SAC-Copula: Quality-Preserving Watermarking for Diffusion Language Models via Smooth Correlated Gumbel Fields

Watermarking methods for language models have been studied extensively in the autoregressive setting, where tokens are generated sequentially. These works largely focus on local-context schemes that perturb the next token's distribution as a function of its preceding tokens. In diffusion language models, distributions over many unresolved positions are jointly sampled, allowing additive statistics of the entire sequence to be tractable during generation. We propose a watermark for masked diffusion language models that controls a global, vector-valued sketch representation of the text. Compared to context-dependent watermarking, the sketch formulation decouples detection from the local contexts seen during generation, resulting in an order-agnostic statistic and a watermarking rule which does not manifest as a simple token bias. We analyze the distortion, soundness, and robustness properties of the method.

Statistical watermarking is a common approach for verifying whether text was written by a language model. Most existing schemes assume autoregressive generation, where tokens are produced left to right and contextual hashing is well defined. Diffusion language models generate text by denoising tokens in arbitrary order, so these schemes cannot be applied directly. A recent watermark by Gloaguen et al. addresses this gap for LLaDA 8B Instruct and reports true positive detection above 99%. This paper studies what happens when watermarked text is rewritten not once but several times. Using the same watermark configuration, 1,605 watermarked completions of about 300 tokens each are produced across five WaterBench domains. Each completion is rewritten by four open weight language models, from 1.5B to 8B parameters, none of which know the watermark key. Five rewrite styles are tested: paraphrase, humanize, simplify, academic, and summarize expand. Each style is chained for up to five hops, producing 160,500 rewritten texts in total. The watermark is detected on 87.9% of the original outputs at the standard significance threshold. After a single rewrite, detection falls to between 14% and 41% depending on the rewriter and style. After five chained rewrites, detection falls to 4.86%, meaning 94.76% of the originally detected texts are no longer flagged. After three rewrites, the detector score has dropped 86% of the way from its watermarked baseline toward the null distribution. Repeated rewriting is therefore a much stronger attack than a single rewrite, and the result holds across all four rewriters tested.

Watermarking should identify language-model output without degrading quality or limiting verification to the model provider. Multilingual deployment makes this harder because morphology, segmentation, and script change where watermark evidence can enter naturally. We introduce LUNA, a linguistically adaptive watermark that combines model-free detection with single-token non-distortion under the standard random-key model. LUNA estimates normalized next-tag entropy from part-of-speech contexts in an external corpus and uses it to set the depth of a non-distortionary binary tournament sampler; the detector reconstructs the same schedule from text, a tokenizer, a tagger, and a secret key. We evaluate six typologically diverse languages and two domains against eight primary baselines. LUNA attains an AUROC of 0.9959 and the lowest mean absolute median perplexity shift of 0.045 across the twelve settings; its 95% bootstrap interval [0.022, 0.073] lies below all baseline intervals. LUNA also records the lowest mean Self-BLEU, Distinct-1, surprisal, and entropy shifts. It is the only method that simultaneously achieves AUROC > 0.99 and an absolute median perplexity shift below 0.1 in a majority of settings, reaching this regime in 9 of the 12 settings while no baseline reaches it in more than 2. Our code is available at: https://github.com/Shinwoo-Park/luna_watermark