Successive Data Injection (SuDaI): Methods & Impact

• Successive Data Injection (SuDaI) is a paradigm involving repeated, temporally structured data injections that cumulatively alter system behavior.
• It is applied across cyber-physical systems, quantum machine learning, federated learning, and solar physics to exploit vulnerabilities or optimize performance.
• Empirical studies demonstrate high detection accuracy and improved system control through matrix decomposition, median-based defenses, and active countermeasures.

Successive Data Injection (SuDaI) is a technical paradigm and attack/analysis modality in which data—malicious, synthetic, or operational—are introduced into a target system or model through sequential, temporally correlated, or iteratively applied insertions. This concept materializes in diverse domains, including cyber-physical systems, quantum machine learning, federated learning, and solar physics, each adopting SuDaI for different objectives ranging from vulnerability exploitation and detection to overcoming fundamental physical or hardware constraints.

1. Core Definition and Conceptual Scope

Successive Data Injection (SuDaI) denotes the repeated, temporally resolved, or layer-wise introduction of data (real, adversarial, or synthetic) into a system in such a manner that the cumulative, time-dependent impact differs critically from a single-shot injection. The term encompasses:

• Adversarial SuDaI: Malicious actors inject forged or perturbed data over multiple system cycles or across several features/sensors, aiming for stealth, persistence, or maximal disruption.
• Algorithmic/Architectural SuDaI: As a constructive mechanism (especially in quantum computing), SuDaI refers to repeated, sequential data encoding in computational architectures, circumventing resource limits (e.g., number of qubits) by increasing circuit depth or operational rounds.
• Physical SuDaI: In space/solar physics, SuDaI describes the physical process whereby injected quantities (e.g., magnetic helicity) are introduced with temporal structure, significantly controlling the system’s stability and eruptivity.

This definition subsumes or generalizes a range of attack models, encoding approaches, and physical modeling schemes unified by their non-trivial, successive operation.

2. Principal Methodologies and Mathematical Formulations

2.1. Quantum Machine Learning (Variational Circuits, GANs)

In quantum networks, SuDaI addresses the challenge of efficiently mapping high-dimensional classical data into low-qubit variational quantum circuits (VQCs) by repeated re-uploading of data at each circuit depth. For a contextual time series window $\{a_1,\ldots,a_K\}$ and $n$ qubits, the quantum state is prepared as: $$\ket{0}^{\otimes n} \xrightarrow{I^{(1)}(a_1)} V^{(1)} \cdots \xrightarrow{I^{(K)}(a_K)} V^{(K)}$$ where $I^{(k)}(a_k)$ is an input/data injection layer (parameterized rotation), $V^{(k)}$ a variational/entangling layer (Kalfon et al., 2023, Hammami et al., 16 May 2025, Hammami et al., 30 Oct 2025). This achieves representation of high-dimensional time series without increasing the physical qubit count, instead growing circuit depth and representational power.

2.2. Cyber-Physical and Control Systems

In attack scenarios, SuDaI typically involves forging sensor values at many or all timesteps during a data-driven learning window: $\tilde{x}[k] = x[k] + a[k],\quad k=0,\ldots,T$ where the injected sequence $a[k]$ is crafted—potentially by simulating a stable/unstable fake system, or maintaining constant biases—to mislead data-driven estimators, causing the operator to learn suboptimal or unstable feedback controllers (Anand, 24 Apr 2025).

2.3. Process Systems and Information-Theoretic Attacks

SuDaI enables sequential, greedy construction of sparse multivariate Gaussian attacks: $$a \sim \mathcal{N}(0, \Sigma_a)\quad \text{with}~|\mathrm{supp}(\Sigma_a)|_0 = k$$ The optimal $k$-sparse attack set is selected in a successive, sensor-by-sensor fashion, maximizing KL divergence between state/estimate distributions under attack and under normal operation, while constraining detection by minimizing divergence on observations (Ye et al., 31 Jan 2025).

2.4. Power Systems and Matrix Decomposition

Persistent, "unobservable" attacks are modeled as a column-sparse matrix structure injected into time-series PMU measurements: $M = \bar{L} + \bar{C}W^T + N$ where $\bar{C}$ is sparse along columns (few buses attacked persistently in time), $W^T$ transforms to the PMU measurement domain, and $N$ is noise (Gao et al., 2016). Convex optimization is used for decomposition and attacker identification.

3. Detection and Defense Strategies under SuDaI

3.1. Cyber-Physical and CAN Bus Networks

Detection of SuDaI attacks leverages stable system-level patterns in message or sensor sequences. For example, in CAN bus intrusion detection, similarity scores (cosine similarity, Pearson correlation) are computed between message-sequence graphs (MSGs) from successive time windows: $$\operatorname{Cosim}(x, y) = \frac{\sum_i x_i y_i}{\sqrt{\sum_i x_i^2} \sqrt{\sum_i y_i^2}}$$ Abrupt drops in similarity are indicative of malicious SuDaI. Thresholding, change point detection, and LSTM-RNN models operating on such similarity sequences allow for robust, unsupervised attack detection up to $98.45\%$ accuracy (Jedh et al., 2021).

