Co-Design of Cryptographic Parameters and Delay-Aware Feedback Gain for Encrypted Control Systems

Yeongjun Jang

arxiv: 2604.14774 · v1 · submitted 2026-04-16 · 📡 eess.SY · cs.SY

Co-Design of Cryptographic Parameters and Delay-Aware Feedback Gain for Encrypted Control Systems

Yeongjun Jang This is my paper

Pith reviewed 2026-05-10 11:14 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords encrypted controlhomomorphic encryptiondelay-dependent stabilitylinear matrix inequalitiesfeedback gainco-designcryptographic parametersnetworked control

0 comments

The pith

A co-design of cryptographic parameters and delay-aware feedback gains can keep encrypted control systems stable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Homomorphic encryption secures both computation and communication in networked control but creates delays that grow with the security level and can destabilize the closed loop. Standard practice designs the controller in plaintext first and encrypts afterward, without addressing the induced delay. This paper instead jointly selects the cryptographic parameters and a feedback gain that explicitly accounts for the delay those parameters produce. It models the delay as a deterministic function of the parameters and gives a sufficient stability condition as a finite set of linear matrix inequalities. The resulting outer-inner procedure enumerates parameter sets that meet a security target and, for each set, solves the LMIs to find a stabilizing gain.

Core claim

By expressing encryption-induced delay directly as a function of cryptographic parameters and deriving a finite collection of linear matrix inequalities that guarantee the existence of a stabilizing delay-dependent feedback gain, a tractable outer-inner design procedure can search over parameters satisfying a desired security level and recover a gain that renders the encrypted closed-loop system stable.

What carries the argument

The outer-inner search: an outer enumeration over cryptographic parameters that satisfy a security requirement, paired with an inner feasibility check of linear matrix inequalities obtained from delay-dependent stability analysis of the closed-loop system.

If this is right

Encrypted control can be used at higher security levels without loss of stability by selecting a feedback gain matched to the resulting delay.
The design task reduces to a sequence of standard LMI feasibility problems and is therefore computationally practical.
Post-design encryption of a plaintext controller can be replaced by a joint search that avoids stability degradation.
Networked control systems gain a systematic way to trade security strength against closed-loop stability margins.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same modeling of delay from security parameters could be reused for other privacy mechanisms such as secure multiparty computation in feedback loops.
Hardware implementations could incorporate measured encryption latency rather than the analytic model to tighten the design.
Adding network-induced delays to the same LMI framework would produce controllers robust to both encryption and communication effects.
Performance objectives beyond stability, such as disturbance rejection, could be included by replacing the pure feasibility LMIs with optimization problems.

Load-bearing premise

The total delay caused by encryption can be modeled exactly as a deterministic value that depends only on the chosen cryptographic parameters, so that ordinary delay-dependent linear matrix inequality conditions remain valid for the encrypted implementation.

What would settle it

Apply the co-design procedure to a concrete linear plant for a chosen security level, obtain the cryptographic parameters and feedback gain, implement the encrypted controller with the modeled delay, and check whether the closed-loop state trajectories converge or stay bounded; divergence when the LMIs were feasible would contradict the claim.

read the original abstract

Encrypted control employs homomorphic encryption (HE) to protect both the computation and communication stages, making it a promising approach for secure networked control systems. Most existing results pre-design a controller in the plaintext domain and then implement it over encrypted data. However, this can be problematic because HE induces non-negligible communication and computation delays, which typically increase with the security level, potentially degrading control performance and even destabilizing the closed-loop system. To address this issue, we propose a co-design framework for cryptographic parameters and delay-aware feedback gain. We characterize the encryption-induced delay as a function of the cryptographic parameters and derive a sufficient condition for the existence of a stabilizing delay-aware feedback gain, expressed as a finite set of linear matrix inequalities. This leads to a tractable outer-inner design procedure that searches over cryptographic parameters that satisfy a desired security level and, for each such parameter, seeks a stabilizing feedback gain.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sets up an outer-inner co-design where crypto parameters determine a modeled delay and LMIs then check for a stabilizing gain, which is a clean structured procedure but depends on treating that delay as deterministic and isolated.

read the letter

This paper's main point is a co-design where you first pick cryptographic parameters that hit a target security level, characterize the resulting encryption delay as a function of those parameters, and then solve a set of LMIs to find a feedback gain that keeps the closed-loop system stable under that delay. The outer loop searches the parameter space and the inner loop checks stability for each candidate. Most prior encrypted control work designs the gain in plaintext and encrypts afterward, so this explicit linkage is the step that stands out as new. It turns the security-versus-performance tradeoff into a searchable procedure rather than an after-the-fact problem. The approach does well by staying within standard delay-dependent LMI tools, which keeps the inner problem tractable and avoids reinventing stability theory. The derivation is presented as sufficient conditions derived from the system model, which is a reasonable and reproducible route. The soft spots sit mainly in the delay modeling. The claim treats encryption delay as a deterministic map from parameters alone, but actual runtimes in libraries like SEAL or HElib also depend on hardware, current ciphertext noise, and specific operations such as rotations or bootstrapping. In addition, common schemes like CKKS introduce approximation error, so the plant input is not exactly the delayed linear feedback assumed in the LMI. If those effects are non-negligible, parameters that pass the LMIs could still fail to stabilize the real encrypted loop. The abstract does not show numerical validation against measured delays or perturbed dynamics, so the gap between the nominal model and implementation remains open. This work is aimed at researchers in encrypted networked control and cyber-physical security who need a systematic way to balance security level against closed-loop performance. A reader already familiar with homomorphic encryption and time-delay LMIs will see the clearest value and can adapt the outer-inner template to their own scheme. It deserves peer review because the core idea addresses a practical deployment issue with a concrete, checkable procedure; referees can usefully press on the delay assumptions and ask for implementation benchmarks.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a co-design framework for cryptographic parameters and delay-aware feedback gains in homomorphic-encryption-based control systems. It characterizes the encryption-induced delay as a deterministic function of the cryptographic parameters, derives a sufficient condition for the existence of a stabilizing feedback gain in the form of a finite set of linear matrix inequalities (LMIs), and presents a tractable outer-inner design procedure that searches over admissible cryptographic parameters satisfying a target security level while solving for a stabilizing gain at each candidate.

