S-R3: Secure and Robust RIS-Assisted RSMA under Finite-Resolution Phase Shifts

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S-R3: Secure and Robust RIS-Assisted RSMA under Finite-Resolution Phase Shifts

arXiv:2611.02662v1 [cs.NI] 2 Nov 2026

Ibrahima Sory Diallo, Elhadj Moustapha Diallo , Wang Mi, Mamadou Aliou Diallo , Amayika Kakati

Abstract—This paper investigates a reconfigurable intelligent surface (RIS)-assisted rate-splitting multiple access (RSMA) downlink with imperfect eavesdropper channel state information and finite-resolution RIS phases. The common stream carries public information, while user-specific private streams carry confidential information. For a conservative security assessment, each eavesdropper is allowed to remove the public stream and is given the other private streams as side information. We formulate a robust design of the transmit precoders, isotropic artificial noise, common-rate allocations, and RIS phases that balances confidential and public throughput under rate and power constraints. Bounded errors in the direct and cascaded eavesdropper channels yield an effective-channel uncertainty bound valid across RIS phase updates. To address the resulting nonconvex problem, we construct conservative leakage bounds and convex inner transmit updates, initialize through a feasibility procedure, and search finite artificial-noise and RIS-phase candidates with explicit feasibility checks. For a fixed artificialnoise power and RIS configuration, the feasible inner iterates have a nondecreasing objective that converges in value, and their successive changes vanish. Accepted outer updates remain feasible and are finite in number. Our code is available at https://github.com/ABM-ADAM/RIS RSMA-secure. Index Terms—Artificial noise, finite-resolution phase shifts, physical-layer security, rate-splitting multiple access (RSMA), reconfigurable intelligent surface (RIS), robust optimization.

stream. This structure can also accommodate multicast and unicast traffic in the same transmission [4]. We assign different security requirements to the two streams: the common stream carries public message parts, whereas each private stream carries the confidential part of one user’s message. Thus, the public common rate contributes to delivered throughput but never to confidential throughput. Unlike RIS–RSMA formulations that allocate a confidential common rate [5], our security model allows an eavesdropper to know and cancel the public common stream. The RIS, transmit precoders, and AN are consequently designed to protect each confidential private stream under that assumption. This paper studies that mixed-traffic downlink with bounded eavesdropper-channel errors and finite-resolution RIS phases. The objective balances a weighted sum of guaranteed confidential private rates against delivered public common rate, subject to explicit minimum-rate and transmit-power constraints. Because the resulting problem couples robust leakage bounds, multiuser decoding, AN, and discrete phase choices, both the security meaning of a rate and the scope of an algorithmic guarantee require careful specification. A. Related Works

I. I NTRODUCTION Reconfigurable intelligent surfaces (RISs) shape reflected propagation paths in a multiuser downlink, affecting the channels of both legitimate users and potential eavesdroppers. Secure RIS-assisted transmission therefore calls for jointly designing the base-station precoders, artificial noise (AN), and RIS phases [1], [2]. Practical RIS elements provide a finite set of phase states, whereas continuous-phase control is an idealization [3]. Furthermore, the eavesdropper channels may be imperfectly known at the transmitter [1]. A security formulation should account for both finite phase resolution and eavesdropper-channel uncertainty. Rate-splitting multiple access (RSMA) splits user messages into common and private parts: the common stream is decoded by all served users before each user decodes its own private I. S. Diallo is Department of Computer Science Engineering, Ahmadou Dieng University, Coyah 130, Guinea. E. M. Diallo is with Ningbo Yinci Technology Co., Ltd. and Beijing SQUID Quantum Technology Co., Ltd., Beijing 100083, China. W. Mi is with Beijing Milestone Technology Development Co.. Ltd., Beijing 100083, China. M. A. Diallo and A. Kakati are with the School of Communications and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing, China Corresponding author: Elhadj Moustapha Diallo, Email: L202110011@stu.cqupt.edu.cn.

RIS-assisted secure transmission has been investigated in several aerial and multiuser settings. Joint transmit beamforming, RIS phases, and unmanned aerial vehicle (UAV) trajectory are optimized in [6], while the impact of UAV jitter and imperfect hardware is addressed in [7]. More recently, hybrid RISs and multihop propagation have been studied for secure indoor–outdoor aerial millimeter-wave links [8]; secure content fetching in the presence of aerial and ground eavesdroppers is considered in [9]. These settings establish the relevance of propagation control and uncertain operating conditions, although their airborne geometry and traffic objectives differ from the fixed, narrowband downlink considered here. Secure RSMA has also been studied in UAV networks through deep unfolding and reinforcement learning [10], quality-of-experience-aware resource allocation [11], and quantum deep reinforcement learning [12]. For RIS-assisted multiuser secrecy, the precoding study in [13] develops an alternating-optimization solution and an unfolded network for its original model. The closer RIS–RSMA work of Wang et al. [5] optimizes a max–min secrecy objective with transceiver hardware impairments and an allocation of confidential common rate. In contrast, the present common stream is public and may be known to the eavesdroppers. Its allocation cannot be inserted into a secrecy sum, and a comparison with [5]

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Legitimate-user links Eavesdropper links

Legitimate users

hd,k

K single-antenna receivers

Passive RIS

M elements; B-bit phases

hr,k

User k

···

User 1

User K

G BS–RIS channel

hr,e Θ = diag(θ),

Multiantenna BS

RSMA transmission with artificial noise

x = wc sc +

X

wk sk + z

k∈K

sc : public

sk : confidential

θm ∈ QB Eve e

hd,e

N transmit antennas

z ∼ CN (0, qIN )

