# WaterMod: Modular Token-Rank Partitioning for Probability-Balanced LLM Watermarking

arXiv: https://arxiv.org/abs/2511.07863 · HTML: https://mlsanity.com/paper/2511.07863/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Watermarking generated content and models, one of 12 directions in LLM safety and security.
- Builds on 19 of its 35 references that are in the map.
- First posted on arXiv 11 Nov 2025.

## Placed in

- LLM safety and security / [Watermarking generated content and models](https://mlsanity.com/safety/watermarking-generated-content-and-models/)

## Builds on (most cited first, 19 of 19)

- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Is Your Code Generated by ChatGPT Really Correct? Rigorous Evaluation of Large Language Models for Code Generation](https://mlsanity.com/paper/2305.01210/) (2023)
- [Robust Distortion-free Watermarks for Language Models](https://mlsanity.com/paper/2307.15593/) (2023)
- [Who Wrote this Code? Watermarking for Code Generation](https://mlsanity.com/paper/2305.15060/) (2023)
- [Watermarking Text Generated by Black-Box Language Models](https://mlsanity.com/paper/2305.08883/) (2023)
- [WaterMax: breaking the LLM watermark detectability-robustness-quality trade-off](https://mlsanity.com/paper/2403.04808/) (2024)
- [k-SemStamp: A Clustering-Based Semantic Watermark for Detection of Machine-Generated Text](https://mlsanity.com/paper/2402.11399/) (2024)
- [Watermarking Conditional Text Generation for AI Detection: Unveiling Challenges and a Semantic-Aware Watermark Remedy](https://mlsanity.com/paper/2307.13808/) (2023)
- [No Free Lunch in LLM Watermarking: Trade-offs in Watermarking Design Choices](https://mlsanity.com/paper/2402.16187/) (2024)
- [CoProtector: Protect Open-Source Code against Unauthorized Training Usage with Data Poisoning](https://mlsanity.com/paper/2110.12925/) (2021)
- [CodeMark: Imperceptible Watermarking for Code Datasets against Neural Code Completion Models](https://mlsanity.com/paper/2308.14401/) (2023)
- [Fine-tuning Large Language Models for Improving Factuality in Legal Question Answering](https://mlsanity.com/paper/2501.06521/) (2025)
- [WatME: Towards Lossless Watermarking Through Lexical Redundancy](https://mlsanity.com/paper/2311.09832/) (2023)
- [CodeIP: A Grammar-Guided Multi-Bit Watermark for Large Language Models of Code](https://mlsanity.com/paper/2404.15639/) (2024)
- [DeepTextMark: A Deep Learning-Driven Text Watermarking Approach for Identifying Large Language Model Generated Text](https://mlsanity.com/paper/2305.05773/) (2023)
- [Can Watermarking Large Language Models Prevent Copyrighted Text Generation and Hide Training Data?](https://mlsanity.com/paper/2407.17417/) (2024)
- [Edit Distance Robust Watermarks via Indexing Pseudorandom Codes](https://mlsanity.com/paper/2406.02633/) (2024)
- [Inevitable Trade-off between Watermark Strength and Speculative Sampling Efficiency for Language Models](https://mlsanity.com/paper/2410.20418/) (2024)

## Built on by (most cited first, 6 of 6)

- [Sequential Behavioral Watermarking for LLM Agents](https://mlsanity.com/paper/2605.11036/) (2026)
- [SSG: Logit-Balanced Vocabulary Partitioning for LLM Watermarking](https://mlsanity.com/paper/2604.22438/) (2026)
- [Provable Model Provenance Set for Large Language Models](https://mlsanity.com/paper/2602.00772/) (2026)
- [Watermarkable Multi-Draft Speculative Sampling via Poisson Processes](https://mlsanity.com/paper/2609.21858/) (2026)
- [Embedding Information in Policy Randomness: Distribution-Preserving Implicit Communication in MDPs](https://mlsanity.com/paper/2610.15976/) (2026)
- [Pin It on Anyone: Certified Targeted Displacement of Multi-Bit LLM Watermarks Without the Key](https://mlsanity.com/paper/2610.18359/) (2026)

## Abstract (arXiv)

Large language models now draft news, legal analyses, and software code with human-level fluency. At the same time, regulations such as the EU AI Act mandate that each synthetic passage carry an imperceptible, machine-verifiable mark for provenance. Conventional logit-based watermarks satisfy this requirement by selecting a pseudorandom green vocabulary at every decoding step and boosting its logits, yet the random split can exclude the highest-probability token and thus erode fluency. WaterMod mitigates this limitation through a probability-aware modular rule. The vocabulary is first sorted in descending model probability; the resulting ranks are then partitioned by the residue rank mod k, which distributes adjacent-and therefore semantically similar-tokens across different classes. A fixed bias of small magnitude is applied to one selected class. In the zero-bit setting (k=2), an entropy-adaptive gate selects either the even or the odd parity as the green list. Because the top two ranks fall into different parities, this choice embeds a detectable signal while guaranteeing that at least one high-probability token remains available for sampling. In the multi-bit regime (k>2), the current payload digit d selects the color class whose ranks satisfy rank mod k = d. Biasing the logits of that class embeds exactly one base-k digit per decoding step, thereby enabling fine-grained provenance tracing. The same modular arithmetic therefore supports both binary attribution and rich payloads. Experimental results demonstrate that WaterMod consistently attains strong watermark detection performance while maintaining generation quality in both zero-bit and multi-bit settings. This robustness holds across a range of tasks, including natural language generation, mathematical reasoning, and code synthesis. Our code and data are available at https://github.com/Shinwoo-Park/WaterMod.