## 3 Proposed LAO-3D Block Cipher
### Encryption Algorithm
The input for the 3 DBitRotation_X-axis is the output of the SubColumn function defined as \(C=\{(c_{3,15},\ldots,c_{3,0}),(c_{2,15},\ldots,c_{2,0}),(c_{1,15},\ldots,c_{1,0}), (c_{0,15},\ldots,c_{0,0}):c_{i,j}\in C\) for \(i=(3,\ldots,0)\) and \(j=(15,\ldots,0)\}\) is permutated using the permutation table and produced output \(D=\{(a_{3,15},\ldots,a_{3,0}),(d_{2,15},\ldots,d_{2,0}),(d_{1,15},\ldots,d_{1,0}),(d_{0,15},\ldots,d_{0,0}):d_{i,j}\in D\) for \(i=(3,\ldots,0)\) and \(j=(15,\ldots,0)\}\) as defined in the following equation. \[D=P(C) \tag{2}\]

AddRoundKey: In this operation, the output from the 3 DBitRotation_X-axis function \(D=\{(d_{3,15},\ldots,d_{3,0}),(d_{2,15},\ldots,d_{2,0}),(d_{1,15},\ldots,d_{1,0 }),(d_{0,15},\ldots,d_{0,0}):d_{i,j}\in D\) for \(i=(3,\ldots,0)\) and \(j=(15,\ldots,0)\}\) is XOR with the RoundKey \(K=\{(k_{3,15},\ldots,k_{3,0}),(k_{2,15},\ldots,k_{2,0}),(k_{1,15},\ldots,k_{1,0 }),(k_{0,15},\ldots,k_{0,0}):k_{i,j}\in K\) for \(i=(3,\ldots,0)\) and \(j=(15,\ldots,0)\}\): The result from the operation, cipher State \(E=\{(e_{3,15},\ldots,e_{3,0}),(e_{2,15},\ldots,e_{2,0}),\\ (e_{1,15},\ldots,e_{1,0}),(e_{0,15},\ldots,e_{0,0}):e_{i,j}\in E\) for \(i=(3,\ldots,0)\) and \(j=(15,\ldots,0)\}\) is obtained from the following equation. \[E=D\oplus K \tag{3}\]

3. 3 DBitRotation_Z-axis: A specific rotation formation that follows a clockwise direction at the \(Z\)-axis. The position of each cipher Slice before and after executing the 3 DBitRotation_Z-axis function is displayed in Figure 8. Permutation values of the function are shown in Table 3.

\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Slic\({}_{0}\)**} & \multicolumn{4}{c}{_Slice\({}_{1}\)_} & \multicolumn{4}{c}{_Slice\({}_{2}\)_} & \multicolumn{4}{c}{_Slice\({}_{3}\)_} \\ \cline{2-13}  & \(\alpha_{0,0}\) & \(\alpha_{0,1}\) & \(\alpha_{0,2}\) & \(\alpha_{0,3}\) & \(\alpha_{1,0}\) & \(\alpha_{1,2}\) & \(\alpha_{1,2}\) & \(\alpha_{2,3}\) & \(\alpha_{2,0}\) & \(\alpha_{2,1}\) & \(\alpha_{2,2}\) & \(\alpha_{2,3}\) & \(\alpha_{0,3}\) & \(\alpha_{3,1}\) & \(\alpha_{3,2}\) & \(\alpha_{3,3}\) \\ \hline \multirow{3}{*}{3 DBitRotation} & \(\alpha_{0,4}\) & \(\alpha_{0,5}\) & \(\alpha_{0,6}\) & \(\alpha_{0,7}\) & \(\alpha_{1,4}\) & \(\alpha_{1,5}\) & \(\alpha_{1,6}\) & \(\alpha_{1,7}\) & \(\alpha_{2,4}\) & \(\alpha_{2,5}\) & \(\alpha_{2,6}\) & \(\alpha_{2,7}\) & \(\alpha_{3,4}\) & \(\alpha_{3,5 The 3 DBitRotation_Z-axis permutes the output of the AddRoundKey function defined as \(E=\{(e_{3,15},\ldots,e_{3,0})\), \((e_{2,15},\ldots,e_{2,0})\), \((e_{1,15},\ldots,e_{1,0})\), \((e_{0,15},\ldots,e_{0,0}):e_{i,j}\in E\) for \(i=(3,\ldots,0)\) and \(j=(15,\ldots,0)\}\) using the permutation table. Equation (4) derived the results from the operation defined as \(F=\{(f_{3,15},\ldots,f_{3,0}),(f_{2,15},\ldots,f_{2,0}),(f_{1,15},\ldots,f_{1,0 })\), \((f_{0,15},\ldots,f_{0,0}):f_{i,j}\in F\) for \(i=(3,\ldots,0)\) and \(j=(15,\ldots,0)\}\). \[F=P(E) \tag{4}\]

### Key Schedule Algorithm

The RoundKeys are derived using the key scheduling algorithm from a secret key. Overall, the total number of RoundKeys bits is equal to the round number plus one and multiplied by the block length. For an algorithm with a 64-bit block and 20 rounds, 1344 RoundKey bits are required. The RoundKeyExtraction process selected 64 bits RoundKey from the revised key after operating the NonceXOR, KeySubColumn, and RowTransformation functions, as shown in Figure 9.