Eavesdroppers

E single-antenna receivers

···

Eve E

Noncolluding; imperfect CSI

Legitimate user k

Conservative eavesdropper model

sc −→ SIC −→ sk

Other private streams and AN remain as noise

sc and {sj }j̸=k assumed known; AN remains

b e (θ) ≤ ϵe he (θ) − h 2

Fig. 1. RIS-assisted RSMA downlink with noncolluding eavesdroppers. ChatGPT-6 Astra is used to plot the figure.

requires adapting the formulations to the same traffic and channel assumptions. Learning-based methods address related but distinct RIS control problems: hierarchical reinforcement learning for terahertz massive MIMO [14], multimodal reinforcement learning for an interference-limited RIS–UAV network [15], and a discrete policy-gradient method for statistical quality of service in RIS-aided holographic MIMO [16]. A quantum-enhanced multi-RIS approach has also been studied for millimeter-wave Internet-of-Things networks using nonorthogonal multiple access [17]. These works illustrate different channel, traffic, or control models; none of their reported performance numbers is used as a surrogate for the robust public-common/privateconfidential problem posed here. B. Motivations and Contributions The above literature motivates a formulation in which the security claim matches the information actually required to be secret. Counting a public common allocation as secrecy, or relying on other users’ private streams to remain undecodable at an eavesdropper, would overstate the confidentiality provided by this model. A second issue is that RIS phase updates change the effective eavesdropper channel, so uncertainty defined only around one fixed RIS configuration need not remain valid after an update. Finally, finite phase resolution and coupled rate constraints call for an implementable procedure with guarantees stated for the problem it actually solves. The contributions are as follows: • A traffic-consistent robust model: We formulate a downlink with a public common stream, confidential private

streams, isotropic AN, and a finite-resolution passive RIS. Eavesdroppers are allowed to cancel the public stream and, conservatively, are given the other private streams as side information. We bound direct-link and cascaded-channel estimation errors and derive an enclosing effective-channel uncertainty ball valid for every admissible RIS configuration. The public and confidential rate constraints and their objective weights remain separate. • A verifiable joint design procedure: We upper-bound worst-case private-stream leakage and construct convex inner transmit updates for fixed AN power and RIS phases using rate minorization, leakage majorization, and a proximal term. A Phase I procedure searches for a feasible starting point; finite AN-power candidates and quantized RIS coordinate changes are then evaluated with direct checks of the physical constraints before acceptance. The method returns a feasible conservative design whenever its initialization and subsequent numerical solves succeed. • Explicit guarantees and evaluation protocol: We provide pseudocode and arithmetic-complexity scaling for the finite-budget procedure. For a fixed AN-power/RIS pair and a feasible initial point, the inner objective is monotone and converges in value, with successive steps vanishing under the stated proximal update. Accepted outer moves remain feasible and improve the objective by a prescribed positive threshold, so their number is finite. The numerical methodology separates confidential and public rates and calls for an adapted RIS–RSMA

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literature baseline evaluated on paired channel draws. Section II specifies the signal model and robust problem. Section III develops the solution and its guarantees, and Section IV describes the simulation methodology and results. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. Network and Channel Model We consider the downlink of a reconfigurable intelligent surface (RIS)-assisted rate-splitting multiple access (RSMA) network, as depicted in Fig. 1. A base station (BS) equipped with N antennas serves K single-antenna legitimate users in the presence of E noncolluding, single-antenna eavesdroppers. The RIS has M passive reflecting elements. The sets of legitimate users and eavesdroppers are denoted by K = {1, . . . , K} and E = {1, . . . , E}, respectively. The BS-to-RIS channel is G ∈ CM ×N . For legitimate user k, let hd,k ∈ CN and hr,k ∈ CM denote the direct BS-to-user and RIS-to-user channels, respectively. Similarly, hd,e ∈ CN and hr,e ∈ CM denote the channels associated with eavesdropper e. A blocked direct link is represented by setting its corresponding direct-channel vector to zero; it is not assigned a nonzero line-of-sight component. The RIS employs B-bit phase control with alphabet n o B QB = ej2πℓ/2 : ℓ = 0, . . . , 2B − 1 . (1) The RIS reflection coefficients are collected in  T θ = ejϕ1 , . . . , ejϕM , Θ = diag(θ),

θm ∈ QB .

H H hH k (θ) = hd,k + hr,k ΘG,

k ∈ K,

(3)

H H hH e (θ) = hd,e + hr,e ΘG,

e ∈ E.

(4)

Both effective channels in (3)–(4) have dimension 1 × N . The channel generation model, including path loss, fading distributions, spatial correlation, and any line-of-sight components, will be specified separately in the simulation methodology. These statistical assumptions are not needed to define the instantaneous signal and optimization models below. We assume that the BS and RIS controller know the legitimate-user channels during the considered transmission interval. For eavesdropper e, write its effective-channel row vector as H T hH e (θ) = hd,e + θ Be , (5) Be = diag(h∗r,e )G ∈ CM ×N . b d,e and B b e , while the underlying The controller has estimates h channel errors obey b e ∥F ≤ ϵB,e . ∥Be − B

Ue ≜ {∆ ∈ CN : ∥∆∥2 ≤ ϵe },

(6)

b H (θ) = h bH + Define the estimated effective channel by h e d,e T b θ Be . For every admissible θ, the effective-channel error satisfies √ b e (θ)∥2 ≤ ϵe ≜ ϵd,e + M ϵB,e . (7) ∥he (θ) − h

b e (θ) + ∆e . (8) he (θ) = h

The ball Ue can contain effective-channel errors that cannot arise from the primitive errors in (6); protecting against the entire ball is conservative. These radii are deterministic bounds, not Gaussian standard deviations. Setting both radii to zero gives perfect estimated eavesdropper CSI. B. RSMA Transmission and Legitimate-User Rates Each user message is split into a public part and a confidential part. The public parts are jointly encoded into one common stream sc , which every legitimate user decodes. The confidential part of user k is encoded into its private stream sk . The streams are independent and satisfy E[|sc |2 ] = E[|sk |2 ] = 1. The BS transmits X x = wc sc + wk sk + z, (9) k∈K N

where wc , wk ∈ C are the common and private precoders. The independently generated artificial noise (AN) obeys z ∼ CN (0, qIN ), where q ≥ 0 is an optimized scalar. Consequently, the total average transmit-power constraint is X ∥wc ∥22 + ∥wk ∥22 + N q ≤ PB . (10) k∈K

