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Watermarking for diffusion language models can be embedded by steering the order in which tokens are unmasked, not by biasing token probabilities.

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2026-08-03 06:17 UTC pith:H2BKSILD

load-bearing objection Decoding-order watermarking for dLLMs is a real new idea, and the experiments are broad; but the reported FPR=0.000 rests on a token-ID parity hash that does not justify the theoretical null. the 3 major comments →

arxiv 2601.22985 v2 pith:H2BKSILD submitted 2026-01-30 cs.LG

dgMARK: Decoding-Guided Watermarking for Diffusion Language Models

Pyo Min Hong , Albert No This is my paper
classification cs.LG
keywords watermarkingdiffusion language modelsdecoding ordermasked diffusionparity hashtext provenancesliding-window detectiondiscrete diffusion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Discrete diffusion language models generate text by iteratively unmasking tokens, and although an ideal model would be indifferent to the unmasking order, practical dLLMs are sensitive to it. dgMARK exploits this sensitivity to embed a watermark: instead of reweighting token probabilities like prior schemes, it guides which position is revealed next, prioritizing positions whose candidate token satisfies a parity constraint tied to the position index. This raises the fraction of parity-matching tokens above the chance level of 0.5, providing a statistical signal that can be detected with a secret key and without access to model internals. The paper argues that this yields strong detectability with smaller text-quality degradation than probability-biasing baselines, and that a sliding-window detector makes it robust to post-editing. If correct, it establishes decoding order itself as a viable watermark channel for dLLMs.

Core claim

The paper claims that the decoding order of a dLLM can serve as a watermark channel. In dgMARK, at each step the decoder restricts attention to unrevealed positions whose sampled candidate token falls in the parity-matching set G_j = {v : f(v, ξ) ≡ j mod 2}, where f is a balanced binary hash keyed by a secret ξ, and selects the highest-reward position among those, falling back to the full set when no match exists. This simple rule systematically elevates the parity-matching rate of generated tokens, so detection reduces to a z-test on whether the observed rate significantly exceeds 0.5. The paper also introduces a one-step lookahead beam variant that preserves future parity-alignment opportu

What carries the argument

The central mechanism is the binary parity hash f: V × Ξ → {0,1} that partitions the vocabulary into parity-matching sets G_i, combined with a decoding algorithm that prioritizes positions whose candidate token lies in G_i. The hash is the carrier of the watermark signal: its balance gives the binomial null (Parity matches ~ Binomial(n, 1/2)), and its secrecy prevents easy spoofing. The one-step lookahead beam search computes, for each top candidate, how many future positions would remain parity-matching after committing, making the embedding stronger while adding computation.

Load-bearing premise

The scheme bets that real dLLMs are sensitive enough to the unmasking order that steering which position is revealed next can shift token statistics without visibly harming text quality, and that the parity hash is balanced and secret enough for the binomial null to hold.

What would settle it

Run dgMARK on a dLLM trained to be fully order-invariant (e.g., by averaging over all unmasking permutations during training); if the parity-matching rate no longer rises above 0.5, the watermark channel disappears. Alternatively, generate text with the token-ID modulo 2 hash but without a secret key and compute the empirical z-score distribution on many sequences: if it deviates from a standard normal, the detection null is mis-specified and the false-positive claims do not hold as stated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Decoding order is a practical watermark channel for dLLMs, complementary to probability-biasing methods, and can be combined with common decoding strategies such as confidence, entropy, and margin-based ordering.
  • Watermark detection only needs the generated text and the secret key, not the model's internals or the prompt, making it suitable for provenance verification in deployed systems.
  • The one-step lookahead variant provides a tunable trade-off between detectability, text quality, and computational cost, with beam sizes of 3-8 yielding near-perfect detection at low false-positive rates.
  • The sliding-window detector extends the watermark's robustness to post-editing, including random insertions, deletions, substitutions, and paraphrasing, so it can survive common text transformations.
  • Longer generations (around 200 tokens or more) are particularly amenable to dgMARK, suggesting practical use in summaries, reports, and stories, while very short outputs would need aggregation or a less stringent detection threshold.

