1
Secure RIS-Aided Multicasting: Globally Optimal Beam Management and Discrete-Phase RIS Configuration
arXiv:2609.21723v1 [eess.SP] 18 Sep 2026
Luis F. Abanto-Leon and Setareh Maghsudi Ruhr University Bochum, Germany {luis.abantoleon, setareh.maghsudi}@ruhr-uni-bochum.de
Abstract—Reconfigurable intelligent surfaces (RISs) are poised to revolutionize wireless multicasting by enabling extended coverage and reliable operation in obstructed environments. These benefits, however, can be undermined by security vulnerabilities arising from practical deployment factors. This work addresses three such critical factors, (i) the discrete nature of RIS phase shifts, (ii) the presence of colluding eavesdroppers, and (iii) the inefficiency of static illumination beams, each threatening security if not properly accounted for in system design. To mitigate these issues, we formulate a joint resource allocation problem that minimizes the wiretap signal-to-noise ratio (SNR) across all eavesdroppers by co-optimizing the RIS configuration and the base station (BS) beam management. This yields a complex, nonconvex mixed-integer nonlinear program (MINLP), which we equivalently reformulate into a tractable mixed-integer quadratically constrained program (MIQCP) solvable to global optimality. Numerical results confirm that the proposed scheme significantly bolsters security, suppressing the wiretap SNR by up to 58% compared to existing baselines. Index Terms—Reconfigurable intelligent surface, discrete phases, beam management, physical-layer security, multicast.
I. I NTRODUCTION To meet soaring capacity demands, 3GPP has prioritized operation in high-frequency bands, such as mmWave and THz, which offer abundant spectrum resources [1]. However, these frequencies are highly susceptible to blockage. To mitigate this, reconfigurable intelligent surfaces (RISs) have emerged as a transformative technology capable of reconfiguring wireless propagation environments to bypass obstacles. In parallel, multicasting is envisioned as a key enabler for 6G, exploiting the wide bandwidths available in high-frequency spectra [2]. The synergy of RISs and multicasting thus holds significant promise for ubiquitous, high-capacity coverage. Despite this potential, ensuring physical-layer security (PLS) in RIS-aided multicast systems remains highly challenging. Existing security-oriented designs often rely on idealized assumptions, overlooking critical practical deployment factors. As discussed next, neglecting these aspects can lead to significant security vulnerabilities. Discrete phases: Most RIS designs assume continuous phases, which lack practical feasibility, e.g., [3], [4]. To address this, continuous-phase relaxation followed by projection has become standard practice [5]–[7]. However, the projection step distorts the intended RIS beampattern by introducing quantization artifacts and generating unintended sidelobes that can be exploited by eavesdroppers, representing a largely overlooked vulnerability in current research. Colluding eavesdroppers: While some works addressed RIS-assisted multicasting under independent eavesdroppers
TABLE I: Comparison of related work. Works
System
RIS phases
[3], [4]
Unicast
Continuous
Beam management Eavesdroppers N/A
Colluding
Solution Suboptimal
[5]
Multicast Discrete via projection
Fixed
N/A
Suboptimal
[6], [7]
Multicast Discrete via projection
N/A
N/A
Suboptimal
[8], [9]
Multicast
Continuous
N/A
Non-colluding
Suboptimal
[11], [12]
Unicast
Discrete via projection
Dynamic
N/A
Suboptimal
[13]
Unicast
Discrete
Fixed
N/A
Suboptimal
[14]
Unicast
Discrete via projection
Fixed
N/A
Suboptimal
Discrete
Dynamic
Colluding
Optimal
Proposed Multicast