(2)

For a fixed RIS configuration, the effective BS-to-receiver channels are

b d,e ∥2 ≤ ϵd,e , ∥hd,e − h

√ √ b e )∥2 ≤ M ϵB,e . Indeed, ∥θ∥2 = M and ∥θ T (Be − B This bound holds uniformly when the RIS phases change. For compactness, define the enclosing uncertainty set

The AN in (9) is a controllable transmission variable; it does not represent transmitter hardware distortion. The signal received by legitimate user k is X H yk = hH hH k (θ)wc sc + k (θ)wj sj + hk (θ)z + nk , (11) j∈K

where nk ∼ CN (0, σk2 ) and σk2 > 0. For compactness, define Dk (θ, q) = q∥hk (θ)∥22 + σk2 .

(12)

Each legitimate user first decodes sc while treating all private streams and AN as noise. Its achievable commonstream rate is ! 2 |hH (θ)w | c k . (13) Rc,k = log2 1 + P H 2 j∈K |hk (θ)wj | + Dk (θ, q) After decoding and removing sc , user k decodes sk , treating the other private streams and AN as noise. Its achievable private-stream rate is Rp,k = log2 (1 + γp,k ), 2 |hH k (θ)wk | . H 2 j̸=k |hk (θ)wj | + Dk (θ, q)

γp,k = P

(14)

The rates in (13)–(14) assume successful common-stream cancellation. The effect of imperfect successive interference cancellation is outside the present signal model. Let Ck ≥ 0 be the portion of the common-stream rate assigned to user k. Because all legitimate users must decode

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the complete common stream, the common-rate allocation must satisfy X Ck ≤ Rc,i , i ∈ K. (15) k∈K

Writing the constraint for every i avoids the nonsmooth mini Rc,i notation and is equivalent to requiring that the aggregate common rate be decodable by the weakest legitimate user.

C. Eavesdropper Model and Confidential Rates The common stream carries public information. We therefore allow an eavesdropper to know and remove sc when attempting to decode a confidential private stream. We also give it the other users’ private streams as side information. This latter assumption gives each eavesdropper a stronger decoding capability than simply treating the other private streams as noise; it avoids relying on multiuser interference as the sole source of confidentiality. For an effective eavesdropper channel he (θ), the resulting information-leakage rate associated with the private stream of user k is bounded by   2
|hH e (θ)wk | , (16) Re,k he (θ) = log2 1 + q∥he (θ)∥22 + σe2 where σe2 > 0 is the receiver-noise variance of eavesdropper e. In (16), the other information streams do not appear in the denominator because they are assumed known to the eavesdropper for this conservative leakage calculation. The eavesdroppers are noncolluding: each acts on its own observation. With independent Gaussian signaling and stochastic wiretap encoding of each confidential private message, the following private-stream rate difference is an achievable confidential-rate bound under the stated noncolluding, genie-aided eavesdropper model:  
+ Rs,k = Rp,k − max Re,k he (θ) , (17) e∈E

where [a]+ ≜ max{a, 0}. The public allocation Ck is not included in (17). Thus, Ck may contribute to a user’s overall delivered rate, but it must not be reported as confidential throughput. Under the bounded uncertainty model (8), define the worstcase leakage rate   b e (θ) + ∆e . Re,k = max sup Re,k h (18) e∈E

∆e ∈Ue

The corresponding robust confidential rate is  + Rs,k = Rp,k − Re,k .

(19)

The supremum in (18) is taken over a bounded channelerror set. It is not a chance constraint, and no secrecy-outage probability is claimed.

D. Robust Joint Precoding and RIS Design Let αk > 0 denote the confidential-rate weight of user k, and let λ ≥ 0 denote the public-throughput weight. We introduce tk ≥ 0 as its designed confidential rate. To ensure that the common stream serves an actual traffic requirement, let rkpub ≥ 0 denote user k’s minimum public rate. An optional minimum confidential-rate requirement is denoted by rksec ≥ 0. These are distinct requirements: public bits cannot satisfy a confidential-rate requirement. The joint public-and-confidential throughput design is parameterized by λ. We maximize a scalarized objective, while always reporting the public and confidential components separately. For λ = 0 the formulation reduces to the confidentialthroughput objective; for λ > 0 the common stream can carry more than its minimum public traffic requirement. A sweep over λ traces the attained public–confidential tradeoff, subject to the limits of the nonconvex solver. Let V ≜ (wc , {wk }, q, θ, {Ck , tk }). The design is X X max αk tk + λ Ck (20a) V

subject to

k∈K

k∈K

  b e (θ) + ∆e , tk ≤ Rp,k − Re,k h ∀k ∈ K, e ∈ E, ∆e ∈ Ue , X Ck ≤ Rc,i , ∀i ∈ K,

(20b) (20c)

k∈K

Ck ≥ rkpub , tk ≥ rksec , ∀k ∈ K, X ∥wc ∥22 + ∥wk ∥22 + N q ≤ PB ,

(20d) (20e)

k∈K

q ≥ 0,

θm ∈ QB ,

m = 1, . . . , M. (20f)

