## 4 Experimental Results and Analysis
### Comparisons with Existing Methods
#### 4.4.3 Efficiency  
While enhancing efficiency was not the primary focus of our study, we conducted an efficiency assessment as it is a crucial consideration for practical usage. For the same set of test images, we ran each of the three zero-watermarking methods 10 times and calculated the average processing time. The efficiency results for these methods are presented in Table 6.  
The efficiency comparison presented in Table 6 clearly shows that the time required to generate a single zero-watermark with ZWNet is significantly lower than that of the other two methods. [PERSON]'s method appears to be less efficient than ZWNet due to the utilization of the CPU for computation without GPU acceleration. In fact, classical zero  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **ZWNet** **(Proposed)** & **[PERSON]’s Method** & **[PERSON]’s Method** & **[PERSON]’s Method** \\ \hline Average cost time & 96 ms & 2100 ms & 2440 ms & 1384 ms \\ \hline \hline \end{tabular}
\end{table}
Table 6: Efficiency comparison.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Attack Description** & \begin{tabular}{c} **ZWNet** \\ **(Proposed)** \\ \end{tabular} & \begin{tabular}{c} **[PERSON]’s** \\ **Method** \\ \end{tabular} & \
\begin{tabular}{c} **[PERSON]’s** \\ **Method** \\ \end{tabular} &
\begin{tabular}{c} **[PERSON]’s** \\ **Method** \\ \end{tabular} \\ \hline Rotation (15\({}^{\circ}\)) & 0.9688 & 0.9943 & 0.9466 & 0.9247 \\ Rotation (30\({}^{\circ}\)) & 0.9258 & 1 & 0.9466 & 0.9058 \\ Pepper and salt noise & & & & \\ (intensity = 0.01) & 0.9922 & 0.9375 & 0.9766 & 0.8935 \\ Gaussian noise & & & & \\ (mean = 0, variance = 0.005) & 1 & 0.9531 & 1 & 0.9340 \\ Random crop (1/6) & 1 & 0.9023 & 0.9522 & 0.8706 \\ Blur (\(\times\) 3\(\times\)) & 1 & 0.9414 & 0.9766 & 0.9172 \\ Blur (11 \(\times\) 11) & 0.9961 & 0.9414 & 0.9302 & 0.8388 \\ Gaussian blur (3 \(\times\) 3) & 1 & 0.9063 & 1 & 0.8902 \\ Gaussian blur (11 \(\times\) 11) & 0.9922 & 0.9375 & 0.9766 & 0.8253 \\ Median blur (3 \(\times\) 3) & 0.9922 & 0.9297 & 0.9961 & 0.9049 \\ Median blur (11 \(\times\) 11) & 0.9766 & 0.9648 & 0.9102 & 0.8138 \\ \hline \hline \end{tabular}
\end{table}
Table 4: NC results of the comparison methods.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Hamming Distance of** & **ZWNet** & **[PERSON]’s** & **[PERSON]’s** & **[PERSON]’s** \\
**Zero-Watermarks** & **(Proposed)** & **Method** & **Method** & **Method** \\ \hline [PERSON] and [PERSON] & 90 & 28 & 59 & 43 \\ [PERSON] and [PERSON] & 88 & 29 & 26 & 35 \\ [PERSON] and [PERSON] & 113 & 40 & 90 & 93 \\ [PERSON] and [PERSON] & 92 & 21 & 61 & 66 \\ [PERSON] and [PERSON] & 97 & 20 & 57 & 21 \\ [PERSON] and [PERSON] & 91 & 37 & 62 & 55 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Hamming distance of comparison methods.  
watermarking methods are generally impractical to run on GPUs, as many of their steps cannot be efficiently implemented with tensor operations.  
In the case of [PERSON]'s method, although both it and ZWNet are based on neural networks and can be accelerated by GPUs, [PERSON]'s method involves learning different style features for various host images. This learning process includes training and multiple epochs, making it more time-consuming compared to ZWNet. In contrast, ZWNet can produce a zero-watermark directly for different images without the need for retraining by passing the host image through the ZWNet's layers only once.