# A Novel Digital Signature Scheme for Advanced Asymmetric Encryption Techniques
## 5 Experimental Results  
Table 2 represents the sizes used in each test. The message size is fixed in all tests where \(size(m)=8\) bits. In this type of public key \(k+r\times p\), generally \(size(p)=2\times size(k)\) because  
\(k^{2}<p\). \(r\) must satisfy the condition \(r>q\); to minimize the complexity, we can choose \(r>q\) with \(size(r)=size(q)\); for example, \(r=725\), \(q=307\). We have \(size(k+r\times p)=size(r\times p)\); thus, the size of the public key is \(size(pk)=size(r)+size(p)\). While \(n=p\times q\) and \(size(r)=size(q)\), then \(size(n)=2\times size(p)\). We note here that, for security measures, we selected \(p\) and \(q\) with similar, but not equal sizes to prevent the square computation on \(n\). The size of \(s_{1}\) and \(s_{2}\) means the average size of many tests. However, we note that the sizes of \(s_{1}\) and \(s_{2}\) are always almost equal to the size of \(n\). Figure 2 shows the execution time in each test, that is the Signing Process Time (SPT) and the Verification Process Time (VPT). In \(T_{1}\), the sizes used are impractical, but used to show the increase in time. In \(T_{2}\), \(T_{3}\), and \(T_{4}\), we noticed that SPT and VPT take a constant ratio between them, where \(VPT=1.33\times SPT\). SPT and VPT increase more rapidly in \(T_{4}\). Of course, the sizes used in \(T_{4}\) exceed the safety recommendations, because the recommended size of primitive \(p\) is 1024 bits, which is used in \(T_{3}\). The results shown in this section, namely signing and verifying time, showed the applicability of the proposed method. Four experiments were presented containing realistic example values; the implementation time was not exceeded, at the highest estimate, 250 milliseconds for signing and 330 milliseconds for verifying represented in Experiment No. 4. Table 3 shows the used notations and the corresponding runtimes for each operation. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Test/Size (bits)** & \(k\) & \(r\) & \(p\) & \(n\) & \(s_{1}\) & \(s_{2}\) \\ \hline \(T_{1}\) & 128 & 256 & 256 & 512 & 512 & 512 \\ \hline \(T_{2}\) & 256 & 512 & 512 & 1024 & 1024 & 1024 \\ \hline \(T_{3}\) & 512 & 1024 & 1024 & 2048 & 2048 & 2048 \\ \hline \(T_{4}\) & 1024 & 2048 & 2048 & 4096 & 4096 & 4096 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter sizes in the different tests. Figure 2: Signing and verifying execution time in the proposed scheme. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Scheme** & **Signing** & **Verification** & **Total (ms)** & **Sig (bit)** & **Suitable for \(pk=k+r\times p\)** \\ \hline
[23] & \(T_{r,s,b,m,a,e}\) & \(T_{2s,c,b,m}\) & 94.8 & 320 & No \\ \hline
[31] & \(T_{h,5m,2a}\) & \(T_{h,4m,2a}\) & 86.4 & 64–512 & No \\ \hline
[26] & \(T_{r,m,3a,e}\) & \(T_{2m,a,e}\) & 44 & 1024 & No \\ \hline
[41] & \(T_{r,3m,h,3c,a}\) & \(T_{h,3c,3m,a}\) & 87.4 & 340 & No \\ \hline
[42] & \(T_{r,3m,4c,b,a}\) & \(T_{4m}\) & 55.6 & 992 & No \\ \hline Ours & \(T_{2e}\) & \(T_{2e,m}\) & 48.4 & 320 & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 4: A comparison of the computation cost and size. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** & **Value (ms)** \\ \hline \(T_{m}\) & multiplication & 4.4 \\ \hline \(T_{a}\) & addition & 2.2 \\ \hline \(T_{h}\) & hash function & 19 \\ \hline \(T_{e}\) & exponentiation & 11 \\ \hline \(T_{r}\) & random number generation & 11 \\ \hline \(T_{c}\) & concatenation & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The used notations with runtimes. Figure 3: Comparison of computational and communication costs of our proposal with Mughal 2018 [23], Kavin 2021 [31], Zaghian 2013 [26], Ullah 2020 [41], and Ali 2020 [42] schemes. This practical comparison, which was presented in this section, shows the effectiveness of the proposed protocol, not only in a theoretical study, but also in an applied study, outperforming many modern techniques, both in execution time and in signature size.