Constraint (20b) limits tk by the legitimate private rate minus the leakage rate for every eavesdropper and every channel in its uncertainty set. Consequently, any P feasible tk is no greater than Rs,k in (19). The term λ k Ck rewards only public throughput; it is never interpreted as a secrecy rate. Constraints (20c) and (20d) ensure commonstream decodability and distinguish public from confidential traffic. The remaining constraints impose the BS power budget, nonnegative isotropic AN power, and finite-resolution passive RIS operation. Problem (20) is nonconvex. In particular, the legitimate and eavesdropper rates are coupled through the precoders, AN covariance, and RIS phases; the semi-infinite secrecy constraints span the channel-uncertainty sets; and the discrete RIS phase decisions are combinatorial. A solution method must therefore specify how it handles each of these features. Neither alternating optimization nor semidefinite relaxation alone establishes global optimality. The next section develops a tractable iterative method and states the guarantees that its individual updates support. III. C ONSERVATIVE ROBUST T RANSMIT AND RIS D ESIGN A. A Uniform Upper Bound on Leakage For a chosen public-throughput weight λ ≥ 0, we seek a feasible, possibly suboptimal solution of (20). For each RIS

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configuration, define   b e (θ)∥2 − ϵe 2 , ae (θ) = ∥h +

Introduce real ve,k , βe,k ≥ 0 and impose [x]+ ≜ max{x, 0}.

(21)

The uniform error bound (7) implies bH |hH e (θ)wk | ≤ |he (θ)wk | + ϵe ∥wk ∥2 , ∥he (θ)∥22 ≥ ae (θ).

(23)

k ∈ K, e ∈ E,

(35)

ub

(n)

log2 (1 + β) ≤ Le,k (β), (n)

B. Convex Transmit Update at Fixed AN Power and RIS Phases be = Fix q = q0 ≥ 0 and θ = θ0 . Write hk = hk (θ0 ), h b he (θ0 ), and ae = ae (θ0 ). The precoder-power budget is P0 = PB −N q0 ; we exclude pairs with P0 < 0. For given precoders, define X 2 2 2 2 Tc,k = |hH |hH (26) k wc | + k wj | + q0 ∥hk ∥2 + σk , 2 2 2 |hH k wj | + q0 ∥hk ∥2 + σk .

(27)

j∈K

For scalar complex equalizers gc,k and gp,k , the common and private mean-square errors (MSEs) are (28)

∗ ep,k = |gp,k | Tp,k − 2 Re{gp,k hH k wk } + 1.

(29)

+

uc,k =

.

(n)

It is value tight at βe,k . Let X collect the real and imaginary parts of all precoders divided by a fixed positive amplitude scale and the real variables {Ck , tk } divided by a fixed positive rate scale. This invertible normalization makes the proximal norm dimensionless. Define F (X) =

X

α k tk + λ

k

X

Ck ,

ρ > 0.

(37)

k

The fixed-pair inner update is the following convex program: max

X,{ve,k ,βe,k }

s.t.

ρ F (X) − ∥X − X(n) ∥22 2 (n) ep,k , tk + Le,k (βe,k ) ≤ R X ec,k , Cj ≤ R

(38a) ∀k, e,

(38b)

∀k,

(38c)

Ck ≥ rkpub , tk ≥ rksec , X ∥wc ∥22 + ∥wk ∥22 ≤ P0 ,

∀k,

(38d) (38e)

k∈K

ec,k = ln uc,k − uc,k ec,k + 1 , R (30) ln 2 ep,k = ln up,k − up,k ep,k + 1 . R (31) ln 2 ec,k ≤ Rc,k and R ep,k ≤ Rp,k . For arbitrary auxiliaries, R Both are value tight at the current precoders under the MMSE updates hH k wc , Tc,k hH wk gp,k = k , Tp,k

(n)

(1 + βe,k ) ln 2

For auxiliary MSE weights uc,k , up,k > 0, set

gc,k =

β − βe,k

j∈K

∗ ec,k = |gc,k |2 Tc,k − 2 Re{gc,k hH k wc } + 1, 2

(36)

(n)

j∈K

X

(n)

Le,k (β) ≜ log2 (1 + βe,k )

(25)

are sufficient, and generally conservative, constraints for (20b). In what follows, the conservative physical problem means (20) with its semi-infinite secrecy constraints replaced by (25).

Tp,k =

de ≜ σe2 + q0 ae > 0.

These convex constraints imply Re,k ≤ log2 (1 + βe,k ). The second inequality has a rotated second-order-cone (SOC) representation because de is constant. If q were also variable here, that convexity argument would not apply. (n) (n) At fixed-pair iteration n, set ve,k and βe,k to their smallest feasible values at the current precoders. Concavity of log2 (1+ β) gives the affine global upper bound

For every admissible primitive-channel error, Re,k (he (θ)) ≤ ub Re,k . The numerator and denominator bounds need not be attained by the same channel error. Therefore, ub

(34)

2 ve,k ≤ de βe,k ,

(22)

Since q ≥ 0 and σe2 > 0, a uniform upper bound on the b H (θ)wk | + ϵe ∥wk ∥2 . leakage rate is Define be,k (θ, wk ) ≜ |h e Then ! 2 be,k (θ, wk ) ub Re,k = log2 1 + 2 . (24) σe + qae (θ)

tk + Re,k ≤ Rp,k ,

b H wk | + ϵe ∥wk ∥2 , ve,k ≥ |h e

1

, ec,k 1 up,k = , ep,k

(34),

(35),

∀k, e.

