Algorithm 1: The embedded watermark model.

Input: The model $\mathcal{F}$ (the model level $l\in (1,2,\dots ,L)$), important parameters $\widehat{\mathsf{\Theta }}$ of $\mathcal{F}$, value of important parameters $p_{\theta }$, neuron number set $N=[n_1,n_2,\dots ,n_l]$, ratio of the number of layers to be embedded c, watermark information B, translation T, precision R, random matrix D, random value k, watermark m in a level. Output: Watermarked model ${\mathcal{F}}_{watermark}$. Step 1: For each level to calculate the unimportance ${\widehat{L}}_{p_{\theta }}=1-\frac{1}{n}{\displaystyle \sum _{i=1}^n}{p_{\theta }}^i$ Step 2: Sort ${\widehat{L}}_{p_{\theta }}$, select the top $c\times L$ layers (a total number of L layers) in the order. Step 3: Estimate the proportion of important parameters $\theta$ in these selected layers $\widehat{L}$ by watermark information B; obtain $wet–indicate$, which represents the wet-block position. Step 4: Convert floating-point $\theta$ into a nonnegative integer b (watermark information B) through $b=I\left(\theta \right)=(\theta +T)·{10}^R$. Then convert nonnegative integer b to floating-point ${\theta }^{\prime }$ through ${\theta }^{\prime }=I^{-1}\left(b\right)=b·{10}^{-R}-T$. Step 5: Generate a random matrix D and a random value k and obtain H by D and k. Record these values. Step 6:b is divided equally, $[b_1,b_2,\dots ,b_m]$ are denoted as m, X is obtained by $wet–indicate$ in $\mathcal{F}$. Step 7: Embed watermark m; after embedding m each time, $b=b-m$. Repeat Steps 5–7 until $b=0$.