[8], [9], practical adversaries may collude to enhance their joint decoding capability [3], [4], [10]. This threat is acute in multicasting, where a common signal is broadcast to multiple users, expanding the attack surface. Yet, collusion-based interception in such settings remains largely unexplored. Dynamic illuminating beam: While beam management has been studied in prior works [11], [12], its role in secure communications remains largely underexplored. When direct base station (BS)-user links are obstructed, a common approach is to use a static BS-to-RIS illuminating beam [13], [14], consistent with fixed infrastructure. However, this imposes a fixed angle of incidence, restricting the spatial degrees of freedom at the RIS and limiting its ability to suppress eavesdroppers, thereby motivating dynamic beam management. Despite their practical relevance, the impact of these critical factors on PLS remains largely unexamined. To bridge this gap, we formulate a comprehensive resource allocation problem that jointly optimizes (i) the RIS configuration and (ii) the BS beam management to minimize the wiretap signal-to-noise ratio (SNR) under colluding eavesdroppers, while ensuring the required quality of service for legitimate users. The resulting problem is a nonconvex mixed-integer nonlinear program (MINLP), which we equivalently reformulate as a tractable mixed-integer quadratically constrained program (MIQCP) solvable to global optimality. Numerical results demonstrate that the proposed joint design significantly suppresses the wiretap SNR compared to existing benchmarks. A detailed comparison with the state-of-the-art is provided in Table I. Notation: The transpose and Hermitian transpose are denoted by (·)T and (·)H . The sets of complex, binary, and nonnegative numbers are denoted by C, B, and R+ , respectively. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider a downlink multicast system consisting of a multiantenna BS, an RIS, U single-antenna users, and E single-
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antenna eavesdroppers. The direct BS-user links are blocked, so communication relies solely on signals reflected by the RIS, as illustrated in Fig. 1. The RIS comprises Mx × Mz uniformly spaced elements in the xz-plane, with total size M = Mx Mz . The BS has N antennas and is located at the left of the RIS, with its antennas placed along the y-axis. Users and eavesdroppers are located in the half-space in front of the RIS. The corresponding index sets are U = {1, . . . , U } and E = {1, . . . , E}. For notational simplicity, the u-th user is denoted by Uu and the e-th eavesdropper by Ee . Beam management model: The BS employs a predefined beam codebook {tk }k∈K comprising K candidate beams for RIS illumination, such that ktk k22 = Ptx , where Ptx is the total transmit power and K = {1, . . . , K}. To model the selection of a single beam, we introduce the following constraints P C1 : αk ∈ B, ∀k ∈ K, C2 : k∈K αk = 1.
In C1 , αk = 1 indicates that beam tk is active, and αk = 0 otherwise. In addition, C2 enforces that exactly one beam is chosen from the codebook. Consequently, the transmit beam used by the BS is given by P (1) b = k∈K αk tk .
RIS phase shift model: The RIS is characterized by w ∈ CM×1 , where each element [w]m corresponds to the phase shift of the m-th reflecting element. The phase shifts are drawn from a finite discrete set, modeled as C3 : [w]m ∈ ejφ1 , . . . , ejφQ , ∀m ∈ M,
where φq denotes the q-th phase value and Q is the number of available phase choices. Communication model: The BS transmits data symbol s ∈ C, modeled as a zero-mean, unit-variance complex random variable, i.e., E {ss∗ } = 1. Thus, after reflection by the RIS, the signal received by Uu is y u = hT u diag(w)Gbs + nu , P = k∈K αk hT u diag(w)Gtk s + nu ,
where hu ∈ CM×1 is the channel between the RIS and Uu , G ∈ CM×N is the channel between the RIS and BS, and nu ∼ CN 0, σu2 is additive white Gaussian noise (AWGN) at Uu . Hence, the SNR of Uu is P 2 2 T (2) SNRu (Ω) = k∈K αk hu diag(w)Gtk /σu ,
where Ω , (α, w) denotes the set of all decision variables in T the resource allocation problem, with α = [α1 , . . . , αK ] . To ensure reliable communication, each user is required to satisfy C4 : SNRu (Ω) ≥ Γth , ∀u ∈ U,
where Γth is the imposed SNR threshold. Eavesdropping model: The signal reflected by the RIS and received by Ee is y e = feT diag(w)Gbs + ne , P = k∈K αk feT diag(w)Gtk s + ne ,
where fe ∈ CM×1 is the channel between the RIS and Ee , while ne ∼ CN 0, σ 2e is AWGN at Ee . The SNR experienced by eavesdropper Ee is P 2 2 T SNRe (Ω) = (3) k∈K αk fe diag(w)Gtk /σ e .