(38f)

Here ve,k , βe,k ≥ 0 are imposed explicitly; Ck , tk ≥ 0 follow from nonnegative rate floors. The proximal term affects only the update; reported performance always uses F , not the penalized objective. The problem is a convex optimization problem (maximization of a concave objective over convex constraints), and its quadratic and norm constraints admit an SOC formulation.

(32) (33)

where the MSEs on the right use the corresponding MMSE ec,k and equalizers. Holding these auxiliaries fixed makes R e Rp,k concave quadratic functions of the precoders.

C. Feasible Initialization and Discrete Outer Search A positive public or secrecy floor can make an all-zero precoder initialization infeasible. For a fixed pair, use the following auxiliary Phase I problem. At iterate n, construct

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the MMSE rates and leakage majorizers exactly as above and solve min

X,{ve,k ,βe,k },s

s.t.

s

(39a)

(n) ep,k + s, tk + Le,k (βe,k ) ≤ R X ec,k + s, Cj ≤ R

∀k, e, (39b) ∀k,

(39c)

j

(38d),

(38e),

(38f),

s ≥ 0.

(39d)

This is convex and always admits a sufficiently large s if the initial precoders satisfy the power bound. After every solve, make the leakage slacks tight again before forming the next majorizer. If s = 0, direct evaluation certifies feasibility of the conservative physical problem at that pair. If a finite initialization budget ends at s > 0, record “no feasible start found”; this does not prove infeasibility. Select a finite AN design grid Qq ⊆ [0, PB /N ],

0 ∈ Qq .

(40)

Grid points restrict the search, not the physical AN variable. Exhaustive search evaluates at most |Qq |2BM pairs. For a larger RIS, a budgeted coordinate search tests alternate phases of one element and alternate AN grid values. Whenever q or θ changes, a fresh feasible start must be certified; the incumbent precoders need not remain feasible at the candidate. An outer move is accepted only if its directly evaluated conservative physical constraints are satisfied and its unpenalized objective improves by more than δout > 0. Feasibility checks include the public rate floors, the exact legitimate rates, power, and (25) for every (e, k). D. Pseudocode, Complexity, and Convergence Algorithms 1–3 specify the update and acceptance tests. In the idealized statements below, “verify” means exact evaluation; a numerical implementation must use tolerances and report its largest constraint residual. a) Arithmetic complexity.: Let L = |Qq | and let TI and TS bound the Phase I and inner iterations per pair. With the cascaded channel matrices precomputed, forming effective channels for a pair takes O(M N (K + E)) arithmetic operations; updating the legitimate MMSE quantities and evaluating all EK leakage amplitudes takes O(N (K 2 +EK)) in a direct implementation. A standard SOC lift has O(N K +K 2 +EK) real scalar and cone coordinates; denote the actual Newton/KKT system dimension by D and the aggregate selfconcordant barrier parameter by ν. For a standard interiorpoint solve with suitable initialization, √ a conservative dense factorization accounting bound is O( ν D3 log(1/εip )) arithmetic operations per solve; this excludes canonicalization and exploits no sparsity. This is a solver model, not an iterationindependent timing prediction. If H pairs are tested, an upper accounting bound is O H[M N (K + E) + (TI + TS )
√ × (N K 2 + νD3 log(1/εip ))] .

Algorithm 1 Feasible start for a fixed (q0 , θ0 ) Require: Pair with P0 ≥ 0; total Phase I solve budget TI ; one or more power-feasible precoder seeds Ensure: A verified conservative feasible X, or N OT F OUND 1: for each seed while total budget remains do 2: Set Ck = rkpub , tk = rksec ; set n = 0 3: while total Phase I budget remains do (n) (n) 4: Set ve,k and βe,k tightly at the current precoders 5: Update the MMSE equalizers and weights by (32)– (33) 6: Solve Phase I (39); retain its physical variables as X(n+1) 7: if direct evaluation certifies (25), common decoding, QoS, and power then 8: return X(n+1) 9: end if 10: Increment n and the total Phase I solve count 11: end while 12: end for 13: return N OT F OUND

(41)

Algorithm 2 Conservative fixed-pair transmit optimization Require: Fixed feasible (q0 , θ0 , X(0) ); ρ > 0; maximum iterations TS ; tolerance εS Ensure: Verified feasible Xout and its unpenalized value F out 1: for n = 0, . . . , TS − 1 do 2: Update the MMSE auxiliaries at X(n) by (32)–(33) (n) (n) 3: Set ve,k and βe,k tightly; form (36) 4: Solve the SOC-representable problem (38) and obtain X(n+1) 5: Evaluate F (X(n+1) ) and all conservative physical constraints 6: if verification fails then 7: return the last verified incumbent and record a numerical failure 8: end if 9: if F (X(n+1) ) − F (X(n) ) ≤ εS then 10: return (X(n+1) , F (X(n+1) )) 11: end if 12: end for 13: return (X(TS ) , F (X(TS ) ))

Exhaustive enumeration has H = L2BM . A complete coordinate sweep tests at most M (2B − 1) + (L − 1) alternatives, although accepted moves and finite budgets determine the number of sweeps. The exponential factor in exhaustive RIS search rules out a general real-time complexity claim for that variant. Proposition 1 (Feasibility and convergence of the inner update). Assume PB < ∞, all receiver-noise variances are strictly positive, and a fixed pair has a feasible starting point for the conservative physical problem. Suppose (38) is solved exactly at each iteration with ρ > 0 and the specified tight MMSE and leakage updates. Then every physical iterate is feasible for (20); F (X(n) ) is nondecreasing and converges to