Where Pith is reading between the lines

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

  • Because dgMARK's signal relies on the gap between ideal order-invariance and practical order-sensitivity, its reliability may weaken as dLLMs become better trained and more order-invariant; the method's long-term viability depends on whether that sensitivity persists in future models.
  • The experiments instantiate the hash as token-ID modulo 2, which is not secret and may correlate with token frequencies, so the paper's detection null (parity matches ~ Binomial(n, 1/2)) is a theoretical ideal rather than a verified property of the actual test; using a true cryptographic PRF would close this gap but was not empirically tested.
  • dgMARK could likely be extended to carry multi-bit payloads (e.g., by using more than one parity class or by keying the hash to the prompt), enabling richer provenance metadata than a single binary watermark bit.
  • An attacker who reorders or rephrases text to restore parity balance might evade the sliding-window detector; since the watermark lives in token positions, adversarial re-sorting attacks are a natural threat model worth testing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. dgMARK is a watermarking method for discrete diffusion language models that embeds a signal by steering the order in which tokens are unmasked, rather than by reweighting token probabilities. A binary hash f(v, ξ) assigns a parity to each vocabulary token; at each decoding step the decoder restricts attention to positions whose sampled candidate token has parity matching the position index, selects the highest-reward such position (falling back to any position if none matches), and reveals that token. Detection counts parity matches and applies a z-test (Eq. 1), with a sliding-window variant for robustness to post-editing. The paper evaluates on LLaDA-8B/1.5/2.0 and Dream-7B over C4 and Writing Prompts, compares to KGW and PATTERN-MARK, and reports high detectability, modest perplexity increase, robustness to token edits and paraphrasing, and computational overhead. The experiments instantiate the hash as token-ID modulo 2 (Appendix D.1), although the design allows keyed PRFs.

Significance. The paper identifies a genuinely new watermark channel for dLLMs — the unmasking order — and provides broad empirical evidence: multiple model families, datasets, sampling strategies, quality benchmarks, robustness attacks, and overhead measurements. The beam-search lookahead variant and the attempt to avoid probability reweighting are useful contributions. The significance is conditional on resolving a calibration gap in the detection statistic: the experiments use a public, non-secret parity map, whereas the detection null and the security argument require a balanced, secret keyed hash. If the authors re-run with a keyed PRF and validate human-text false-positive rates, the contribution would be solid and timely.

major comments (3)
  1. [§3.5 (Eq. 1), Appendix D.1] Eq. (1) derives the Binomial(n,1/2) null from a balanced and secret f(·,ξ). Appendix D.1 states that all experiments use f(v)=token_ID mod 2, which is public and not necessarily balanced over the natural-token distribution. Any correlation between token-ID parity and token frequency shifts the null mean, so the FPR=0.000/TNR=1.000 values in Tables 1 and 2 are calibrated only on non-watermarked model outputs under this non-secret hash. They do not establish the behavior on human-authored text, and they do not transfer to the keyed-PRF scheme the paper says it supports. This is load-bearing for the provenance claim; please report parity balance of the tokenizer, human-text FPR, and preferably re-run detection with a keyed PRF.
  2. [§3.5 (robust detection)] The sliding-window statistic zwin=(1/S)Σ z_s^2 is introduced without a null distribution or a threshold calibration. Figures 2–3 report ROC/AUC, but no fixed-FPR operating points are given under insertion, deletion, substitution, or paraphrasing. Because absolute-position parity is disrupted by insertions/deletions, the two-sided aggregation is heuristic, and it is not clear what false-positive rate corresponds to the advertised robustness. Please provide the null/calibration for zwin or report TPR/FPR at the chosen operating point for each attack budget.
  3. [§3.3, Appendix D.1] The security claim is unverified: the paper argues that a keyed PRF would make the watermark resilient to adversarial reverse-engineering, but no experiments or analysis consider an adversary who knows the token-ID parity map used in the evaluation. Since embedding and detection use exactly the same public statistic, an adversary can forge or strip the watermark. Please state the threat model explicitly (secret vs. public f) and provide at least one adversarial evaluation for the PRF version, e.g., best-effort parity flipping or reordering.
minor comments (5)
  1. [Abstract] First sentence in the main text reads "We proposedgMARK"; should be "We propose dgMARK".
  2. [Eq. (1)] The denominator appears as "p n/4"; it should be sqrt(n/4).
  3. [Tables 1–2] Please clarify in the captions that FPR/TNR are computed on non-watermarked dLLM outputs at z=4.0, not on human-authored text.
  4. [Tables 9–10] Some cells appear internally inconsistent; e.g., Table 9 with +5-beam at z=4.0 lists TPR=0.987 and FNR=0.127, which do not sum to 1. Please double-check all error-rate columns.
  5. [Figure 15] The stacked-bar labels are difficult to read; adding a legend or text annotations would improve clarity.