Fig. 1: Secure RIS-assisted multicast system. Under a collusive strategy, the eavesdroppers can enhance information decoding through collaboration. Thus, we employ the wiretap SNR in [10], [15] as our performance metric, which represents the collective signal interception capability of all eavesdroppers, and is defined as g (Ω) = P (4) SNR e∈E SNRe (Ω) . q χ LoS + Channel model: Let G = ω χ+1 · G q 1 NLoS , where GLoS and GNLoS are the line-of-sight χ+1 G (LoS) and non-LoS (NLoS) components, ω is the large-scale fading coefficient, and χ is the Rician fading factor. Here, d
az el GLoS = e−j2πf0 v aris (γ az , γ el ) aT bs (γ , γ ), where d is the RIS-BS distance, f0 is the carrier frequency, and v is the speed of light. The RIS is centered at (0, 0, zris ) and the BS is centered at (ẋbs , ẏbs , żbs ). The m-th element of aris (γ az , γ el ) 2πf0
T
ris
is ej v v pm , where pris m is its coordinate relative to the T el RIS and v = sin(γ ) cos(γ az ), sin(γ el ) sin(γ az ), cos(γ el ) . 2πf0
T
bs
Similarly, the n-th element of abs (γ az , γ el ) is ej v v pn , where pbs n is its coordinate relative to the BS. The channelsq hu and fe are defined analogously, specifically, q hu = q χ̂u χ̌e 1 LoS NLoS and f = ω̌ · h + h ω̂u e e u χ̂u +1 χ̌e +1 · χ̂u +1 u q 1 NLoS feLoS + , with parameters ω̂u , ω̌e , χ̂u , and χ̌e +1 fe dˆu
χ̌e . Here, hLoS = e−j2πf0 v aris (θuaz , θuel ) and feLoS = u ďe
e−j2πf0 v aris (βeaz , βeel ), where (θuaz , θuel ) and (βeaz , βeel ) are the LoS angles of Uu and Ee , respectively, while dˆu and dˇe denote their respective distances from the RIS. The NLoS components hNLoS , feNLoS , and GNLoS are modeled as zero-mean, unitu variance complex Gaussian random variables1 . Problem formulation: To safeguard multicast transmission against interception, we jointly optimize the RIS phase configuration and the BS beam management, ensuring a target SNR for legitimate users while minimizing the wiretap SNR. The corresponding resource allocation problem is formulated as g (Ω) s.t. C1 − C4 , P : minimize f (Ω) , SNR Ω
1 We assume user channel state information (CSI) is obtained via uplink pilots, whereas eavesdropper CSI is estimated using sensing or locationbased methods. To establish an optimal performance bound, we assume perfect CSI for both users and eavesdroppers. In practice, however, CSI is inherently imperfect, which inevitably leads to a secrecy performance degradation relative to the ideal case. While a rigorous treatment of CSI uncertainty is crucial, it remains beyond the scope of the current work and is therefore deferred to future investigation.