7

Algorithm 3 Budgeted AN-grid and finite-phase coordinate search Require: Qq , QB ; candidate budget H; δout > 0; Phase I and inner budgets Ensure: Best verified incumbent found, or N OT F OUND 1: Run Algorithm 1 at chosen seed pairs until a verified pair is found 2: if no verified pair is found then 3: return N OT F OUND 4: end if 5: Run Algorithm 2 at that pair; store the verified incumbent and value Finc 6: repeat 7: Set improved ← false 8: Form the one-element phase and alternate-q neighbors of the current incumbent 9: Set Fbest ← Finc and best candidate to empty 10: for each neighbor, subject to remaining budget H do 11: Run Algorithm 1 at the candidate pair 12: if the candidate has a verified feasible start then 13: Run Algorithm 2; directly verify its physical constraints 14: if verification succeeds and Fcand > Fbest + δout then 15: Store this candidate as best; set Fbest ← Fcand 16: end if 17: end if 18: end for 19: if a best candidate was stored then 20: Replace the incumbent; set Finc ← Fbest and improved ← true 21: end if 22: until improved = false or budget H is exhausted 23: return the verified incumbent with public and confidential rates reported separately

IV. S IMULATION R ESULTS We evaluate the robust RIS–RSMA design on one 360kHz NR physical resource block (PRB) at 3.5 GHz. The direct BS–terminal links use the 3GPP urban-microcell (UMi) LOS/NLOS path-loss and LOS-probability expressions [18]. The RIS–terminal links use the same expressions with the RIS treated as a 10-m node, while the clear BS–RIS hop uses ITU-R P.525-5 free-space loss [19]. The latter RIS–terminal construction, independent small-scale fading across links, and independent shadowing are explicit modeling assumptions; the generator is not a spatially consistent 3GPP channel simulator. The 20-MHz carrier power is allocated uniformly to the modeled 12-subcarrier PRB at 30-kHz subcarrier spacing [20], [21]. Thus, a 30-dBm carrier budget corresponds to PB = 18 mW (12.55 dBm) on the PRB. Table I gives the nominal settings; the sweep ranges are stated with the corresponding figures. Comparison and reporting protocol.: The matched controls are no AN (q = 0), fixed RIS (all RIS phases zero), and no RIS (the cascaded channel suppressed); the remaining variables are optimized by the same finite-budget implementation. Each plotted comparison uses common channel seeds and, except for Fig. 10, conditions on the draws for which all displayed methods return directly verified solutions. We use 2000 bootstrap resamples of these paired draws for the reported 95% intervals; the sample sizes and intervals are retained in the accompanying numeric files. A failed feasiblestart search is recorded as not_found, not as a certificate of physical infeasibility. The AO–SCA/SDR method of Wang et al. [5] addresses a different max–min secrecy formulation with a confidential common stream and hardware impairments. An adaptation of that method is not implemented in these experiments, so the present figures support comparisons with the three matched controls only. A. Confidential Rate, CSI Uncertainty, and RIS Aperture

a finite value; and ∥X(n+1) − X(n) ∥2 → 0. Proposition 2 (Verified outer acceptance). Under the same finite-power and positive-noise assumptions, suppose each tested pair is processed for a finite number of iterations and accepted only after exact physical verification and an improvement greater than δout > 0. Every retained incumbent is feasible for (20), and there are finitely many accepted outer moves even if the search repeatedly revisits pairs. With a finite candidate budget, the procedure terminates. It does not assert optimality over untested or unsuccessfully initialized pairs. Proofs are given in Appendix A. The first proposition proves objective-value convergence and vanishing physical step lengths; it does not, by itself, prove convergence of the iterates to a unique point or KKT stationarity. The outer result does not establish optimality for continuously varying q or a globally optimal RIS phase configuration.

Fig. 2 shows that both methods gain from increased transmit power, while the joint design maintains a confidential-rate advantage over no RIS at every tested power. At 30 dBm, the paired-draw means are 18.95 and 16.41 bit/s/Hz, respectively; their paired difference is 2.54 bit/s/Hz with a 95% bootstrap interval of [2.03, 3.14]. The observed gain reflects the assumed reflective corridor and should not be extrapolated to arbitrary RIS placements. The CSI-error fraction ρ in Fig. 3 affects both the direct and cascaded primitive eavesdropper channels. As ρ increases from zero to 0.03, the joint confidential rate decreases from 25.27 to 12.62 bit/s/Hz. AN provides essentially no gain at ρ ≤ 0.005 but increases the mean by 2.86 bit/s/Hz over no AN at ρ = 0.03 (paired interval [1.94, 3.81]). Thus, the benefit of AN is conditional on uncertainty; it does not compensate for the overall loss due to a larger uncertainty set. Fig. 4 uses nested 8 × 8, 12 × 12, and 16 × 16 central subarrays of the same channel draw. The joint rate increases from 16.94 to 18.89 bit/s/Hz between M = 64 and 256, whereas the fixed-RIS rate remains near 16.2 bit/s/Hz. The paired joint-minus-fixed-RIS gain consequently rises from

8

TABLE I N OMINAL SIMULATION PARAMETERS FOR THE BLOCKED - CORRIDOR CASE .