Circularity Check

0 steps flagged

No significant circularity: the watermark embedder and detector are a designed matched pair, and the paper's empirical evaluations are independent.

full rationale

dgMARK is a watermarking scheme, so its embedder and detector are intentionally matched: Algorithm 2 prioritizes positions whose sampled candidate satisfies a parity condition, and Section 3.5 detects watermarks by counting parity matches. This is a designed signal channel, not a derivation of a prediction from the same input. The paper's actual claims—detectability on LLaDA/Dream, PPL and benchmark quality, and robustness to editing/paraphrasing—are tested against non-watermarked model outputs and external metrics, so the evidence is independent of the construction. The Eq. (1) null relies on a balanced secret hash; the Appendix's use of public token-ID parity is an external-validity limitation (the null may be miscalibrated for human text or adversarially knowable), not a circular reduction. There are no fitted parameters masquerading as predictions, and no load-bearing self-citations. Score 0.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 0 invented entities

The central claim rests on the order-sensitivity of real dLLMs, on the balanced-hash null hypothesis, and on the choice of a concrete hash for experiments. The paper introduces no new physical or mathematical entities; its free parameters are operating-point and implementation choices rather than fitted physical constants.

free parameters (4)
  • Hash mapping f = token_id mod 2
    Experiments use a deterministic non-keyed mapping instead of the keyed PRF described in the framework. This choice determines the null distribution and any residual parity bias.
  • Detection threshold z = 4.0
    The main tables use z=4.0 as the one-sided test threshold; 'negligible error rates' is with respect to this chosen operating point, though TPR@FPR curves are also given.
  • Sliding-window size w = 8
    Robustness results use window size 8; the appendix shows AUC varies modestly with w ∈ {8,16,32}.
  • Beam size k = 1,3,5,8
    The lookahead strength is a user-controlled hyperparameter; larger k improves detectability at higher compute cost.
axioms (3)
  • domain assumption Practical dLLMs are sufficiently order-sensitive that changing the unmasking order changes generated-text statistics.
    Section 3.1 relies on the gap between ideal order-invariance and practical order-sensitivity to create the watermark channel; this is cited from Kim et al. 2025 and is not proven in this paper.
  • domain assumption A balanced hash f implies E[m_i] ≈ 1/2 for non-watermarked text.
    Section 3.5 uses this for the Binomial null. It holds for a keyed random hash over text-key pairs, but not for the token-ID-mod-2 implementation in D.1, where token frequency and ID parity can correlate.
  • domain assumption The two-sided sliding-window statistic z_win has a stable null distribution that needs no calibration.
    Section 3.5 defines z_win as an average of squared window z-scores but gives no null distribution or calibration procedure; empirical FPR is measured only on same-model non-watermarked outputs.

pith-pipeline@v1.3.0-alltime-deepseek · 26993 in / 19001 out tokens · 208172 ms · 2026-08-03T06:17:05.961053+00:00 · methodology