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where P is a nonconvex MINLP and thus challenging to solve. In particular, obtaining its global optimum would require exhaustive enumeration over all RIS phase configurations and BS candidate beam selections, leading to a prohibitive worstcase computational complexity of O(KQM ). III. P ROPOSED MIQCP R EFORMULATION To circumvent the high complexity of P, we reformulate it as an equivalent MIQCP, denoted P ′ , via Proposition 1 to Proposition 4. These propositions transform intractable expressions into convex equivalents while preserving the original feasible set. The proofs are provided in the Appendix. A. Transformation of the objective function To handle the intractable objective function, we introduce an auxiliary variable to convert the objective into an equivalent constraint. We then decompose this constraint into elementary expressions, yielding a tractable reformulation that facilitates subsequent manipulation. g (Ω) Proposition 1. The objective function f (Ω) , SNR ′ can be recast as a new function g Ω , µ by introducing additional constraints D1 , D2 , D3 , and D4 , g Ω′ , µ, D1 : µ ∈ R+ , P 2 ∈ R+ , ∀e ∈ E, D3 : f (Ω) ⇔ D2 : δeP e∈E δe ≤ µ, H k∈K αk pe,k w ≤ δe , ∀e ∈ E, D4 : σe
where µ and δe are new variables, pe,k = diag (G∗ t∗k ) fe∗ , and Ω′ denotes the set of decision variables of the reformulated problem P ′ . This set is progressively augmented as additional variables are introduced during the transformation. B. Transformation of constraint D4 The summation inside the absolute value, together with the multiplicative coupling between αk and w, makes constraint D4 intractable. To enable efficient optimization, we reformulate D4 into an equivalent form that eliminates these couplings while preserving the original feasible solution set. Proposition 2. Constraint D4 can be equivalently rewritten as constraint E1 pH e,k w D5 ⇔ E1 : ≤ δe + (1 − αk )Le , ∀e ∈ E, k ∈ K, σe √ where Le = M Ptx kfe k2 kGkF /σe . C. Transformation of constraint C3 To circumvent the combinatorial complexity of C3 , we adopt a one-hot encoding approach. This replaces the multiple-choice selection with a set of linear constraints while preserving global optimality within the reformulated solution space. Proposition 3. Constraint C3 can be equivalently rewritten as constraints F1 , F2 , and F3 Q×1 , ∀m ∈ M, F1 : zm ∈ B T C3 ⇔ F2 : 1 zm = 1, ∀m ∈ M, F3 : [w]m = qT zm , ∀m ∈ M, T where zm are new variables and q = ejφ1 , . . . , ejφQ .
D. Transformation of constraint C4 The nonconvexity and summation over bilinear terms αk w inside the absolute value make constraint C4 challenging to optimize. To address this, we develop an exact reformulation that eliminates these nonconvexities through variable augmentation while preserving the original solution space. Proposition 4. Constraint C4 can be equivalently rewritten as constraints G1 , G2 , G3 , G4 , G5 , G6 , and G7 2 G1 : dH u,k Rdu,k ≥ αk Γth σu , ∀u ∈ U, k ∈ K, ∈ M, G2 : [R]m,m = 1, ∀m, ∗ ′ ′ G3 : [R]m,m′ = [R]m′ ,m , ∀m, m ∈ M, m > m, C4 ⇔ G4 : [R]m,m′ = qT Ym,m′ q∗ , ∀m, m′ ∈ M, m′ > m, G5 : Ym,m′ 1 = zm , ∀m, m′ ∈ M, m′ > m, T ′ ′ G6 : Ym,m ′ 1 = zm′ , ∀m, m ∈ M, m > m, Q×Q ′ , ∀m, m ∈ M, m′ > m, G7 : Ym,m′ ∈ B
where Ym,m′ and R are new variables, whereas du,k = diag (G∗ t∗k ) h∗u . E. Reformulated problem From the above propositions, the problem becomes P ′ : minimize g Ω′ , µ ′ Ω
C1 − C2 , D1 − D3 , E1 , F1 − F3 , G1 − G7 , where Ω , µ, α, δ, w, Z, R, Y , δ = [δ1 , . . . , δE ]T , Z = [z1 , . . . , zM ], and Y = [Y1,1 , . . . , YM,M ]. Our reformulation maps the intractable problem P into an equivalent MIQCP, denoted by P ′ , replacing the original combinatorial search with a structured optimization problem, which is solved via interior point methods (IPMs) within a branch-and-cut (BnC) framework. The BnC framework handles the discrete variables via hierarchical branching, where each node induces a continuous QCP subproblem by relaxing a subset of the binary variables. Since each subproblem is convex, it is solved to global optimality using IPM, yielding rigorous dual bounds that enable systematic pruning of the search space via relaxation, bounding, branching, and cutting [16], [17], ultimately converging to the global optimum while avoiding the prohibitive cost of exhaustive enumeration. Deriving an exact expression for the computational complexity of P ′ is challenging due to solver-specific mechanisms within BnC. Nevertheless, an approximate characterization can √ worst-case 3 2 + Nvar Ncon ) , where be obtained as O Nnodes · Nvar (Nvar Nvar = 1 + K + E + M Q(Q + 1) + M(M−1) and Ncon = 2 3+E+M +K(2+E+U )+2M Q(Q+1)+M (M +Q)(M −1) denote the number of variables and constraints, respectively, and Nnodes is the number of explored BnC nodes, i.e., the number of convex QCP relaxations solved. s.t.
