## 4 Materials and Methods
### Analysis of Distinguishability of OTP Extraction Function
#### 4.3.1 Distinguishable Security Model for the Proposed OTP Extraction Function  
When comparing an arbitrary function family to a family of random functions with the same domain and range, if the two families are indistinguishable, then the arbitrary family satisfies pseudorandomness, which is called a pseudorandom function (PRF) [21]. Based on this, we define a family of random functions and a family of extraction functions with weak properties. The domain and range of the random and weak functions are presented in Table 2. To distinguish between the family of weak extraction functions \(J\), a set of functions, called a function family, was constructed, which generalizes the extraction function defined by TTA and has the same properties. Figure 4: Truncation of OTP strings to varying lengths. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline
**Range** & \(\mathbf{E[X]}\) & \multicolumn{2}{c}{\(\mathbf{MAX(X)}\)} & \(\mathbf{H}\) & \multicolumn{2}{c}{\(\mathbf{\#(X>H)}\)} & \multicolumn{2}{c}{\(\mathbf{\#(X\leq H)}\)} & \multicolumn{2}{c}{\(\mathbf{e}\)} \\ \hline \([0,10^{1}-1]\) & \(\approx\)10\({}^{5}\) & - & & - & - & - & - & - \\ \hline \multirow{3}{*}{\([0,10^{2}-1]\)} & \(\approx\)10\({}^{4}\) & static & 10,714 & & static & 2 & static & 98 & 1.001 \\  & dynamic & 10,679 & 10,700 & dynamic & 0 & dynamic & 100 & 0.998 \\  & improved & 10,821 & & improved & 10 & improved & 90 & 1.011 \\ \hline \multirow{3}{*}{\([0,10^{3}-1]\)} & \(\approx\)10\({}^{3}\) & static & 1151 & & static & 61 & static & 939 & 1.046 \\  & dynamic & 1154 & 1100 & dynamic & 59 & dynamic & 941 & 1.050 \\  & _improved_ & **1358** & & _improved_ & **256** & improved & 744 & **1.235** \\ \hline \multirow{3}{*}{\([0,10^{4}-1]\)} & \(\approx\)10\({}^{2}\) & static & 148 & & static & 0 & static & 10,000 & 0.740 \\  & dynamic & 151 & 200 & dynamic & 0 & dynamic & 10,000 & 0.755 \\  & _improved_ & **413** & & _improved_ & **256** & improved & 9744 & **2.065** \\ \hline \multirow{3}{*}{\([0,10^{5}-1]\)} & \(\approx\)10\({}^{1}\) & static & 33 & & static & 0 & static & 100,000 & 0.33 \\  & dynamic & 29 & 100 & dynamic & 0 & dynamic & 100,000 & 0.29 \\  & _improved_ & **333** & & _improved_ & **256** & improved & 99,744 & **3.33** \\ \hline \multirow{3}{*}{\([0,10^{6}-1]\)} & \(\approx\)10\({}^{0}\) & static & 9 & & static & 0 & static & 1,000,000 & 0.257 \\  & dynamic & 8 & 35 & dynamic & 0 & dynamic & 1,000,000 & 0.229 \\ \cline{1-1}  & _improved_ & **320** & & _improved_ & **256** & improved & 999,744 & **9.143** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the experiment in Ref. [1]: This table numerically shows how often certain numbers occur when the improved algorithm is used. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Glossary of Terms** & **Symbols** & **Description** \\ \hline Domain of Extraction Function & \(D\) & \(D=Z_{24}\), 24-bit representation area \\ Range of Extraction Function & \(R\) & \(X\)_mod_\(10^{6}\), \((X\in D)\) \\ Family of Random Functions & \(Func(D,R)\) & A set that includes all functions with domain \(D\) and range \(R\) \\ Family of Vulnerable & \(J\) & A set that includes all functions with the domain \(D\) and range \(R\), \\ Extraction Functions & \(J\) & and has the vulnerable characteristics mentioned in “Section 4” \\ Randomly Selected Instance & g & A function randomly selected from \(J\) or \(Func(D,R)\), \\  & & (i.e., \(g\gets Func(D,R)\) or \(g\gets J\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definition of function families for analysis. a total of 4 bits (i.e., \(2^{4}\) possible number representations), so there are a total of 16 extraction locations that can be used for OTP random number extraction. If the four indices that cannot be selected as extraction positions from the 20-byte sequence are listed as \((0,1,2,3),(0,1,2,4),\cdots,(16,17,18,19)\), the total number of cases is \(\binom{20}{4}=\binom{20}{16}=4845\).