Covert Verifiable Learning

• Covert verifiable learning is a design paradigm that conceals elements of communication, computation, or data acquisition while incorporating mechanisms for cryptographic or formal validation.
• It integrates diverse methodologies such as covert communication in federated networks, learner-private optimization, zero-knowledge proofs, and ledger-attested protocols to secure learning systems.
• Research in this area highlights trade-offs between enhanced privacy and verifiability, often resulting in increased latency or computational overhead in exchange for robust security guarantees.

Searching arXiv for papers on covert verifiable learning, covert learning, and verifiable/decentralized ML. “Covert verifiable learning” (Editor’s term) denotes the conjunction of covertness and verifiability in learning systems: the learning process, data-acquisition strategy, communication event, or target is concealed from an eavesdropper or adversary, while the resulting computation, update, or evidence remains subject to cryptographic, formal, or protocol-level validation. In the literature represented here, the term spans several technically distinct lines of work: covert communication in federated learning networks, learner-private optimization, selective ignorance under partial verifiability, zero-knowledge-proof-based verifiable decentralized machine learning, secure and verifiable collaborative transfer learning, ledger-attested formal learning, and covert verifiable quantum learning. The unifying concern is to limit what can be inferred from observation while preserving a mechanism for correctness, auditability, or detection of tampering (Xie et al., 2021, Xing et al., 2023, Anand et al., 8 Oct 2025).

1. Conceptual scope and semantics

In this literature, “covert” does not have a single meaning. In wireless federated learning, covert communication—also known as Low Probability of Detection (LPD) communication—aims to ensure that adversaries cannot determine, with probability better than chance, whether legitimate wireless communication is happening. In learner-private optimization, covert or learner-private optimization means minimizing an objective function $f$ while preventing an eavesdropper from inferring $\arg\min f$ by observing the learner’s queries. In the economic model of covert learning and disclosure, the sender can covertly and costlessly acquire information by designing and observing an experiment whose choice and outcome are private. In quantum learning, covert learning is decomposed into strategy-covertness, which hides the learner’s strategy, and target-covertness, which hides the identity of the object or state being learned (Jain et al., 2023, Escudé, 2023, Anand et al., 8 Oct 2025).

“Verifiable” is likewise instantiated in several ways. In Zero-Knowledge Proof-based Verifiable Machine Learning, correctness is established by a prover that produces a proof and a verifier that checks the proof without learning anything else. In secure multiparty collaborative transfer learning, verifiability is realized through SPDZ computation and MAC-based checks. In split learning, zk-SNARK proofs are generated for both sides in forward propagation and backward propagation on the server side. In ledger-attested formal learning, verifier outcomes are committed to an immutable, append-only ledger, and only ledger-attested outcomes can drive subsequent learning (Xing et al., 2023, Ma et al., 2020, Alaa et al., 3 Nov 2025, Abdullah, 22 Dec 2025).

A plausible implication is that covert verifiable learning is best understood not as a single protocol family but as a cross-cutting design objective. The observable object varies—wireless transmissions, stochastic-gradient queries, message sets, proof transcripts, measurement settings, or ledger updates—but the recurrent architectural pattern is a separation between what is hidden from an adversary and what is made checkable to an authorized verifier.

2. Covertness in distributed and federated learning

In federated learning networks, the paper “Securing Federated Learning: A Covert Communication-based Approach” proposes Covert Communication-based Federated Learning (CCFL), whose central idea is to hide the very existence of communication between mobile devices and the federated learning server. The system model includes $N$ devices, a federated learning server at the base station, an adversary (“Willie”), and a friendly jammer with $N$ antennas; devices use orthogonal wireless channels; model updates from mobile devices to the server are transmitted using covert communication methods; and the main focus is on uplink updates. The security constraint is the covert probability condition

$P_{MD} + P_{FA} \ge 1 - \epsilon,$

where $P_{MD}$ is the probability of missed detection and $P_{FA}$ is the probability of false alarm by Willie. The optimization objective is to minimize federated learning latency, subject to covert communication constraints, device and jammer power constraints, the server’s budget constraint, and required local model accuracy. Because the problem is non-convex, the solution uses alternating descent algorithms and successive convex approximation (Xie et al., 2021).