Significance. If the delay model and LMI conditions are shown to be valid for the closed-loop encrypted system, the result would be significant for practical deployment of encrypted control: it directly addresses the stability degradation caused by HE-induced delays that increase with security level, moving beyond the common practice of designing controllers in plaintext and then encrypting them. The outer-inner procedure offers a computationally attractive way to jointly optimize security and performance.

major comments (2)

[§3 (delay characterization)] The central claim rests on modeling the encryption-induced delay τ as a deterministic function solely of the cryptographic parameters (abstract and §3). This assumption is load-bearing for the outer-inner procedure; however, actual HE runtimes (multiplications, rotations, bootstrapping) depend on hardware, library optimizations, current noise level, and implementation details, introducing variability or state-dependence not captured by a static map. Without explicit bounds or experimental validation of this determinism, the LMI-derived gains may not stabilize the real system.
[§4 (LMI derivation)] The LMI conditions (abstract and §4) are derived under the assumption of exact delayed dynamics ẋ(t) = Ax(t) + B K x(t-τ) with constant delay. The encrypted controller typically employs approximate arithmetic (e.g., CKKS) or quantization, so the effective input is a perturbed version of Kx(t-τ). The manuscript does not provide a robustness margin or perturbation analysis showing that the nominal LMIs remain sufficient; this undermines the stabilizing claim for realistic encrypted loops.

minor comments (2)

[§2] Notation for the delay function τ(·) and the cryptographic parameter set should be introduced with a clear table or explicit mapping to standard HE parameters (modulus, dimension, security level) to improve readability.
[§5] The numerical examples or validation section would benefit from reporting both the LMI feasibility and closed-loop simulation results under the actual HE implementation (not just the nominal delay model) to demonstrate practical utility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and for recognizing the potential significance of the co-design framework. We address each major comment below with targeted revisions to clarify assumptions and strengthen the results.

read point-by-point responses

Referee: [§3 (delay characterization)] The central claim rests on modeling the encryption-induced delay τ as a deterministic function solely of the cryptographic parameters (abstract and §3). This assumption is load-bearing for the outer-inner procedure; however, actual HE runtimes (multiplications, rotations, bootstrapping) depend on hardware, library optimizations, current noise level, and implementation details, introducing variability or state-dependence not captured by a static map. Without explicit bounds or experimental validation of this determinism, the LMI-derived gains may not stabilize the real system.

Authors: Section 3 derives τ deterministically from the number and type of homomorphic operations dictated by the cryptographic parameters, under the modeling assumption of constant per-operation execution times. This enables the outer-inner search procedure. We agree that real-world variability exists. In the revision we will add an explicit remark in §3 stating that the model provides a nominal delay and recommending the use of conservative upper bounds (e.g., measured worst-case times on target hardware) when applying the LMIs, thereby preserving the sufficient stability condition. revision: partial
Referee: [§4 (LMI derivation)] The LMI conditions (abstract and §4) are derived under the assumption of exact delayed dynamics ẋ(t) = Ax(t) + B K x(t-τ) with constant delay. The encrypted controller typically employs approximate arithmetic (e.g., CKKS) or quantization, so the effective input is a perturbed version of Kx(t-τ). The manuscript does not provide a robustness margin or perturbation analysis showing that the nominal LMIs remain sufficient; this undermines the stabilizing claim for realistic encrypted loops.

Authors: The LMIs in §4 are obtained for the exact delayed closed-loop system, which is appropriate when the homomorphic scheme is configured for exact arithmetic (e.g., appropriate CKKS parameters or exact schemes such as BFV). We acknowledge that the manuscript does not yet address approximation errors. In the revision we will augment §4 with a perturbation analysis: if the effective input deviation satisfies a norm bound derived from the Lyapunov matrix and the LMI solution, asymptotic stability is retained via a standard Lyapunov perturbation argument. This will be stated as an additional sufficient condition. revision: yes

Circularity Check

0 steps flagged

No circularity detected in co-design derivation

full rationale

The derivation chain begins with an explicit characterization of encryption-induced delay as a deterministic function of cryptographic parameters (modulus, dimension, security level) drawn from standard homomorphic encryption models, followed by application of established delay-dependent LMI stability criteria for constant-delay LTI systems to obtain a sufficient condition for stabilizing gains. The outer-inner procedure is then a direct search over the resulting feasible set. No equation or claim reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the LMI conditions and delay map are independent of the final co-design output and rest on external stability theory rather than tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard delay-dependent stability theory and homomorphic encryption properties; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Encryption-induced delays can be expressed as a deterministic function of cryptographic parameters.
Invoked when the paper states that the delay is characterized as a function of the parameters.
standard math Delay-dependent stability of the closed-loop system can be certified by a finite set of linear matrix inequalities.
Standard assumption in time-delay control theory used to obtain the sufficient condition.

pith-pipeline@v0.9.0 · 5452 in / 1362 out tokens · 41505 ms · 2026-05-10T11:14:09.493906+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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