CertMark: Distortion-Free Multi-Bit Watermarking with Certified Decoding
Organizations: Heinrich Heine University Düsseldorf · Jagiellonian University · IDEAS Research Institute
Abstract
Leading multi-bit watermarking methods for language models encode messages by biasing the model's next-token probabilities, creating a trade-off between message recovery and text quality. Their decoders typically return the highest-scoring candidate from accumulated token-level evidence, without a certified abstention rule that bounds the probability of outputting an incorrect message. We introduce CertMark, a distribution-preserving multi-bit watermark with certified decoding. Rather than modifying probabilities, CertMark uses the embedded message to seed an exact Gumbel-max sampler, thereby preserving the model's original sampling distribution. We propose two scalable decoders: a model-agnostic, text-only decoder and a model-aware variant that leverages the original next-token distributions for stronger recovery. Both support certified abstention with mathematical bounds on the probability of returning an incorrect message. Across text completion, summarization, and story generation, CertMark matches the perplexity of unwatermarked text while reliably recovering multi-bit messages. The model-aware decoder further achieves higher bit accuracy than probability-biasing baselines. Our code is publicly available at https://github.com/Batorskq/CertMark.
Figures & tables
| Model and text | Encoder | ||
|---|---|---|---|
| vocabulary | context length in tokens | ||
| prompt | context of step | ||
| , , | text, its -th token, its prefix | chunk carried at step | |
| emitted text as a random variable | contexts used so far | ||
| deployed sampler’s law at step | argument tuple of step | ||
| Key and randomness | uniform attached to token at step | ||
| Method | Avg. | Distortion (avg. over ) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| BA | PPL | BA | PPL | BA | PPL | BA | PPL | BA | PPL | Top-1 | Top-5 | R-1 | R-L | |
| Text Completion | ||||||||||||||
| CycleShift | 100.00 | 7.97 | 100.00 | 7.58 | 100.00 | 7.05 | 100.00 | 7.44 | 100.00 | 7.51 | 55.02 | 84.96 | 0.290 | 0.170 |
| DepthW | 95.75 | 7.40 | 100.00 | 6.96 | 100.00 | 6.52 | 97.25 | 6.68 | 98.25 | 6.89 | 57.39 | 85.87 | 0.290 | 0.168 |
| StealthInk | 89.75 | 6.15 | 90.50 | 5.79 | 95.25 | 5.73 | 94.25 | 5.61 | 92.44 | 5.82 | 62.81 | 89.16 | 0.308 | 0.179 |
| MPAC | 97.25 | 7.70 | 98.00 | 7.52 | 98.75 | 7.03 | 99.00 | 7.09 | 98.25 | 7.33 | 55.60 | 85.34 | 0.293 | 0.170 |
| Qwen3.5-4B | Llama-3.1-8B | |||||
| Method | BA | PPL | BLEU | BA | PPL | BLEU |
| RSBH | 59.38 | 2.51 | 20.64 | 57.62 | 3.49 | 24.84 |
| MPAC | 70.00 | 2.58 | 20.98 | 64.62 | 3.50 | 26.80 |
| XMark | 74.38 | 2.49 | 20.32 | 71.25 | 3.46 | 25.99 |
| StealthInk | 64.88 | 2.22 | 23.00 | 55.75 | 3.19 | 27.08 |
| CertMark model-agnostic 16-bit | 61.50 | 2.18 | 23.15 | 48.88 | 3.08 | 26.34 |
