# Mathematical Background and Algorithms of a Collection of Android Apps for a Google Play Store Page
## 3 Problem Solving
### Private File Manager App
Consider a B-tree node \(x\) containing \(n\) keys; the properties listed below are satisfied:  
1. \(n\leq 2m-1\). 2. If \(x\) is an internal node that is not the root, it must have a minimum of \(\lceil m/2\rceil\) children. Securing data involves encrypting files. One popular encryption technique is the Advanced Encryption Standard (AES). The AES encryption was implemented in the file manager app. AES encryption comprises several transformation rounds. Each round \(r\) of AES-128 encryption includes the following:  
1. SubBytes: A substitution step with non-linear properties. 2. ShiftRows: A permutation of rows. 3. MixColumns: A mixing operation performed on the columns. 4. AddRoundKey: An XOR operation using the round key. In round \(r\), the state matrix \(S\) is transformed by \(S_{r}\), as demonstrated in Equation (42). \[S_{r}=\text{AddRoundKey}(\text{MixColumns}(\text{ShiftRows}(\text{SubBytes}(S)))) \tag{42}\]  
File compression minimizes file size for optimal storage. Huffman coding is a widely used algorithm for lossless file compression. Huffman encoding allocates codes of varying lengths to symbols according to their occurrence rates. Consider a set of characters \(C=\{c_{1},c_{2},\ldots,c_{n}\}\) with associated frequencies \(f(c_{i})\). The expected length \(E(L)\) of the encoded message is depicted in Equation (43). \[E(L)=\sum_{i=1}^{n}P(c_{i})\cdot l(c_{i}) \tag{43}\]  
\(P(c_{i})=\frac{f(c_{i})}{\sum_{i=1}^{n}l(c_{i})}\), where \(l(c_{i})\) denotes the length of the code for \(c_{i}\). Synchronizing files guarantees consistency between different devices. The rsync algorithm is a popular technique for this purpose. The rsync algorithm employs a rolling checksum to identify differences between files. The rolling checksum, denoted as \(R\), for a block of size \(b\) beginning at position \(i\), is illustrated in Equation (44). \[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]0  
To detect changes, the hash of this block is compared later with the hashes of blocks in the target file. Let the total storage capacity be denoted by \(S\), and let \(s_{i}\) represent the size of the \(i\)-th file. The total storage utilized, \(U\), is defined as:  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]1  
Where:  
* \(n\) indicates the complete count of files. The proportion of storage in use, denoted as \(P\), is determined by:  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]2The rate at which a file of size \(s\) is transferred over a duration \(t\) is represented by \(R\):  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]3  
Where:  
* \(s\) represents the size of the file in megabytes (MB). * \(t\) refers to the duration of the transfer measured in seconds (s). Figure 11 shows the file transfer rates. The effectiveness of arranging files can be measured by the mean number of files per directory \(E\), calculated as:  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]4  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]5 File transfer rates. \[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]6 Storage usage over time. Where:  
* \(n\) represents the complete count of files. * \(f\) denotes the aggregate number of directories. A mobile file manager equipped with a concealment feature allows users to handle their files while ensuring that certain ones remain hidden for privacy purposes. This app incorporates encryption, file storage structuring, energy consumption regulation, and enhanced access speed. This paper presents a mathematical model that uses techniques from differential equations, linear algebra, analytic geometry, and mathematical analysis. Figure 12 shows the file organization efficiency. The process of hiding data generally involves encrypting files. Consider \(\mathbf{F}\in\mathbb{R}^{n}\), which denotes the file content as a vector of dimension \(n\) (e.g., a vector format of the file's binary data).