Towards Anytime-Valid Statistical Watermarking

As large language models (LLMs) are increasingly deployed, reliable and efficient mechanisms for distinguishing AI-generated text from human-written content have become essential. Statistical watermarking has emerged as a promising solution, yet most existing methods are typically fixed-horizon procedures, precluding valid early stopping in streaming generation. In this paper, we develop an efficient online watermark detection framework with anytime-valid inference based on Rao-Blackwellized e-processes, enabling recursive token-level evidence updates without storing the full history. In particular, we instantiate the framework for the Gumbel-max watermark and reduce the original token-level dependence testing problem to a pivot-induced sequential testing problem with an explicit null distribution. Theoretically, we prove anytime-valid Type I error control under arbitrary optional stopping and establish positive asymptotic log-growth under watermarking, implying consistency of the proposed stopping rules. Simulations and experiments on real LLM-generated text demonstrate efficient online detection with rigorous anytime-valid guarantees.

Watermarking has emerged as a key tool for identifying text generated by large language models (LLMs), with current schemes embedding statistical signals into the generation process to enable later detection. In this work, we identify a shared weakness across a broad class of popular LLM watermarking methods, including KGW (and its variants), GumbelSoft, DipMark, ITS, and SynthID. We show that these schemes exhibit a fundamental link between their detection scores and the token probability shifts they introduce during generation. This link, which we call score-ratio alignment, allows an attacker with only black-box query access to the model--and no knowledge of secret keys or model weights--to produce text that evades detection while remaining fluent and natural. We prove that the resulting attack is optimal across all covered watermark families, and show that the weakness is intrinsic: any watermark designed to maximize statistical detection power necessarily induces the very structure that attackers can exploit. We evaluate our attack on seven popular generative LLM watermarking schemes, three LLMs, and tasks, which reduce detection rates to at most 0.05 across all settings while preserving text quality. Our findings show that current watermarking methods can be reliably defeated under realistic attack conditions, highlighting the need for fundamentally different watermarking designs that decouple detection scores from token probability shifts.

We introduce Stateless Bernoulli Watermarking (SBW), a new statistical watermark for Large Language Models that determines green list membership through independent per-token Bernoulli trials. Unlike KGW's vocabulary permutation or SynthID's multi-layer tournament, SBW requires only a single comparison per token against a counter-based random number generator, reducing membership complexity to $O(1)$ and enabling single-kernel execution with zero intermediate allocations. We prove that this formulation preserves the same detection guarantees as fixed-size green lists: the z-score test remains $\mathcal{N}(0,1)$ under the null. The stateless architecture enables capabilities unavailable to existing methods: full-vocabulary self-salt watermarking (over 6000$\times$ faster than KGW's self-salt and 2$\times$ faster than SynthID despite biasing the entire vocabulary with candidate-dependent seeding) and architectural compatibility with distributed inference. In end-to-end generation benchmarks, SBW adds less than 1% overhead at all batch sizes. We additionally identify hash function design as a previously unexplored axis for watermark quality, showing that a GPU-native Jenkins hash improves null calibration by 1.8$\times$ while producing more diverse text. Experiments across two seeding schemes and eight $(γ, δ)$ configurations confirm statistical equivalence with ROC-AUC differences below 0.01.