Value

Carrier frequency; carrier bandwidth; subcarrier spacing Modeled resource; receiver noise figure BS antennas; served users; noncolluding eavesdroppers RIS elements and phase resolution BS and RIS positions (m) User positions (m) Eavesdropper positions (m) Conditioned direct/RIS link states BS–RIS channel; RIS element efficiency LOS/NLOS small-scale Rician factors; shadowing standard deviations Carrier BS power; per-PRB budget PB Per-PRB receiver noise power Primitive eavesdropper CSI-error fraction ρ AN fraction N q/PB Minimum public/confidential rate per user Objective weights Screened candidates; Phase-I/inner update budgets Independent seeds per operating point

3.5 GHz (n78); 20 MHz; 30 kHz One 360-kHz PRB; 7 dB N = 8; K = 4; E = 2 M = 256 (16 × 16); B = 2 bits (0, 0, 10); (40, 0, 10) [70, 120] × [−10, 10] × {1.5}
[60, 130] × [−35, −22] ∪ [22, 35] × {1.5} Users: NLOS/LOS; eavesdroppers: LOS/NLOS LOS free-space link; 0.8 5/0; 4/7.82 dB; BS–RIS Rician factor 10

Joint RIS/AN RSMA No RIS

20

15

10 20

25 30 Carrier BS power (dBm)

30 dBm; 12.55 dBm (18 mW) σk2 = σe2 = −111.44 dBm b d,e ∥2 , ϵB,e = ρ∥B b e ∥F 0.01; ϵd,e = ρ∥h {0, 0.05, 0.10, 0.20, 0.30} rkpub = 0.1; rksec = 0 bit/s/Hz αk = 1; λ = 0.2 12; 6/4 100 for each main case; 50 for power; 40 for other sweeps

Confidential sum rate (bit/s/Hz)

Confidential sum rate (bit/s/Hz)

Parameter

35

25

Joint RIS/AN RSMA No AN

20

15

10 0.0

0.5

1.0

1.5

2.0

2.5

Primitive CSI error fraction

3.0 ×10−2

Fig. 2. Confidential sum rate versus carrier BS power in the blocked corridor. Power is allocated uniformly to one 360-kHz PRB.

Fig. 3. Confidential sum rate versus primitive eavesdropper CSI-error fraction ρ. The no-AN control uses the same channel draws.

0.70 to 2.66 bit/s/Hz. In this geometry, a larger aperture is useful primarily when its phases are adapted, rather than merely increased in number.

The empirical distributions in Fig. 6 use the intersection of 97 verified nominal draws across all four methods. The paired mean gain of the joint design is 2.59 bit/s/Hz over no RIS and 2.78 bit/s/Hz over fixed RIS, with respective 95% intervals [2.10, 3.12] and [2.22, 3.41]. Its mean gain over no AN is only 0.25 bit/s/Hz and occurs in a small subset of draws; the joint design exceeds no AN on 7 of the 97 paired draws. This distributional view avoids interpreting a small average AN gain as a typical per-channel gain. Fig. 7 explains this behavior: of the 99 verified nominal joint-design draws, 92 select zero AN and 7 select N q/PB = 0.05; none select a larger fraction. At ρ = 0.03 in the uncertainty sweep, positive AN is instead selected on 23 of 39 verified draws. Hence, the AN grid is available to the optimizer, but its use is selective under the stated conservative

B. Public Traffic, Rate Distribution, and AN Selection Fig. 5 varies the public-rate weight over λ ∈ {0, 0.1, 0.2, 0.5, 1}. For the joint design, increasing the delivered public sum rate from 0.40 to 1.07 bit/s/Hz changes the confidential sum rate from 18.89 to 18.57 bit/s/Hz. The common stream contributes only to the horizontal coordinate: its public bits are excluded from confidential throughput. The limited movement of the points at intermediate weights is an outcome of the finite candidate search, rather than a claim that the plotted points trace a globally optimal rate region.

19.0

1.0

Joint RIS/AN RSMA Fixed RIS No RIS

0.8 Empirical CDF

18.5 18.0 17.5 17.0 16.5 16.0

0.4 Joint RIS/AN RSMA No AN Fixed RIS No RIS

0.0 64

144 RIS elements M

256

Fig. 4. Confidential sum rate versus RIS aperture size M on nested physical channel draws. Fixed RIS uses zero phases.

Confidential sum rate (bit/s/Hz)

0.6

0.2

0

5 10 15 20 25 Confidential sum rate (bit/s/Hz)

30

Fig. 6. Empirical CDF of confidential sum rate on the 97 nominal draws verified by all four methods.

1.0

19.0 18.5 18.0 Joint RIS/AN RSMA No RIS

17.5 17.0 16.5 16.0

Selection frequency

Confidential sum rate (bit/s/Hz)

9

0.8 0.6 0.4 0.2 0.0

0.4

0.6 0.8 1.0 Public sum rate (bit/s/Hz)

Fig. 5. Delivered public common rate and confidential private rate as the public-rate weight λ varies.

leakage bound. C. Algorithm Behavior and Propagation Sensitivity Fig. 8 reports one fixed-pair diagnostic: zero RIS phases, N q/PB = 0.05, and the prespecified median-rate nominal channel draw. The directly evaluated, unpenalized objective rises from 11.14 to 14.54 bit/s/Hz and levels off within approximately ten inner updates. This diagnostic uses a 20-update budget, whereas the Monte Carlo campaign uses four inner updates; it neither depicts full alternating RIS/AN convergence nor establishes finite-run global optimality. Fig. 9 gives a paired joint-minus-no-RIS gain of 1.40 bit/s/Hz under random UMi LOS states and 2.59 bit/s/Hz in the conditioned blocked corridor. The respective bootstrap intervals are [1.03, 1.80] and [2.10, 3.12]. The two cases also differ in terminal geometry and link-state conditioning; this

0

0.05 0.1 0.2 Total AN power fraction Nq/PB

0.3

Fig. 7. Selection frequency of the total AN power fraction N q/PB on 99 verified nominal draws.

comparison therefore measures the gain within each specified case, not the isolated causal effect of blockage. Fig. 10 imposes per-user minimum confidential rates from 0 to 1.5 bit/s/Hz. Here the ordinate is verified confidential goodput: a search that does not return a verified design contributes zero service. At the largest requirement, the joint and no-RIS goodputs are 13.77 and 12.23 bit/s/Hz, while verifiedsolution counts fall to 30/40 and 29/40, respectively. The decline therefore reflects both the tighter service requirement and the reduced rate of successful feasible-start searches; it must not be described as a measured physical infeasibility probability. D. Search Time and Phase Resolution Fig. 11 reports wall-clock time for the finite-budget implementation. The joint-design mean rises from 1.27 to 1.45 s as M increases from 64 to 256, but the fixed-RIS curve

14

12

10 0

5

10 Iteration

15

20

Fig. 8. Direct objective trace for a single fixed zero-phase RIS and N q/PB = 0.05 pair.

