As a valuable digital product, deep neural networks (DNNs) face increasingly severe threats to the intellectual property, making it necessary to develop effective technical measures to protect them. Trigger-based watermarking methods achieve copyright protection by embedding triggers into the host DNNs. However, the attacker may remove the watermark by pruning or fine-tuning. We model this interaction as a game under conditions of information asymmetry, namely, the defender embeds a secret watermark with private knowledge, while the attacker can only access the watermarked model and seek removal. We define strategies, costs, and utilities for both players, derive the attacker's optimal pruning budget, and establish an exponential lower bound on the accuracy of watermark detection after attack. Experimental results demonstrate the feasibility of the watermarked model, and indicate that sparse watermarking can resist removal with negligible accuracy loss. This study highlights the effectiveness of game-theoretic analysis in guiding the design of robust watermarking schemes for model copyright protection.

Existing watermarking methods for large language models (LLMs) mainly embed watermark by adjusting the token sampling prediction or post-processing, lacking intrinsic coupling with LLMs, which may significantly reduce the semantic quality of the generated marked texts. Traditional watermarking methods based on training or fine-tuning may be extendable to LLMs. However, most of them are limited to the white-box scenario, or very time-consuming due to the massive parameters of LLMs. In this paper, we present a new watermarking framework for LLMs, where the watermark is embedded into the LLM by manipulating the internal parameters of the LLM, and can be extracted from the generated text without accessing the LLM. Comparing with related methods, the proposed method entangles the watermark with the intrinsic parameters of the LLM, which better balances the robustness and imperceptibility of the watermark. Moreover, the proposed method enables us to extract the watermark under the black-box scenario, which is computationally efficient for use. Experimental results have also verified the feasibility, superiority and practicality. This work provides a new perspective different from mainstream works, which may shed light on future research.