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Secure Integrated Sensing and Communications with Movable Beyond-Diagonal RIS

Chandan Kumar Sheemar, Symeon Chatzinotas

arXiv:2610.34088v1eess.SPcs.ITmath.IT

TL;DR

When the sensed target is an eavesdropper, the illumination needed for sensing conflicts with secrecy. The paper characterizes the fields a lossless reciprocal BD-RIS can realize and uses a two-stream zero-forcing design; it reports that this design is within 0.06 bit/s/Hz of the unrestricted two-stream optimum in tested settings.

  • Problem

    When the target is also an eavesdropper, sensing illumination can reveal confidential information, creating a conflict between target illumination and secrecy.

    Source: §I · p. 1, B. Open Problem and Contributions · p. 1

  • Method

    The paper characterizes realizable field transformations for a lossless reciprocal BD-RIS and uses a rank-two, two-stream zero-forcing design for monostatic sensing.

    Source: A. Main Result · p. 4, C. Incident Fields and the Rank of the… · p. 4, §VIII · p. 11

  • Results

    The closed-form zero-forcing design lies within 0.06 bit/s/Hz of the unrestricted two-stream optimum in all tested settings.

    Source: §VIII · p. 11

  • Takeaways & Limitations

    A rank-two link lets the surface null data at the target and sensing at the user, so sensing costs only an SNR offset rather than an unavoidable rate loss.

    Source: §VIII · p. 11

Abstract

from arXiv · show

In integrated sensing and communication (ISAC), when the sensed target is itself an eavesdropper, the illumination required for sensing coincides with the leakage that secrecy must suppress. This paper shows how a beyond-diagonal reconfigurable intelligent surface (BD-RIS) with movable elements decouples the two tasks in a monostatic setting, where the echo returns to the base station through the surface. We prove that a lossless reciprocal BD-RIS maps $K$ incident fields onto $K$ prescribed fields if and only if the two sets share the same Gram matrix and their bilinear Gram matrix is symmetric; unlike a diagonal RIS, it can thus steer a two-dimensional incident subspace onto any prescribed subspace. With a single stream, sensing incurs an unavoidable rate loss, whereas a closed-form zero-forcing design with a confidential and a Gaussian sensing stream nulls the data at the target and the sensing at the user, so that sensing costs only an SNR offset. Since all variables follow in closed form for any placement, the joint design reduces to a monotone search over the element positions, without alternating optimization. Simulations show that the dual-stream design loses only $1.1$ bit/s/Hz when the sensing requirement grows by $30$ dB, against $8.3$ bit/s/Hz for a single stream, while movement, interconnection, and the robust derivative-null design add up to $1.9$, $0.9$, and $2.6$ bit/s/Hz, respectively.

I. INTRODUCTION

When the sensed target is also an eavesdropper, the illumination needed for sensing conflicts with secrecy. The paper addresses this conflict with a movable BD-RIS, characterizes its realizable fields, and develops secure-ISAC designs and a position-search method.

A. Related Work

B. Open Problem and Contributions

  • A lossless reciprocal BD-RIS realizes prescribed fields if their Gram matrices match and their bilinear Gram matrix is symmetric.The characterization enables fully connected BD-RISs to steer a two-dimensional incident subspace onto a prescribed subspace; diagonal RISs cannot do so.Source: B. Open Problem and Contributions · p. 2
  • With a rank-two link, the proposed zero-forcing design nulls data at the target and sensing at the user, making sensing cost only an SNR offset.For a single stream, sensing instead costs at least log2(1 + Γ̄/2) bit/s/Hz for any placement and surface.Source: B. Open Problem and Contributions · p. 2
  • Element movement provides user–target decorrelation, while the proposed derivative-null design reduces leakage from O(δ^2) to O(δ^4) under target-angle error.The paper also reports that the same element-position variance that renders the nominal null fragile motivates the derivative-null design.Source: B. Open Problem and Contributions · p. 2
  • The joint design uses a monotone coordinate search over element positions, with all remaining variables in closed form and finite-sweep convergence.Numerical results are presented under Rician fading against fixed and diagonal benchmarks.Source: B. Open Problem and Contributions · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The setup uses a base station, a legitimate user, and a target carrying an eavesdropper, with blocked direct links and a movable BD-RIS.The movable elements follow a lossless reciprocal impedance network; the BS–RIS channel is modeled with two dominant paths.Source: §II · p. 2, A. Movable BD-RIS · p. 2, B. Channel Model · p. 3

C. Signal Model and Sensing Metric

The model separates confidential communication and sensing into independent Gaussian streams and defines monostatic sensing through the surface-mediated echo. It then formulates secrecy-rate maximization subject to sensing and power requirements, with target knowledge either exact or uncertain.