| Qwen3.5-4B | Llama-3.1-8B | |||||
| Method | BA | PPL | BLEU | BA | PPL | BLEU |
| RSBH | 59.38 | 2.51 | 20.64 | 57.62 | 3.49 | 24.84 |
| MPAC | 70.00 | 2.58 | 20.98 | 64.62 | 3.50 | 26.80 |
| XMark | 74.38 | 2.49 | 20.32 | 71.25 | 3.46 | 25.99 |
| StealthInk | 64.88 | 2.22 | 23.00 | 55.75 | 3.19 | 27.08 |
| CertMark model-agnostic 16-bit | 61.50 | 2.18 | 23.15 | 48.88 | 3.08 | 26.34 |
| Qwen3.5-4B | Llama-3.1-8B | |||||
| Method | BA | PPL | BSc. | BA | PPL | BSc. |
| RSBH | 85.03 | 5.38 | .8208 | 89.25 | 4.63 | .8341 |
| MPAC | 87.28 | 5.46 | .8170 | 88.38 | 4.94 | .8279 |
| XMark | 93.38 | 5.48 | .8199 | 96.00 | 4.74 | .8351 |
| StealthInk | 79.81 | 4.52 | .8222 | 80.69 | 3.75 | .8392 |
| CertMark model-agnostic 16-bit | 99.75 | 4.29 | .8262 | 99.44 | 3.66 | .8365 |
| CertMark model-agnostic | CertMark model-aware | |||||||||||||
| Qwen3.5-4B | ||||||||||||||
| cert. error ( ) | .0000 | .0000 | .0063 | .0221 | .0305 | .0537 | .1168 | .0000 | .0011 | .0074 | .0147 | .0168 | .0274 | .0474 |
| null cert. rate ( ) | .0021 | .0063 | .0137 | .0621 | .1200 | .2147 | .4074 | .0000 | .0000 | .0074 | .0442 | .1000 | .1895 | .3863 |
| abst., completion | .178 | .129 | .082 | .056 | .044 | .027 | .013 | .040 | .022 | .009 | .004 | .004 | .002 | .000 |
| abst., summarization | .902 | .840 | .727 | .627 | .582 | .498 | .324 | .604 | .509 | .387 | .251 | .207 | .158 | .069 |
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
| Qwen3.5-4B | Llama-3.1-8B | |||
| Method | Gen. | Dec. | Gen. | Dec. |
| CycleShift | 19.6 | .15 | 15.0 | .07 |
| DepthW | 19.0 | 11.56 | 14.9 | 12.03 |
| StealthInk | 20.0 | .18 | 15.3 | .10 |
| MPAC | 18.9 | .03 | 14.8 | .03 |
| RSBH | 19.7 | 2.18 | 15.1 | 1.18 |
| Method | Avg. | Distortion (avg. over ) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| BA | PPL | BA | PPL | BA | PPL | BA | PPL | BA | PPL | Top-1 | Top-5 | R-1 | R-L | |
| Text Completion | ||||||||||||||
| CycleShift | 100.00 | 5.63 | 100.00 | 5.75 | 100.00 | 5.59 | 100.00 | 5.51 | 100.00 | 5.62 | 57.62 | 88.55 | 0.321 | 0.186 |
| DepthW | 98.25 | 5.28 | 99.50 | 5.12 | 99.00 | 5.12 | 100.00 | 5.19 | 99.19 | 5.18 | 60.75 | 89.56 | 0.337 | 0.195 |
| StealthInk | 90.00 | 4.61 | 91.00 | 4.35 | 94.25 | 4.58 | 96.25 | 4.28 | 92.88 | 4.46 | 64.98 | 92.37 | 0.347 | 0.202 |
| MPAC | 96.75 | 5.84 | 98.00 | 5.48 | 99.25 | 5.45 | 99.00 | 5.67 | 98.25 | 5.61 | 57.66 | 88.34 | 0.329 | 0.188 |
| Method | Avg. | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| BA | PPL | BA | PPL | BA | PPL | BA | PPL | BA | PPL | |
| Text Completion | ||||||||||
| StealthInk | 89.75 | 6.15 | 90.50 | 5.79 | 95.25 | 5.73 | 94.25 | 5.61 | 92.44 | 5.82 |
| BiMark | 98.25 | 5.79 | 99.75 | 5.81 | 99.75 | 5.40 | 99.25 | 5.26 | 99.25 | 5.56 |
| MirrorMark | 100.00 | 5.83 | 100.00 | 5.66 | 100.00 | 5.12 | 100.00 | 5.27 | 100.00 | 5.47 |
| CertMark model-agnostic 8-bit | 100.00 | 5.65 | 100.00 | 5.23 | 100.00 | 5.29 | 100.00 | 5.11 | 100.00 | 5.32 |