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read the original abstract

We propose dgMARK, a decoding-guided watermarking method for discrete diffusion language models (dLLMs). Unlike autoregressive models, dLLMs can generate tokens in arbitrary order. While an ideal conditional predictor would be invariant to this order, practical dLLMs exhibit strong sensitivity to the unmasking order, creating a new channel for watermarking. dgMARK steers the unmasking order toward positions whose high-reward candidate tokens satisfy a simple parity constraint induced by a binary hash, without explicitly reweighting the model's learned probabilities. The method is plug-and-play with common decoding strategies (e.g., confidence, entropy, and margin-based ordering) and can be strengthened with a one-step lookahead variant. Watermarks are detected via elevated parity-matching statistics, and a sliding-window detector ensures robustness under post-editing operations including insertion, deletion, substitution, and paraphrasing. Project website: https://dgmark-watermarking.github.io

Figures

Figures reproduced from arXiv: 2601.22985 by Albert No, Pyo Min Hong.

Figure 1
Figure 1. Figure 1: Overview. (Left) Existing autoregressive watermarking methods generate green/red token sets by hashing the preceding context and embed watermark signals by biasing the sampling distribution toward green tokens. (Middle) In contrast, decoding in dLLMs does not follow the traditional left-to-right generation process; instead, the model selects high-reward tokens at each position even in the absence of prior … view at source ↗
Figure 2
Figure 2. Figure 2: ROC curves under post-editing attacks. Illustration of the sliding-window strategy against random deletion, insertion, and substitution with modification budget ϵ. Watermarks are generated by standard dgMARK (k = 1) using multinomial sampling. 0.0 0.1 0.2 0.3 0.4 Lexical Modification Ratio 0.5 0.6 0.7 0.8 0.9 1.0 AUC DIPPER-1 dgMARK 3-beam 5-beam 8-beam KGW PM 0.0 0.1 0.2 0.3 0.4 Lexical Modification Ratio… view at source ↗
Figure 3
Figure 3. Figure 3: Detection AUC under paraphrasing attacks. Results for dgMARK with DIPPER (Krishna et al., 2023): (Left) paraphrasing at predefined ratios via lexical modification; (Middle) paraphrasing with ratio-adjusted lexical modification and an additional 10% order diversity. Comparative results with KGW and PATTERN-MARK are included to assess relative robustness. (Right) paraphrasing generated by Llama 3-8B (Instruc… view at source ↗
Figure 4
Figure 4. Figure 4: Watermark detectability vs. sequence length. Results under multinomial sampling with beam sizes k ∈ {1, 3, 5, 8}, reported as TPR at FPR levels of 10%, 1%, 0.1%, and 0.01%. Generation lengths are set to {16, 32, 64, 128, 256}, where the 256 setting includes sequences with length ≥ 200. of 256 tokens using block sizes {8, 16, 32, 64, 128} and report TPR@FPR and perplexity in [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the distribution of parity alignment. At window size w = 32, comparison of (1) non-watermarked texts (Non WM), (2) intact watermarked texts (WM), and (3) watermarked texts (WM) with “random token insertions”, where the number of inserted tokens increases from left to right. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Non WM WM (intact) WM (deletion) 0.25 0.00 0.25 0.50 0.75 1.00 1.25 Non WM WM (intact) WM … view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the distribution of parity alignment. At window size w = 32, comparison of (1) non-watermarked texts (Non WM), (2) intact watermarked texts (WM), and (3) watermarked texts (WM) with “random token deletion”, where the number of deleted tokens increases from left to right. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the distribution of parity alignment. At window size w = 32, comparison of (1) non-watermarked texts (Non WM), (2) intact watermarked texts (WM), and (3) watermarked texts (WM) with “random token substitution”, where the number of substituted tokens increases from left to right. 0.0 0.2 0.4 0.6 0.8 1.0 False Positive Rate 0.0 0.2 0.4 0.6 0.8 1.0 True Positive Rate =0.0 (AUC = 0.976) =0.1 (A… view at source ↗