CCFL is explicitly framed as protection against attacks that depend on observing or manipulating communications. Eavesdropping is rendered ineffective if Willie cannot reliably distinguish between noise and legitimate update transmissions; inference attacks that require access to observable gradients or updates are nullified if those transmissions are hidden; external attackers cannot inject false models or synchronize poisoning or backdoor manipulations with legitimate protocol events they cannot detect; and insider devices can still poison. The numerical setting reported in the paper uses 50 devices, a $500\text{m} \times 500\text{m}$ area, and 20 MHz bandwidth, and the reported qualitative result is an explicit trade-off between security and efficiency: stricter security, represented by smaller $\epsilon$, increases latency, while a larger jamming budget can reduce latency only up to a point (Xie et al., 2021).

A distinct notion of covertness appears in “Controlling Federated Learning for Covertness,” which formulates covert or learner-private optimization as a control problem over stochastic gradient descent. The learner dynamically chooses between learning and obfuscation by exploiting stochasticity, and the problem is modeled as a Markov decision process. The optimal policy has a monotone threshold structure: $$u^*(y) = \begin{cases} 1 & y^L \geq \theta(y^O)\ 0 & \text{else}, \end{cases}$$ where $\arg\min f$0 means learn and $\arg\min f$1 means obfuscate. For unknown transitions, a Structured Policy Gradient Algorithm is proposed. In the reported hate speech classification task, an eavesdropper can achieve a validation accuracy of $\arg\min f$2 with no information and $\arg\min f$3 when it has a public dataset with 10\% positive samples under the optimal policy, compared to $\arg\min f$4 when the learner employs a greedy policy, while learner accuracy remains $\arg\min f$5 versus $\arg\min f$6 or $\arg\min f$7 in the greedy baseline (Jain et al., 2023).

These two strands conceal different objects. CCFL conceals whether communication is occurring at all; learner-private optimization conceals which observed query trajectory corresponds to genuine learning. This suggests that “covertness” in learning is a systems-level notion rather than a single cryptographic primitive.

3. Verifiability mechanisms: zero-knowledge proofs, MPC, and attested ledgers

The survey “Zero-Knowledge Proof-based Verifiable Decentralized Machine Learning in Communication Network” formalizes ZKP-VML as a tuple of algorithms. The principal algorithms are

$\arg\min f$8

$\arg\min f$9

$N$0

$N$1

together with

$N$2

and

$N$3

The survey organizes existing schemes by transforming ML into ZKP, improving ZKP efficiency, and handling different ML models; and it identifies modularity, efficiency via specialization, maintaining privacy, and scalability awareness as common design principles (Xing et al., 2023).

“VerifyTL: Secure and Verifiable Collaborative Transfer Learning” realizes verifiable collaborative transfer learning under a covert adversary model using SPDZ secure computation. Its architecture consists of a cross transfer unit for two domains and a weave transfer unit for $N$4 domains. In the multi-domain setting, for each domain $N$5, the new representation is

$N$6

Every secret share is accompanied by a share of a Message Authentication Code value satisfying

$N$7

and MACCheck verifies correctness before outputs are opened. The reported results give 98.2\% on MNIST with $N$8 domains versus 74.6\% for Original, and 97.6\% on FashionMNIST versus 73.6\% for Original; for $N$9 to $N$0, training times are 44.8–117.7 minutes for 10 epochs; and VectorMul for $N$1 domains takes 6.93 ms in the covert security model (Ma et al., 2020).

“Verifiable Split Learning via zk-SNARKs” inserts zk-SNARKs into the split-learning workflow. The client computes forward propagation up to the cut layer, sends smashed data, and generates a proof that activations were honestly computed from local data and current client model parameters. The server then generates a proof for forward and backward propagation on its side. Proof generation and verification are written as

$N$2

The paper states that the zk-SNARK test achieves verifiability and correctness, while blockchains are lightweight but unverifiable (Alaa et al., 3 Nov 2025).