Verified confidential goodput (bit/s/Hz)

Unpenalized objective (bit/s/Hz)

10

17 16 15 14 13 12 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Minimum per-user confidential rate (bit/s/Hz)

Fig. 10. Verified confidential goodput versus the minimum confidential rate per user; unsuccessful searches count as zero delivered service.

1.8 Measured search time (s)

3.0

RIS gain (bit/s/Hz)

Joint RIS/AN RSMA No RIS

18

2.5 2.0 1.5 1.0

1.7 1.6

Joint RIS/AN RSMA Fixed RIS No RIS

1.5 1.4 1.3

0.5

64

0.0 Random LOS

Blocked corridor

Fig. 9. Paired confidential-rate gain of joint RIS/AN over no RIS in the random-LOS and conditioned blocked-corridor cases; whiskers show 95% bootstrap intervals.

is nonmonotonic (1.69, 1.80, and 1.38 s). Variable solver paths and runtime variability preclude fitting an asymptotic complexity law to these three operating points; the values characterize this implementation and computing environment only. Finally, Fig. 12 shows that increasing the phase resolution from one to two bits raises the joint confidential rate from 18.35 to 18.89 bit/s/Hz; four bits yield 19.06 bit/s/Hz. The fixed-zero-phase control is independent of B. Thus, two-bit phases capture most of the observed four-bit performance under this channel generator and candidate budget, without establishing a universal hardware-resolution optimum. V. C ONCLUSION This paper formulated a robust RIS-assisted RSMA downlink for mixed public and confidential traffic. The common

144 RIS elements M

256

Fig. 11. Measured finite-budget search time versus RIS size for the joint, fixed-RIS, and no-RIS designs.

stream carries public information, while confidential information is conveyed through private streams. The secrecy model allows each eavesdropper to cancel the common stream and treats the other private streams as known side information. Bounded errors in the direct and cascaded eavesdropper channels lead to an effective-channel uncertainty bound that remains valid as the RIS phases change. We developed a conservative joint design of the transmit precoders, artificial noise, common-rate allocations, and finiteresolution RIS phases. The procedure combines feasible initialization, convex inner transmit updates, and a discrete outer search with direct feasibility checks. For a fixed artificialnoise power and RIS configuration, feasible inner iterates have a nondecreasing objective that converges in value; accepted outer updates remain feasible and are finite in number. These guarantees establish what the procedure delivers under the stated uncertainty model without implying global optimality.

Confidential sum rate (bit/s/Hz)

11

19.0 18.5 18.0 Joint RIS/AN RSMA Fixed RIS

17.5 17.0 16.5

Fmax < ∞ be the uniform objective bound from the preceding proof and F0 be the initial incumbent value. After J accepted moves, FJ > F0 + Jδout ; therefore J < (Fmax − F0 )/δout . Each sweep checks a finite set of alternatives and each candidate uses finite Phase I and inner budgets. Thus the number of accepted moves is finite, and a finite overall candidate budget also forces termination. A failed Phase I run conveys no infeasibility certificate and cannot be used to infer optimality.

R EFERENCES 1

2 3 RIS phase resolution B (bits)

4

Fig. 12. Confidential sum rate versus RIS phase resolution B on paired physical channel draws.

A PPENDIX A P ROOFS OF THE A LGORITHMIC G UARANTEES Proof of Proposition 1. At iteration n, choose the MMSE auxiliaries at the physical incumbent. The resulting rate lower bounds equal the exact legitimate rates at X(n) . Choose the (n) leakage slacks at their smallest feasible values: then βe,k = (n) (n) b H w |+ϵe ∥w ∥2 )2 /de . The affine leakage bound equals (|h e k k the conservative leakage rate at these slacks. Consequently, (X(n) , v(n) , β (n) ) is feasible for (38) whenever the physical incumbent is feasible for (25), common decoding, QoS, and power. ub For any feasible point of (38), Re,k ≤ log2 (1 + βe,k ) ≤ (n) ep,k ≤ Rp,k and R ec,k ≤ Rc,k . Hence, its Le,k (βe,k ), whereas R physical variables satisfy (25) and the exact common, QoS, and power constraints. By induction, all iterates are feasible for the conservative physical problem and therefore also for (20). Use the old incumbent and tight slacks as a feasible point in the optimization at iteration n. Optimality of the new point gives ρ F (X(n+1) ) − F (X(n) ) ≥ ∥X(n+1) − X(n) ∥22 ≥ 0. (42) 2 Because RIS phases belong to a finite alphabet, each Hk ≜ maxθ∈QM ∥hk (θ)∥2 is finite. The power budget and posiB tive noise give the uniform bound Rc,k , Rp,k ≤ log2 (1 + PB Hk2 /σk2 ). The secrecy inequalities imply 0 ≤ tk ≤ Rp,k ; P common decoding gives C ≤ max i Rc,i . Thus F has k k a finite upper bound. A nondecreasing bounded sequence of converges. Summing (42) over n bounds P real values (n+1) ∥X − X(n) ∥22 by a finite constant, proving that n≥0 its individual terms tend to zero. Proof of Proposition 2. Algorithm 3 stores only physically verified designs. Proposition 1 and the direct candidate check imply that every accepted incumbent satisfies the conservative constraints and hence the original robust constraints. Let

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