Figure 8
Figure 8. Figure 8: ROC curves under post-editing attacks. Illustration of the sliding-window strategy against (A) random deletion, (B) insertion, and (C) substitution with modification budget ϵ. The comparison includes (1) KGW, (2) PATTERN-MARK, and (3) dgMARK with 3-beam search. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of text perplexity using the “entropy strategy”: (1) Non-watermarked texts (Non WM) and (2) Watermarked texts generated by dgMARK with beam sizes k ∈ {1, 3, 5, 8}. Lighter green represents k = 1 and darker green represents k = 8 0.0 0.2 0.4 0.6 0.8 1.0 False Positive Rate 0.0 0.2 0.4 0.6 0.8 1.0 True Positive Rate Window size = 8 dgMARK(intact) (AUC = 1.00) 1-beam (AUC = 0.80) 3-beam (AUC = 0.88… view at source ↗
Figure 10
Figure 10. Figure 10: ROC curves under the “DIPPER-1” setting. Illustration of the sliding-window strategy for detection performance against paraphrasing attacks, evaluated at window sizes w ∈ {8, 16, 32}. 0.0 0.2 0.4 0.6 0.8 1.0 False Positive Rate 0.0 0.2 0.4 0.6 0.8 1.0 True Positive Rate Window size = 8 dgMARK(intact) (AUC = 1.00) 1-beam (AUC = 0.73) 3-beam (AUC = 0.83) 5-beam (AUC = 0.89) 8-beam (AUC = 0.90) 0.0 0.2 0.4 0… view at source ↗
Figure 11
Figure 11. Figure 11: ROC curves under the “DIPPER-2” setting. Illustration of the sliding-window strategy for detection performance against paraphrasing attacks, evaluated at window sizes w ∈ {8, 16, 32}. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: ROC curves under post-editing attacks. Illustration of the sliding-window strategy against “random token insertion” attacks with modification budget ϵ, when texts are generated with beam sizes {3, 5, 8}. 0.0 0.2 0.4 0.6 0.8 1.0 False Positive Rate 0.0 0.2 0.4 0.6 0.8 1.0 True Positive Rate 3-Beam Search =0.0 (AUC = 1.000) =0.1 (AUC = 1.000) =0.2 (AUC = 0.993) =0.3 (AUC = 0.955) =0.4 (AUC = 0.899) 0.0 0.2 … view at source ↗
Figure 13
Figure 13. Figure 13: ROC curves under post-editing attacks. Illustration of the sliding-window strategy against “random token deletion” attacks with modification budget ϵ, when texts are generated with beam sizes {3, 5, 8}. 0.0 0.2 0.4 0.6 0.8 1.0 False Positive Rate 0.0 0.2 0.4 0.6 0.8 1.0 True Positive Rate 3-Beam Search =0.0 (AUC = 1.000) =0.1 (AUC = 1.000) =0.2 (AUC = 1.000) =0.3 (AUC = 0.998) =0.4 (AUC = 0.970) 0.0 0.2 0… view at source ↗
Figure 14
Figure 14. Figure 14: ROC curves under post-editing attacks. Illustration of the sliding-window strategy against “random token substitution” attacks with modification budget ϵ, when texts are generated with beam sizes {3, 5, 8}. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Distribution of generated sequence lengths. Text sequences are produced using the block-wise generation strategy with multinomial and greedy decoding under block sizes {8, 16, 32, 64, 128}. Prompt The most saddest love story you can ever think of Non-watermarked Watermarked (dgMARK) Parity-Matching Ratio = 0.55 ... Parity-Matching Ratio = 0.74 ... In a small village, there lived a young woman [PITH_FULL_… view at source ↗
Figure 16
Figure 16. Figure 16: Non-watermarked vs. watermarked text. Generated by LLaDA-1.5 on the Writing Prompts dataset. Tokens highlighted in green indicate parity matches. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗

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