“MathLedger: A Verifiable Learning Substrate with Ledger-Attested Feedback” introduces a different verification substrate. The learning loop is

$N$3

At epoch $N$4, the system commits to two 32-byte roots, $N$5 for canonicalized proof artifacts and $N$6 for UI/interface state, and binds them by

$N$7

The contribution is explicitly infrastructural: Phase I experiments validate measurement infrastructure and fail-closed governance triggers, and no convergence or capability claims are made (Abdullah, 22 Dec 2025).

4. Formal guarantees, adversary models, and auditability

The most explicit generic security vocabulary is given by ZKP-VML. Its core properties are completeness, soundness, zero-knowledge, and auditability. Completeness requires that if the computation is done correctly, the verifier always accepts. Soundness requires that a malicious prover cannot make the verifier accept a false result except with negligible probability. Zero-knowledge requires that the verifier learns nothing but the validity of the statement. Auditability requires that the commitment $N$8 binds the prover to specific data, so the verifier can check that the data used in the proof matches the committed value (Xing et al., 2023).

VerifyTL places these guarantees in a covert adversary model. The security statement is given by the indistinguishability of real and ideal executions: $N$9 The system is stated to be secure against a covert adversary that can compromise up to $P_{MD} + P_{FA} \ge 1 - \epsilon,$0 out of $P_{MD} + P_{FA} \ge 1 - \epsilon,$1 data domains, with privacy and correctness maintained as long as one domain is honest, and MAC-based verification detects manipulations with overwhelming probability (Ma et al., 2020).

MathLedger adds governance-bound auditability. Verifier outcomes are $P_{MD} + P_{FA} \ge 1 - \epsilon,$2, epistemic risk is

$P_{MD} + P_{FA} \ge 1 - \epsilon,$3

and the Reflexive Formal Learning update is

$P_{MD} + P_{FA} \ge 1 - \epsilon,$4

Rejected or failed updates do not disappear silently; they are recorded as negative knowledge, including refuted artifacts, abstentions, and inadmissible updates. Governance triggers such as F5.2 (variance ratio) and F5.3 (windowed drift) block knowledge claims and update only negative-knowledge evidence when variance or drift is out of bounds (Abdullah, 22 Dec 2025).

The game-theoretic paper “Covert learning and disclosure” offers a different formal account of verifiability. A verifiability structure $P_{MD} + P_{FA} \ge 1 - \epsilon,$5 maps signal realizations to available messages, and under the condition that the sender can prove news better than the prior, the equilibrium value is

$P_{MD} + P_{FA} \ge 1 - \epsilon,$6

with

$P_{MD} + P_{FA} \ge 1 - \epsilon,$7

The central equilibrium statement is that the sender never actively deceives; instead, she influences the receiver by choosing to remain selectively ignorant. More verifiable structures help the sender only under the specific lowest-consistent ordering and the PNBP condition; otherwise, “more verifiability” can hurt ex ante if it forces revelation of bad news (Escudé, 2023).

Taken together, these works show that “verifiability” can mean proof soundness, MPC consistency, audit binding, governance traceability, or constrained truthful disclosure. The common thread is not a single proof system, but an enforceable relation between hidden internal state and publicly checkable evidence.

5. Quantum covert verifiable learning

“Covert Quantum Learning: Privately and Verifiably Learning from Quantum Data” extends covert verifiable learning to quantum learning theory and emphasizes that the quantum setting permits information-theoretic constructions in remote data access scenarios without computational hardness assumptions. The paper separates two privacy notions. Strategy-covertness hides the learner’s analysis strategy; target-covertness hides the identity of the object or state being learned. For covert quantum statistical queries, encoder and decoder are required to satisfy completeness, privacy, and efficiency conditions, with perfect privacy $P_{MD} + P_{FA} \ge 1 - \epsilon,$8 or statistical privacy possible. For target-covertness, the learner combines a small number of public examples or public quantum queries with a private weaker oracle such as private quantum statistical queries (Anand et al., 8 Oct 2025).