C. Signal Model and Sensing Metric

  • The base station transmits independent Gaussian confidential and sensing streams, with the sensing sequence known to the BS for echo processing but unknown to other nodes.The streams have separate precoders and share a total transmit-power constraint.Source: C. Signal Model and Sensing Metric · p. 3
  • Monostatic sensing assumes ideal self-interference suppression and a blocked direct target link, so the echo returns over the target–RIS–BS path.The echo SNR combines forward-path target illumination with the surface-dependent return gain.Source: C. Signal Model and Sensing Metric · p. 3

E. Problem Formulation

  • The design maximizes secrecy rate under the sensing requirement, with channels known in Sections II–IV and target direction uncertain by δ in Section V.The sensing stream acts as artificial noise at both receivers, and the problem is nonconvex in the precoders, scattering matrix, and element positions.Source: D. Secrecy Metric and Threat Model · p. 3, E. Problem Formulation · p. 3

III. FIELD REALIZABILITY OF A BD-RIS

A lossless reciprocal BD-RIS can realize field mappings subject to energy conservation and a symmetric bilinear cross-product condition. Its two-field steering capability depends on surface connectivity, while the rank of the BS–RIS link constrains the incident fields and the return echo.

A. Main Result

  • Theorem 1 characterizes realizable field mappings through energy conservation and a symmetric bilinear cross-product condition, with a constructive sufficiency proof.The bilinear condition is the signature of reciprocity; the construction uses Takagi factorization and a real embedding.Source: A. Main Result · p. 4

B. Consequences

  • A fully connected BD-RIS can steer a two-dimensional incident subspace onto any prescribed subspace, whereas a diagonal RIS generally cannot steer two generic fields in the target subspace.For two fields, the BD-RIS condition is u1^T v2 = u2^T v1; a diagonal RIS instead has 3M real conditions but only M phases.Source: B. Consequences · p. 4
  • Group-connected BD-RISs must satisfy the bilinear condition independently within each group, which generically excludes subspace steering when there are at least two groups.A fully connected surface imposes the weakest conditions; for groups of size one, the conditions reduce to those of a diagonal RIS.Source: B. Consequences · p. 4

C. Incident Fields and the Rank of the BS–RIS Link

  • With one BS–RIS path, all incident and reflected fields are parallel, so data and sensing cannot be separated; a weak second path makes its mode expensive to use.Element movement changes the mode gains through the path correlation, and the same modes govern the echo.Source: C. Incident Fields and the Rank of the… · p. 4
  • The frame shaping the forward fields also determines the return gain, which is bounded by the strongest link mode and has a lower bound when the target channel lies in the frame span.For an isotropic incident subspace under LoS, the return gain is λ_iso β1 M.Source: C. Incident Fields and the Rank of the… · p. 4

IV. SECURE ISAC WITH PERFECT TARGET INFORMATION

With a single stream, the confidential signal must also illuminate the target, creating an inherent sensing–secrecy conflict. The resulting secrecy-rate penalty has a lower bound that applies across placements and surface architectures.

A. Single Stream: An Intrinsic Conflict

  • Theorem 2 analyzes a single stream sent through the strongest incident mode, which provides field energy E = λ1P, and gives its optimum for fixed element positions.This is a design through the strongest mode, not the single-stream optimum of the full problem, although the stated penalty bound applies to every single-stream design.Source: A. Single Stream: An Intrinsic Conflict · p. 5
  • The target’s required illumination imposes a minimum Eve SNR, while Bob’s SNR is bounded by aE, producing the unavoidable rate gap.The bound holds for any placement and architecture.Source: A. Single Stream: An Intrinsic Conflict · p. 5
  • For every single-stream design, the secrecy rate is at least log2(1 + Γ̄β0NM/λ1) bit/s/Hz below Bob’s rate bound, because the information-bearing illumination must meet the target’s sensing requirement.Under LoS, the loss is at least log2(1 + Γ̄/2) bit/s/Hz.Source: A. Single Stream: An Intrinsic Conflict · p. 5

B. Two Streams: Decorrelation Serves Both Tasks

A rank-two link lets the BD-RIS shape two streams to null confidential data at the target and sensing interference at the user. This decoupling makes sensing an additive cost, although the relative advantage over one stream depends on SNR and the design conditions.

