"supporting_citations":[{"why":"The AN aided beam focusing approach this work extends and the main performance benchmark.","marker":"[10]"},{"why":"Supplies the Fresnel-integral approximation of steering-vector correlations that turns each beam focusing design into a one-dimensional search.","marker":"[16]"},{"why":"Defines the near-field secrecy rate problem and the distance-disparity insight the decoupling relies on.","marker":"[6]"},{"why":"Provides the max-min signal beam focusing benchmark compared in the results.","marker":"[7]"},{"why":"Provides the null-space AN benchmark compared in the results.","marker":"[9]"},{"why":"Provides the experimental parameter values used in the numerical evaluation.","marker":"[17]"}],"fun_headline_variants":["Two radial searches beat cubic THz secrecy design","Near-field THz secrecy: decouple and search radially","O(NM) AN beam focusing nearly matches cubic optimal","Two 1D searches for near-optimal THz secrecy rates"],"cache_read_input_tokens":12672,"weakest_assumption_plain":"The closed-form power allocation rests on an unproven sign claim: two expressions are asserted to be always negative because of the beam focusing choices, yet they contain terms whose sign is not fixed by those choices.","fun_headline_variants_meta":{"raw":{"variants":["Two radial searches beat cubic THz secrecy design","Near-field THz secrecy: decouple and search radially","O(NM) AN beam focusing nearly matches cubic optimal","Two 1D searches for near-optimal THz secrecy rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000649,"raw_usage":{"total_tokens":2946,"prompt_tokens":882,"completion_tokens":2064,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":1997}},"tokens_in":498,"tokens_out":2064,"duration_ms":16072,"temperature":1.0,"reasoning_tokens":1997,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:03:56.551014+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the correlation values $\\rho_i$ for a geometry with the user at $(5\\text{ m}, 0)$ and the eavesdropper at $(3.5\\text{ m}, 0.01\\text{ rad})$, substitute them into $F_1$ and $F_2$ in Eqs. (29)-(30), and check whether either coefficient is nonnegative. If either is, the derivative sign argument behind Lemma 1 fails, and the closed-form $\\alpha$ is not shown to maximize the exact secrecy rate; a full grid search over $\\alpha$ would then settle whether the claimed optimality still holds.","supporting_citations":[{"cited_title":"Physical layer security for near-field communications via directional modulation,","cited_arxiv_id":null,"evidence_quote":"Provides the null-space AN benchmark compared in the results."},{"cited_title":"The teranova platform: An integrated testbed for ultra- broadband wireless communications at true terahertz frequencies,","cited_arxiv_id":null,"evidence_quote":"Provides the experimental parameter values used in the numerical evaluation."},{"cited_title":"Performance Analysis and Low-Complexity Beamforming Design for Near-Field Physical Layer Security","cited_arxiv_id":"2407.13491","evidence_quote":"The AN aided beam focusing approach this work extends and the main performance benchmark."},{"cited_title":"Hierarchical codebook design for near-field mmWave MIMO communications systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the Fresnel-integral approximation of steering-vector correlations that turns each beam focusing design into a one-dimensional search."},{"cited_title":"Physical layer security in near-field communications,","cited_arxiv_id":null,"evidence_quote":"Defines the near-field secrecy rate problem and the distance-disparity insight the decoupling relies on."},{"cited_title":"Max-min secrecy rate optimization through beam focusing in near-field communications,","cited_arxiv_id":null,"evidence_quote":"Provides the max-min signal beam focusing benchmark compared in the results."}],"review_version":1}