′
IV. S IMULATION R ESULTS Throughout the numerical simulations, large-scale fading coefficients are obtained following the UMa model [18], assuming a BS antenna gain of 10 dBi, a user antenna gain of 3 dBi, and an eavesdropper antenna gain of 5 dBi. Unless otherwise stated, the parameters are set as χ = χ̂u = χ̌e = 100, Ptx = 10 W, f0 = 26 GHz, σu2 = σ 2e = −110 dB, N = 12,
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(a) Varying number of eavesdroppers.
(b) Varying transmit power.
(c) Varying number of beams.
Fig. 2: Wiretap SNR under varying numbers of eavesdroppers, numbers of users, and transmit power levels. M = 100, and Mx = Mz = 102 . The BS-RIS distance is 10 m, with angles γ az ∈ [20◦ , 70◦ ] and γ el = 90◦ . The userRIS distance du ranges from 10 to 20 m with angles θuaz ∈ [110◦ , 160◦] and θuel = 90◦ . The eavesdropper-RIS distance de ranges from 18 to 40 m with angles βeaz ∈ [100◦ , 170◦] and βeel = 90◦ . A 2-bit phase control is considered, where the admissible phases are {0◦ , 90◦ , 180◦ , 270◦}. The number of candidate BS-RIS beams is K = 15, uniformly covering the range [10◦ , 80◦ ] (relative to the RIS) with a resolution of 5◦ . All problems are solved using CVX with the MOSEK solver, and the results show the average over 100 random realizations. A. Ablation study In Scenarios I-III, we evaluate the impact of neglecting each practical factor in the resource allocation design through an ablation study. The following schemes are considered. • ORBE: The proposed approach developed in Section III. • N-beam: Assumes a fixed BS-RIS beam, exposing the security inefficiency of static illumination. • N-eave: Treats eavesdroppers as non-colluding entities, exposing the “false sense of security” caused by neglecting potential collaboration among adversaries. • N-phase: Treats RIS phase shifts as continuous during optimization, then projects onto the discrete set, revealing security risks induced by projection. • N-all: Combines all the aforementioned assumptions. Scenario I: Fig. 2(a) illustrates the impact of an increasing number of users, U , with E = 5 and Γth = 1. As U increases, the colluding eavesdroppers enhance their decoding capability, since a wider distribution of users increases the likelihood of signal interception. In this setting, ORBE can serve up to seven users without allowing the eavesdroppers to achieve the decoding threshold Γth = 1. In contrast, N-beam maintains secrecy up to six users, N-eave up to four, N-phase up to three, and N-all up to two. At U = 7, ORBE achieves a wiretap SNR reduction of 48% compared to N-all. Scenario II: Fig. 2(b) illustrates the impact of an increasing transmit power, Ptx , with U = 3 and E = 7. Notably, ORBE maintains a nearly power-invariant wiretap SNR, a direct consequence of the joint optimization between the illumination beam and RIS configuration. This framework effectively increases the null-depth at adversarial locations, ensuring that increased transmit power does not translate into 2 To reflect practical RIS deployments with reduced signaling overhead, we employ column-wise control. This yields an azimuthal fan-shaped beam with fixed elevation, aligning with realistic deployment constraints.
