## 6 Conclusions  
In this study, we propose DDP--a deep learning-based end-to-end smartphone user authentication method using sequential data obtained from drawing a character or freestyle pattern on a smartphone touchscreen. The proposed method uses raw sequential human behavioral data instead of handcrafted features. The behavioral data are transformed using a statistical method. We collected raw data from 40 participants who drew 13 patterns each 20 times. The patterns comprised five English, five freestyle, and three Korean patterns. We set one of the most unfavorable experimental environments possible for the user-authentication process, wherein all potential intruders know the authenticated user's drawing pattern. In other words, we assumed threat scenarios such as SSA and SA for our experimental design. Figure 14: Training and inference time ratios of RNN to TCN by sequence length. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Valid Index** & **E1** & **E2** & **E3** & **E4** & **E5** & **F1** & **F2** & **F3** & **F4** & **F5** & **K1** & **K2** & **K3** & **AVG** \\ \hline FAR & 0.81 & 0.76 & 2.02 & 0.93 & 1.15 & 1.73 & 1.86 & 1.08 & 2.38 & 1.22 & 2.77 & 0.93 & 0.67 & 1.41 \\ FRR & 0.50 & 0.75 & 0.50 & 2.25 & 0.75 & 1.25 & 2.25 & 0.75 & 1.00 & 1.75 & 1.75 & 0.75 & 1.75 & 1.23 \\ AVG & 0.66 & 0.76 & 1.26 & 1.59 & 0.95 & 1.49 & 2.06 & 0.92 & 1.69 & 1.49 & 2.26 & 0.84 & 1.21 & 1.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Average FAR and FRR values obtained from proposed TCN-based DDP model for each pattern based on the Coordinate-A method (E, F, and K denote English, freestyle, and Korean patterns, respectively, and AVG denotes average value). Based on the experimental results, it was observed that the Coordinate-A-based user-authentication model yielded the best authentication performance in most cases. Additionally, the Coordinate-A-based TCN model produced excellent user-authentication performance, i.e., 0.99%, 1.41%, and 1.23% in terms of average AUROC, FAR, and FRR, respectively. We expect superior user-authentication performance in real-world user-authentication scenarios than that shown by our experimental results, because in the former, the authenticated user's drawing pattern may not be known to the intruder. In other words, the unauthorized user's login attempts can be blocked if the number of attempts for the pattern strokes exceeds its maximum allowable threshold. Based on the result, it can be seen that the Coordinate-A-based TCN model exhibited superior user-authentication performance and robustness for threat scenarios, such as SSA and SA. Next, we compared the authentication performance between the TCN- and RNN-based models. The experimental results demonstrated that the proposed TCN-based DDP model provided statistically superior authentication performance than the RNN-based model in most cases. Additionally, both training and inference time consumption for the TCN-based model were relatively smaller than those for the RNN-based model. Therefore, the proposed TCN-based DDP model not only exhibited superior authentication performance but proved to be more computationally efficient than the RNN-based model in most cases. Finally, we made our dataset publicly available to research and industrial communities. Therefore, we hope that the results of our study and the shared data will contribute to further research in behavioral biometric-based user authentication for smartphones. Despite the promising experimental results, the current study faces the following limitations that lead us to future research directions. First, although the proposed model has a sufficiently small number of parameters, it is beneficial to compress the network without performance loss. Second, this study used 10 training datasets for building an authentication model. Therefore, whether high performance can be preserved even with fewer training data for the purpose of user convenience must be studied.