## 3 Proposed LAO-3D Block Cipher
### Encryption Algorithm
For the LAO-3D block cipher to be immune to linear cryptanalysis, there should be no linear propagation with an amplitude greater than \(2^{-32}\). Table 18 represents the linear trails with the optimal bias for seven rounds of LAO-3D that are depicted in Figure 18.

The highest probability of linear propagation at the seventh round is \(2^{-38}\), which is lower than \(2^{-32}\). Linear cryptanalysis results of block ciphers are shown in Table 17. The clustering of linear trails of LAO-3D is limited and it is impossible to construct an effective linear propagation over six rounds. Hence, the 20-round LAO-3D is sufficient to resist linear cryptanalysis. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Algorithm**} \\  & **LAO-3D** & **GIFT** & \multicolumn{3}{c}{**PRESENT**} & \multicolumn{3}{c}{**RECTANGLE**} \\ \hline  & & \multicolumn{8}{c}{**Attack**} \\
**Round** & **DC** & **LC** & **DC** & **LC** & **DC** & **LC** & **DC** & **LC** \\ \hline
1 & \(2^{-2}\) & \(2^{-2}\) & \(2^{-6}\) & \(2^{-1}\) & \(2^{-2}\) & \(2^{-1}\) & \(2^{-2}\) & \(2^{-1}\) \\
2 & \(2^{-8}\) & \(2^{-8}\) & \(2^{-10}\) & \(2^{-2}\) & \(2^{-4}\) & \(2^{-2}\) & \(2^{-4}\) & \(2^{-2}\) \\
3 & \(2^{-15}\) & \(2^{-14}\) & \(2^{-16}\) & \(2^{-3}\) & \(2^{-8}\) & \(2^{-4}\) & \(2^{-7}\) & \(2^{-4}\) \\
4 & \(2^{-27}\) & \(2^{-20}\) & \(2^{-20}\) & \(2^{-5}\) & \(2^{-12}\) & \(2^{-6}\) & \(2^{-10}\) & \(2^{-6}\) \\
5 & \(2^{-44}\) & \(2^{-26}\) & \(2^{-26}\) & \(2^{-7}\) & \(2^{-20}\) & \(2^{-8}\) & \(2^{-14}\) & \(2^{-8}\) \\
6 & \({}^{*}\,2^{-64}\) & \(2^{-32}\) & \(2^{-30}\) & \(2^{-10}\) & \(2^{-24}\) & \(2^{-10}\) & \(2^{-18}\) & \(2^{-10}\) \\
7 & & \({}^{*}\,2^{-38}\) & \(2^{-36}\) & \(2^{-13}\) & \(2^{-28}\) & \(2^{-12}\) & \(2^{-25}\) & \(2^{-13}\) \\
8 & & & \(2^{-40}\) & \(2^{-16}\) & \(2^{-32}\) & \(2^{-14}\) & \(2^{-31}\) & \(2^{-16}\) \\
9 & & & \(2^{-46}\) & \(2^{-20}\) & \(2^{-36}\) & \(2^{-16}\) & \(2^{-36}\) & \(2^{-19}\) \\
10 & & & \(2^{-50}\) & \(2^{-25}\) & \(2^{-41}\) & \(2^{-18}\) & \(2^{-41}\) & \(2^{-22}\) \\
11 & & & \(2^{-56}\) & \(2^{-29}\) & \(2^{-46}\) & \(2^{-20}\) & \(2^{-46}\) & \(2^{-25}\) \\
12 & & & \(2^{-60}\) & \(2^{-31}\) & \(2^{-52}\) & \(2^{-22}\) & \(2^{-51}\) & \(2^{-28}\) \\
13 & & \({}^{*}\,2^{-64}\) & \({}^{*}\,2^{-34}\) & \(2^{-56}\) & \(2^{-24}\) & \(2^{-56}\) & \(2^{-31}\) \\
14 & & & & \(2^{-62}\) & \(2^{-26}\) & \(2^{-61}\) & \({}^{*}\,2^{-34}\) \\
15 & & & & \({}^{*}\,2^{-66}\) & \(2^{-28}\) & \({}^{*}\,2^{-66}\) & \\
16 & & & & & & \(2^{-30}\) & & \\
17 & & & & & & \(2^{-32}\) & & \\
18 & & & & & & \({}^{*}\,2^{-34}\) & & \\ \hline \hline \end{tabular}

* No effective trail from the encryption round onward.

\end{table} 
Table 18: Linear Trails with Optimal Bias. For a comparison with other existing lightweight algorithms, Table 19 shows the number of active differential and linear S-boxes of seven encryption rounds. The results show that the LAO-3D block cipher has higher active S-boxes from the second round onward. Referring to Table 17[35; 36] and Table 19[37], LAO-3D recorded the lowest allowable probabilities of differential trails (\(2^{-44}\)) with 17 active S-boxes at the fifth round. In addition, LAO-3D achieved the lowest allowable correlation potentials of linear trails

Figure 18: 7-Round Linear Characteristic of LAO-3D. (\(2^{-32}\)) at the sixth round with 11 active S-boxes. Therefore, LAO-3D has a high-security strength and is resistant to differential and linear attacks. ### Performance Test

The proposed LAO-3D was implemented using C++ on an Intel(R) Core(TM) if 2.70 GHz CPU with 8 GB RAM on Windows 10. Speed tests were conducted on the execution process of the proposed LAO-3D and compared with other existing algorithms. The speed tests were carried out on the full rounds of each encryption algorithm to observe the time required to process a ciphertext block that consisted of 64-bit data.