3.2. Data-Driven Control

For repeated attacks during system identification, mitigation is achieved via:

• Active: Preventing adversary knowledge of operator actions (e.g., encrypted control, input watermarking).
• Passive: Exploiting expected system response to input stimuli—constant-bias attacks are detected by failure of system trajectories to decay to equilibrium under null input (Anand, 24 Apr 2025).

3.3. Federated Learning

In federated learning, successively injected malicious updates are detected at the server using a median-based, round-averaged deviation statistic: $$U_j(t) = ||\Delta W_j(t) - \text{median}_{\ell\neq j}(\Delta W_\ell(t))||_\infty$$ After periods of deviation above a threshold, an agent is ignored, with periodic re-evaluation allowing for adaptation to time-varying SuDaI scenarios (Shalom et al., 2023).

3.4. Matrix Decomposition in Power Systems

Attacks persistent over consecutive times are identified via convex minimization: $$\min_{L,C}~||L||_* + \lambda||C||_{1,2} \;\;\text{s.t.}\;\; ||M-L-CW^T||_F \leq \eta$$ Guarantees on exact recovery of the attack support and clean data are established under system-specific incoherence and sparsity conditions (Gao et al., 2016).

4. Experimental Results and Empirical Impact

SuDaI mechanisms have demonstrated significant, context-dependent impacts:

• Quantum GANs with SuDaI: Effective anomaly detection in high-dimensional, multivariate time-series with as few as 4–6 qubits and low parameter count; achieves F1 scores of $0.98$–$0.99$ and MSEs around $0.0013$ (substantially better than classical GANs) (Hammami et al., 16 May 2025, Kalfon et al., 2023, Hammami et al., 30 Oct 2025).
• CAN Bus IDS: Real driving datasets show threshold and LSTM-based SuDaI detection yields $>97\%$ accuracy, comparable to proprietary/ID-dependent industry solutions (Jedh et al., 2021).
• Data-Driven Control: Successive injection during controller training can induce unstable closed-loop dynamics or arbitrarily degrade LQR cost, with effectiveness increasing for higher-order plants (Anand, 24 Apr 2025).
• Power Systems: In both synthetic and real PMU datasets, matrix decomposition robustly identifies SuDaI even under incomplete observations and noise, outperforming classical robust PCA (Gao et al., 2016).
• Federated Learning: Median-based online defense provably isolates attackers in finite time, with empirical verification on MNIST under various SuDaI attack models ensuring convergence to correct models when honest agents are in the majority (Shalom et al., 2023).

5. Physical and Theoretical Manifestations

In solar physics, SuDaI can refer to the injection of magnetic helicity with successive, possibly opposite sign, phases. Observational evidence, such as a transition in the sign of helicity injection rate $dH/dt$ and mean force-free twist parameter $\alpha_{av}$, confirms that such temporal structure suppresses the formation of eruptive, twisted flux rope structures and thus precludes CME (coronal mass ejection) activity. Quantitative analysis for four non-erupting active regions reveals:

• Alternating sign injection, no CME, and only sub-C-class flares.
• Coronal helicity accumulation is canceled by subsequent opposite injections, never crossing the eruption threshold (Vemareddy, 2022).

This establishes a direct link between physical SuDaI and system-level stability and eruptivity, with broader predictive consequences for space weather modeling.

6. Summary Table: Domain-Specific SuDaI Manifestations

Domain	SuDaI Manifestation	Impact/Use
Cyber-physical/can	Repeated message/sensor forgeries	Detection via graph similarity, LSTM, change-point
Data-driven control	Recurrent false data in system ID/learning	Controller degradation, instability, countermeasures
Process resilience	Sparse, greedy, sequential sensor attacks	Stealthy KL-divergence-maximizing DIAs, vulnerability mapping
Power systems	Persistent "unobservable" channel attacks	Column-sparse+low-rank matrix decomposition for attacker recovery
Quantum ML	Sequential/iterated quantum encoding of classical data	Overcomes qubit constraints, boosts anomaly detection, maintains expressivity
Solar physics	Sign-alternating, time-resolved helicity injection	Suppresses flux rope/CME formation, conserves net helicity
Federated learning	Successive malicious updates from dynamic agents	Online median-based exclusion and re-integration, provable convergence

7. Theoretical and Practical Significance

SuDaI constitutes a unifying principle underlying both sophisticated attack strategies (where persistence and stealth are paramount) and resource-optimizing computational design (where temporal/data decomposition is central). The theoretical frameworks developed allow for quantifiable resilience/vulnerability assessment, exact identification, and robust, provably convergent mitigation under broad classes of system models and adversarial conditions. In physical systems, SuDaI-centric temporal signatures provide a basis for new eruptivity diagnostics and preventive strategies.

A plausible implication is that as more complex, distributed, and temporally evolving systems become central to infrastructure, SuDaI—both as a vulnerability and as a technique to exploit system structure—will remain a focal point of analytical, algorithmic, and architectural research.