proportional leakage. While N-eave exhibits a similar trend, its wiretap SNR remains above the decoding threshold (Γth = 1) because its non-colluding eavesdropper assumption fails to account for aggregated leakage. Conversely, N-beam and Nphase suffer from a rapid, monotonic increase in wiretap SNR. For the former, the fixed illumination angle restricts the spatial degrees of freedom, while for the latter, phase quantization errors create leakage floors that scale with Ptx . N-all suffers the most pronounced degradation due to the combined effect of all impairments. At Ptx = 30 W, ORBE achieves up to a 73% reduction in wiretap SNR compared to N-all, and is the only scheme that maintains the wiretap SNR below Γth = 1. Scenario III: Fig. 2(c) illustrates the impact of the number of beams, K, with U = 3 and E = 6. As K increases, ORBE achieves a progressive reduction in the wiretap SNR, demonstrating that finer angular granularity in the illumination beam design significantly expands the spatial degrees of freedom for effective null-steering. Conversely, N-eave exhibits only a marginal response to K. This stems from its objective of minimizing the maximum individual wiretap SNR, typical of non-colluding eavesdropper models, which fails to exploit beam granularity for aggregate leakage suppression. While N-phase exhibits a similar downward trend to ORBE due to improved angular resolution, N-beam and N-all remain invariant, as their use of a fixed beam prevents them from exploiting a larger codebook. At K = 19, ORBE achieves up to a 70% reduction in wiretap SNR compared with N-all. B. Baseline study In Scenarios IV-V, we compare ORBE against three baselines, which are detailed below. • BL1: Assumes non-colluding eavesdroppers (e.g., [8], [9]) with projected phases (e.g., [6], [7]) and dynamic illumination (e.g., [11], [12]). It is optimized via ADMM and supplemented by phase randomization to enforce constraint feasibility. • BL2: Assumes non-colluding eavesdroppers (e.g., [8], [9]), discrete phases (e.g., [13]), and fixed illumination (e.g., [5]). It is optimized via a tailored variant of our framework. • BL3: Assumes colluding eavesdroppers (e.g., [3], [4]) under a fixed illumination beam (e.g., [5]) and projected phases (e.g., [6], [7]). It is optimized via SDR. Scenario IV: Fig. 3(a) illustrates the impact of an increasing number of eavesdroppers, E, with U = 3, Γth = 1, and Q = 4. As E grows, the wiretap SNR rises due to the heightened collusive gain. Nevertheless, ORBE demonstrates strong resilience, maintaining the wiretap SNR below the
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(a) Q = 4.
(b) Q = 2.
Fig. 3: Wiretap SNR under two quantization levels. Power [dBm]
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BS
14
ea
Position in Y [m]
14
us
ea
ers
er s
6 ro pp
Position in Y [m]
(b) BL1
RIS
BS
18
Position in X [m]
(a) ORBE
10
C. Complexity reduction
sd ve
18 ] SNR(Ω) = 1.70 −9 0 9
Position in X [m] 2
us
sd ve
18 ] SNR(Ω) = 1.15 −9 0 9
18
RIS
2 6 10
BS
us
ers
ea
18 ] SNR(Ω) = 0.83 −9 0 9
6 10
er s
14
sd ve
−39
ro pp
us
−98 RIS
2
ers
Position in Y [m]
BS
ers
er s
6 10
−157
er s
−39
ro pp
−98 RIS
2
ro pp
Position in Y [m]
−157
Power [dBm]
d es ea v
14
18 ] SNR(Ω) = 1.69 −9 0 9
Position in X [m]
Position in X [m]
(c) BL2
(d) BL3
BS is located at γ az = 50◦ , the users at θ az = [120◦ , 142◦], and the eavesdroppers at β az = [112◦ , 126◦ , 132◦, 148◦ ], with distances fixed at dˆu = 14 m and dˇe = 17 m. ORBE g achieves SNR(Ω) = 0.83, outperforming BL1 (1.70), BL2 (1.15), and BL3 (1.69). This gain stems from its effective and spatially consistent energy suppression, as evidenced by the pronounced low-energy regions (deep blue) aligned with the eavesdropper directions. In contrast, BL1, BL2, and BL3 exhibit less consistent spatial suppression, with localized nulls at some eavesdropper directions but elevated leakage at others, resulting in higher overall security risk. Notably, ORBE is the only scheme that maintains the wiretap SNR below the decoding threshold (Γth = 1). This underscores the importance of its joint design and globally optimal formulation in reliably safeguarding multicast transmissions.
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Fig. 4: Received energy heatmaps.