The paper also introduces covert verifiable quantum data acquisition. In this setting, completeness requires that if the public quantum oracle is honest, the learner accepts and produces a state $P_{MD} + P_{FA} \ge 1 - \epsilon,$9 with high overlap; soundness requires rejection when tampering occurs unless fidelity remains high; and privacy is given either by a strong factorization condition or by a cheat-sensitive guarantee under which information extraction causes rejection with high probability. The protocols use classical shadow overlap certification and private classical queries to test whether received quantum data are as promised. Against unidirectional adversaries, private randomness or entanglement is used to mask requests; against bidirectional ancilla-free adversaries, masking via maximally entangled states ensures that attempts to measure for information break entanglement and reduce fidelity to at most $P_{MD}$0, which is then efficiently detected (Anand et al., 8 Oct 2025).

The reported learning tasks include strategy-covert algorithms for making quantum statistical queries via classical shadows; target-covert algorithms for learning quadratic functions from public quantum examples and private quantum statistical queries; target-covert protocols for Pauli shadow tomography and stabilizer state learning; and covert verifiable solutions for Forrelation and Simon’s problem. The paper states that the exponential separation between classical and quantum queries for Forrelation and Simon’s problem survives under covertness constraints, and for quadratic function learning the adversary’s guessing probability of the target is exponentially small, specifically $P_{MD}$1 for diagonal entries in the described construction (Anand et al., 8 Oct 2025).

This body of results makes a specific technical point: in quantum settings, privacy and verifiability are not merely compatible but can be co-designed through information–disturbance relations. At the same time, the paper states a limit: full information-theoretic covertness is impossible in some quantum scenarios, so the strongest guarantees are obtained only under physically reasonable adversary constraints such as unidirectional, ancilla-free, or i.i.d. eavesdroppers.

6. Trade-offs, limits, and research directions

A recurrent theme is that covert verifiable learning is governed by trade-offs rather than monotone improvements. In CCFL, tighter security, represented by smaller $P_{MD}$2, increases federated learning latency; increasing the server’s jamming power budget helps only up to a point; and covert communication blocks external poisoning but not insider poisoning. In learner-private optimization, learning and obfuscation are explicitly balanced by policy design: the learner can preserve its own performance while reducing eavesdropper effectiveness, but only by scheduling a substantial fraction of actions for obfuscation rather than direct learning (Xie et al., 2021, Jain et al., 2023).

Verifiability also has nontrivial costs. In split learning, zk-SNARK-based verifiability introduces significant computational cost at proof generation, especially as the number of clients and circuit complexity increase, whereas blockchain-only recording has much lower overhead but does not verify correctness. The paper’s conclusion is precise: applying the zk-SNARK test achieves verifiability and correctness, while blockchains are lightweight but unverifiable (Alaa et al., 3 Nov 2025).

The disclosure model adds a conceptual caution. More verifiability does not uniformly strengthen the informed party. Under PNBP, increasing the set of lowest-consistent types weakly benefits the sender, but outside that condition, additional verifiability can worsen ex ante outcomes because the receiver’s interpretation of absent evidence becomes more skeptical. The equilibrium outcome is determined not by successful lying but by what information the sender chooses to acquire (Escudé, 2023).

The survey on ZKP-VML identifies open challenges in generalizability, efficiency and scalability, privacy and property integration, real-world deployment, and standardization. The stated future directions include broader compiler support for general ML programs, specialized proof systems customized to ML, hardware acceleration with FPGAs, ASICs, and GPUs, integration with DP, HE, and TEE, support for federated and decentralized variants, and common frameworks, APIs, and benchmarks (Xing et al., 2023).

MathLedger frames a related agenda around infrastructure rather than capability. Its Phase I result is a working prototype of ledger-attested learning that enables auditability at scale, validates measurement infrastructure such as $P_{MD}$3 computation and variance tracking, and confirms fail-closed governance triggers under out-of-bounds conditions, while explicitly making no convergence or capability claims. A plausible implication is that future covert verifiable learning systems may need not only privacy-preserving proof systems, but also governance substrates in which negative evidence, blocked updates, and abstentions are first-class audit objects (Abdullah, 22 Dec 2025).

Across these works, the field’s central tension is stable: hiding observations that empower an adversary tends to reduce immediate efficiency, observability, or interoperability, whereas strengthening verifiability tends to increase circuit complexity, proof cost, or disclosure pressure. Covert verifiable learning therefore emerges as the study of how much can be concealed, to whom, and under which formal mechanism of correctness or accountability.