B. Two Streams: Decorrelation Serves Both Tasks

  • The ZF two-stream design sets γE = 0 and gives Bob gain proportional to (1 − ρ²)E1, with the sensing constraint fixing E2 and the power constraint determining E1.The ZF fields are feasible, so the resulting rate is an achievable lower bound on the two-stream optimum, not necessarily the optimum itself.Source: B. Two Streams: Decorrelation Serves… · pp. 5–6
  • The phase ϕ trades data-stream power cost against sensing-stream cost through the frame’s return gain; a grid search over ϕ has rate error O(N^-2).The phase-dependent objective is a smooth 2π-periodic function composed of three sinusoids.Source: B. Two Streams: Decorrelation Serves… · p. 5
  • With two streams, sensing costs only an additive offset and a factor 1 − ρ², rather than the single-stream design’s multiplicative SNR loss.The Gaussian sensing stream carries the illumination, while decorrelation serves secrecy and sensing together.Source: B. Two Streams: Decorrelation Serves… · pp. 5–6

C. When Do Two Streams Win?

  • Under LoS with resolvable equal-gain paths and D ≥ Ddn, two streams do at least as well as one at high SNR whenever Γ̄ ≥ 2 (3 dB).At low SNR, the comparison can reverse because orthogonal target illumination costs 1/(1 − ρ²) power, while single-stream illumination is a by-product of the beam aimed at Bob.Source: C. When Do Two Streams Win? · p. 6

D. Role of Placement

Placement controls the channel correlations that determine whether the surface can null interference, and movement can produce large gains when the target is nearby. Under target-angle uncertainty, the same position spread that benefits secrecy sharpens the null, motivating a robust derivative-null design.

D. Role of Placement

  • When D ≥ D_dn, a feasible four-block placement can set both channel correlations to zero; movement then makes the dual-stream design interference-free.With resolvable equal-gain paths, the double-null placement has λ_iso = β₀NM.Source: D. Role of Placement · p. 6
  • At high SNR, movement gains at least 1.3 bit/s/Hz in the stated setting, but the required double-null track length grows as ρ_c approaches 1.The required D_dn scales as 1/|∆s|.Source: D. Role of Placement · p. 6
  • For a nearby Eve, two compact clusters at opposite track ends maximize placement variance; for M = 16 and D = 16λ, the movement gain approaches 7.6 (2.9 bit/s/Hz) when κ|∆s|D ≪ 1.The gain scales as 3D²/(M²d₀²) when D ≫ Md₀.Source: D. Role of Placement · p. 6

V. ROBUST DESIGN UNDER TARGET-ANGLE UNCERTAINTY

  • The position spread that decorrelates Bob and Eve also sharpens Eve’s null, increasing sensitivity to target-angle errors and motivating robust nulling.The robust derivative-null direction removes first-order sensitivity at the cost of one additional spatial degree of freedom, which placement can reduce.Source: §V · p. 6

A. Uncertainty Model and Robust Problem

  • Because the target direction is estimated from echoes, the robust problem requires the echo constraint to hold across an uncertainty sector and evaluates secrecy at its worst angle.The model takes the target as an LoS point scatterer within the sector around the estimated direction; Bob’s channel is assumed known.Source: A. Uncertainty Model and Robust Problem · pp. 6–7

B. Sensitivity of the Nominal Null

A nominal zero-forcing data direction leaks quadratically with target-angle error, while a derivative-null direction suppresses that sensitivity at a data-gain cost. The sensing stream also acts as artificial noise for Eve, and the robust design evaluates secrecy and sensing over the uncertainty sector.