This scenario evaluates the complexity reduction achieved by ORBE relative to exhaustive search, enabled by the BnC. Scenario VI: Fig. 5 shows the number of explored nodes for ORBE as a function of K, following the settings of Scenario III for U = 3 and U = 6. The number of explored nodes, Nnodes (as defined in the computational complexity analysis of P ′ in Section III-E), increases with K, exhibiting an approximately linear trend on average. Importantly, the number of explored nodes remains orders of magnitude lower than that required by exhaustive search, which spans from 7.3 × 106 candidate solutions at K = 7 to 19.9 × 106 at K = 19. In contrast, ORBE evaluates less than 1% of the potential candidates on average, and remains below 2% in the worst-case scenario. These results validate that the proposed MIQCP formulation effectively prunes vast regions of the search space, ensuring computational tractability and global optimality with significantly reduced overhead. V. C ONCLUSIONS
Fig. 5: Explored nodes by ORBE. decoding threshold Γth = 1 for up to eight eavesdroppers, highlighting its robustness in dense adversarial environments. In comparison, BL1 and BL2 maintain secrecy only up to E = 5, while BL3 fails beyond E = 4. Notably, ORBE achieves a 58% reduction in wiretap SNR relative to BL3. Fig. 3(b) considers the same setup as Fig. 3(a) but with a coarser phase resolution of Q = 2, i.e., phase shifts {0◦ , 180◦ }. This coarser quantization increases the wiretap SNR across all schemes and reduces resilience to eavesdropping due to the more constrained phase design space, which limits the available spatial degrees of freedom. Nevertheless, ORBE maintains a clear advantage, securely tolerating up to E = 4 eavesdroppers, whereas BL1, BL2, and BL3 support at most E = 3. This highlights that joint optimization becomes particularly critical under restricted hardware capabilities. Scenario V: Fig. 4 illustrates the received energy heatmaps in a rectangular setting with Γth = 1. Relative to the RIS, the
This work showed that widely used assumptions, such as non-colluding eavesdroppers, fixed BS-RIS illumination, and phase relaxation followed by projection, can severely underestimate secrecy risks in RIS-assisted multicasting. We proposed ORBE, a globally optimal framework that jointly optimizes the BS beam and discrete RIS phase shifts while accounting for colluding eavesdroppers. Results confirm that properly modeling these practical factors is critical for suppressing wiretap SNR and ensuring secure RIS-assisted multicasting. ACKNOWLEDGMENT This research was supported by the Federal Ministry of Research, Technology and Space (BMFTR) under Grants 16KIS2411. A PPENDIX Proof of Proposition 1 We introduce a new variable µ, specified as constraint D1 : µ ∈ R+ , to bound f (Ω). The objective is then expressed as g (Ω) ≤ µ, g Ω′ , µ by introducing constraint Daux,1 : SNR ′ where Ω denotes the decision variables in P ′ . To bound each eavesdropper’s SNR individually, we introduce variables δe with D2 : δe ∈ R+, ∀e ∈ E. Constraint Daux,1 can
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P 2 then be equivalently decomposed into D3 : e∈E δe ≤ 2 µ, and Daux,2 : SNRe (Ω) ≤ δe , ∀e ∈ E. By defining pe,k = diag (G∗ t∗k ) fe∗ , we can transform Daux,2 into Daux,3 : P 2 H 2 2 k∈K αk pe,k w /σ e ≤ δe , ∀e ∈ E. Since both sides of Daux,3Pare nonnegative, we take the square root, yielding H D4 : k∈K αk pe,k w /σ e ≤ δe , ∀e ∈ E.
Proof of Proposition 2 For any Ee , applying Jensen’s inequality to the quadratic term of D4 yields P P on Hleft-hand side H α p w ≤ Eaux,1 : k e,k k∈K αk pe,k w . Let k∈K ′ k denote the index of the selected transmit beam tk′ . Then, Eaux,1 can be expressed as Eaux,2 : αk′ pH e,k′ w + P P H H H ′ p α p w ≤ α w + α ′ ′ ′ k e,k k e,k k pe,k w . k6=k k6=k ′ Since αk = 0 for all k 6= k , the inequality becomes tight. Consequently, D4 can be equivalently rewritten P H ≤ δe , ∀e ∈ E. as Eaux,3 : k∈K αk pe,k w /σe Using the Pbinary nature of αk , this simplifies to H Eaux,4 : k∈K αk pe,k w /σ e ≤ δe , ∀e ∈ E. Since only one beam is selected at the BS, the summation in Eaux,4 can be decoupled using the big-M method. This removes the bilinear coupling between αk and w, yielding E 1 : pH /σ e ≤ δe + (1 − αk )Le , ∀e ∈ E, k ∈ K, e,k w√ where Le = M Ptx kfe k2 kGkF /σ e is an upper bound for pH e,k w /σ e , computed via the Cauchy-Schwarz inequality. Proof of Proposition 3 Following the one-hot encoding method in [13], we represent the phase selection for each RIS element by introducing the binary vectors F1 : zm ∈ BQ×1 , ∀m ∈ M. To ensure physical consistency, we enforce F2 : 1T zm = 1, ∀m ∈ M, which guarantees that exactly one discrete phase shift is selected per element. The mapping between the admissible phase shifts, collected in q = [ejφ1 , . . . , ejφQ ]T , and the RIS configuration is established via F3 : [w]m = qT zm , ∀m ∈ M.
Proof of Proposition 4 Note that C4 is equivalent to constraint Gaux,1 : P 2 H ≥ Γth σu2 , ∀u ∈ U, where du,k = k∈K αk du,k w diag (G∗ t∗k ) h∗u . By applying Jensen’s inequality and following the procedure in the proof of Proposition 2, Gaux,1 P 2 H ≥ Γth σu2 , ∀u ∈ U. reduces to Gaux,2 : k∈K αk du,k w Since only one variable αk is equal to 1, we can decompose Gaux,2 into the intersection of multiple constraints, collectively 2 defined as Gaux,3 : dH ≥ αk Γth σu2 , ∀u ∈ U, k ∈ K. u,k w Furthermore, Gaux,3 can be recast as G1 : dH u,k Rdu,k ≥ αk Γth σu2 , ∀u ∈ U, k ∈ K, subject to including a new constraint Gaux,4 : R = wwH , as in [1]. Adopting an entrywise notation, Gaux,4 is equivalent to Gaux,5 : [R]m,m′ = [w]m [w∗ ]m′ , ∀m, m′ ∈ M. Leveraging F3 , we further trans∗ form Gaux,5 into Gaux,6 : [R]m,m′ = qT zm qT zm′ = ∗ ∗ = qT zm zT qT zm zT m′ q . By exploiting the conjugate m′ q symmetry of R, we reformulate Gaux,6 as the following three constraints: G2 : [R]m,m = 1, ∀m, ∈ M, G3 : [R]m,m′ = ∗ [R]m′ ,m , ∀m, m′ ∈ M, m′ > m, and G4 : [R]m,m′ = T q Ym,m′ q∗ , ∀m, m′ ∈ M, m′ > m, subject to introducing ′ ′ Gaux,7 : Ym,m′ = zm zT m′ , ∀m, m ∈ M, m > m. Note that Gaux,7 couples zm and zm′ multiplicatively, complicating tractability. To address this, we multiply both sides of Gaux,7
′ by 1, yielding Gaux,8 : Ym,m′ 1 = zm zT m′ 1, ∀m, m ∈ ′ T M, m > m. Leveraging F2 , which states that z 1 = 1, then Gaux,8 reduces to G5 : Ym,m′ 1 = zm , ∀m, m′ ∈ M, m′ > T m. Similarly, from Gaux,7 we can also obtain G6 : Ym,m ′1 = ′ ′ ′ ′ zm , ∀m, m ∈ M, m > m. Given that Ym,m yields from multiplying zn and zm , each of which has one element 1, then Ym,m′ is binary. This conditions is enforced through G7 : Ym,m′ ∈ BQ×Q , ∀m, m′ ∈ M, m′ > m.
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