B. Sensitivity of the Nominal Null

  • For the nominal ZF direction, leakage under angle error grows quadratically, with an upper bound proportional to κ²cos²θ̂ MVar(p).The expansion is ζ(θ) = ν²|ȧᴴd₁|² + O(ν³).Source: B. Sensitivity of the Nominal Null · p. 7
  • A derivative-null direction suppresses leakage to fourth order in angle error, but incurs a smaller data gain and uses one additional spatial degree of freedom.The placement can reduce the added degree-of-freedom cost; position variance otherwise sharpens the nominal null and increases its sensitivity.Source: B. Sensitivity of the Nominal Null · p. 7

C. The Sensing Stream as Artificial Noise

  • The sensing stream masks residual Eve leakage by at least a factor of 1 + Γ̄/2, or 1 + Γ̄ for an isotropic incident subspace.This applies when the sensing stream alone meets the echo constraint throughout the uncertainty sector.Source: C. The Sensing Stream as Artificial… · p. 7

D. Robust Design

  • The robust design evaluates the worst-case secrecy rate over a grid of target angles, while the nominal design is preferable for small δ and the position search remains unchanged.The data stream’s illumination also adds margin to the sensing constraint.Source: D. Robust Design · p. 7

VI. JOINT PLACEMENT AND SCATTERING DESIGN

For any fixed placement, the remaining design variables follow in closed form, so joint placement and scattering design becomes a position search. A coordinate search optimizes the rate directly without alternating between placement and the other variables.

A. Closed-Form Design for Given Positions

  • The fixed-position design obtains the rank-two channel decomposition, ZF directions, surface costs, stream energies, precoders, and scattering matrix through closed-form steps.The rate-search steps are performed per placement; the final precoder and scattering-matrix synthesis is one-time.Source: A. Closed-Form Design for Given… · p. 7, §0 · p. 7

B. Position Optimization

  • Because the rate is highly oscillatory and gradient methods can stall at local maxima, each coordinate is moved to the best grid point allowed by its neighbors.The search uses compact, double-null when feasible, and random initial placements, then keeps the best result.Source: B. Position Optimization · pp. 7–8
  • The search evaluates a merit function equal to the rate for feasible placements and a negative power deficit otherwise, guiding infeasible placements toward feasibility.The current position remains a candidate at each update.Source: B. Position Optimization · p. 8
  • For each candidate placement, closed-form steps evaluate the exact optimum of the ZF-constrained inner problem, so no alternation between positions and the remaining variables is needed.The final synthesis of Θ and precoders is performed once after the search.Source: §VI · p. 7, B. Position Optimization · p. 8

C. Convergence and Complexity

The coordinate search has a nondecreasing, bounded merit sequence and polynomial complexity. In the reported field-pattern comparison, movement separates the data null and sensing peak and raises the secrecy rate relative to a compact placement.

C. Convergence and Complexity

  • The algorithm’s merit is nondecreasing and bounded; for positive tolerance, it terminates within ⌈(R_max − R̃(0))/ϵ⌉ + 1 sweeps.With zero tolerance and d₀/δx rational, it reaches a placement coordinatewise optimal on the grid, provided tied positions are retained.Source: C. Convergence and Complexity · p. 8
  • For a fixed track, the search complexity is quadratic in M; the robust objective adds O(MNθ) per candidate, and one sweep takes about 0.1 s for M = 16 in the simulations.The reported simulations used at most eight sweeps and three initial placements.Source: C. Convergence and Complexity · p. 8

D. Benchmarks

  • The benchmarks include fixed and movable BD-RIS and D-RIS designs, plus single-stream BD-RIS; the movable D-RIS alternates position and phase optimization.D-RIS cannot realize the ZF fields, so its benchmark applies ZF to the effective channels.Source: D. Benchmarks · p. 8

A. Simulation Setup

  • The simulations compare six schemes under Rician fading, with results averaged over 100 realizations except for 60 in Fig. 8 and 50 in Fig. 3.The setup uses resolvable BS–RIS paths and ∆θ = 3°, giving D_dn = 15.6λ.Source: A. Simulation Setup · p. 8, §VII · p. 8

B. Field Patterns

  • 9.53 to 11.54 bit/s/Hz: the movable surface separates the data null and field peak by 3° and raises Rs relative to the compact array.The compact array has ρ = 0.79, while the movable placement has ρ = 0.05.Source: B. Field Patterns · p. 9, Figure 2 · p. 9

C. Convergence

Across the evaluated settings, movable dual-stream BD-RIS mitigates the sensing–secrecy trade-off, while movement and interconnection provide gains that depend on geometry and propagation. Under target-angle uncertainty, the derivative-null design improves robustness, and dual-stream operation remains dependent on a rank-two BS–RIS link.

C. Convergence; VIII. CONCLUSION

  • The coordinate search increases rate monotonically and reaches 99.8% of its final value within three outer iterations.The conclusion bounds the closed-form ZF design to within 0.06 bit/s/Hz of the unrestricted two-stream optimum in tested settings, while noting reliance on a rank-two link and an idealized lossless, fully connected network.Source: C. Convergence · p. 9, §VIII · p. 11

D. Impact of the Sensing Requirement

E. Impact of the Reference SNR

F. Impact of the Angular Separation

G. Impact of the Track Length

  • Under LoS, the double-null placement reaches 11.0 bit/s/Hz for track lengths D ≥ Ddn, and the rate saturates beyond Ddn.The placement is 1.2 bit/s/Hz lower at D = 8λ; the single-stream design saturates 4.3 bit/s/Hz lower. The coordinate search reaches 99.8% of its final value within three outer iterations.Source: C. Convergence · p. 9, Figure 7 · p. 10, G. Impact of the Track Length · p. 10

H. Impact of the Number of Elements

  • Doubling M adds 2.0 bit/s/Hz for the LoS closed form, while movement’s gain shrinks from 3.3 bit/s/Hz at M = 8 to 0.5 at M = 32.With M = 8, the movable BD-RIS achieves 10.1 bit/s/Hz, exceeding the fixed BD-RIS with M = 16 at 9.9 bit/s/Hz.Source: Figure 8 · p. 10, H. Impact of the Number of Elements · p. 10

I. Impact of the Propagation Environment; J. Impact of the Rank of the BS–RIS Link

  • Movement adds 2.0 bit/s/Hz at KR = 30 dB but 1.3 bit/s/Hz at KR = −10 dB; with a weakened second BS–RIS path, dual-stream designs collapse.The movable BD-RIS keeps 10.6 bit/s/Hz down to −10 dB second-path gain and remains above its single-stream counterpart down to −20 dB; the paper recommends switching to one stream when the second path is over 20 dB weaker.Source: §I · p. 10, Figure 9 · p. 10, J. Impact of the Rank of the BS–RIS Link · p. 11

K. Robustness to Target-Angle Uncertainty

APPENDIX A PROOF OF THEOREM 1

The appendix constructs the realizability criterion by reducing the field mapping to a unitary extension and then to Gram-matrix conditions. It also derives the two nonzero eigenvalues used to characterize the incident channel modes.

APPENDIX A PROOF OF THEOREM 1

  • A symmetric unitary scattering matrix is written as Θ = QQT, reducing ΘU = V to finding a unitary Q that maps the transformed incident fields to V.The unitary extension exists when the two field sets have matching Gram matrices.Source: Appendix A · pp. 11–12
  • The mapping condition becomes equality of the incident and prescribed Gram matrices plus symmetry of the bilinear Gram matrix, established by embedding the complex fields as real vectors.The proof factors the resulting real symmetric matrix to construct the vectors and then recovers Q and Θ.Source: Appendix A · p. 12

APPENDIX B PROOF OF LEMMA 1; 0 BHBD1/2

  • The two nonzero eigenvalues are determined from trace NM(β0,1 + β0,2) and determinant N²M²β0,1β0,2(1 − ρ0²).This characterizes the two incident modes through their path gains and correlation.Source: Appendix B · p. 12, §0 · p. 12

APPENDIX C PROOF OF LEMMA 3

The appendix bounds the unit-norm reflected-field cost, constructs a placement that nulls both relevant correlations, and relates position variance to decorrelation and null sensitivity.

APPENDIX C PROOF OF LEMMA 3

  • For the unit-norm reflected-field direction, the cost is a sinusoid whose extreme values lie between 1/λ1 and 1/λ2.The direction depends on the phase difference ϕ = θ2 − θ1.Source: Appendix C · p. 12

APPENDIX D PROOF OF PROPOSITION 2

  • The constructed four-block placement gives ρ = ρ0 = 0 and attains the closed-form rate when D ≥ Ddn.Its blocks satisfy the minimum-spacing constraint, and the converse bounds residual correlation when the track is shorter.Source: Appendix D · p. 12

APPENDIX E PROOF OF PROPOSITION 4

  • At small separation, ρ² = 1 − κ²∆s²Var(p) + O(x⁴), linking position variance to decorrelation.For even M, maximum variance is attained by two compact element clusters at opposite track ends.Source: Appendix E · p. 12

APPENDIX F PROOF OF PROPOSITION 5

  • The nominal null’s leakage begins at second order in angular error, while the derivative-null direction removes first-order sensitivity.The derivative-null design uses an additional spatial degree of freedom; its achievable separation from Bob depends on the derivative-direction correlation.Source: Appendix F · p. 13
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