## 3 Implementation and Evaluation of the Method
### Collision Evaluation  
A collision refers to the event in which two different inputs (voice signals) produce the same hash code. Since it is mathematically impossible to design a hash function without collisions, it is important to test the proposed function to determine its variability. The theoretical value of the collision probability can be calculated through Equation (9) [24],  
\[P=\left(\frac{1}{P}\right)^{L}, \tag{9}\]  
where \(L\) is the number of digits of the hash code and \(P\) is the base of the numeral system used to represent the digits in the hash code. For the proposed scheme, \(L=64\) and \(P=16\). Thus, the theoretical probability of collision is given by,  
\[P=\left(\frac{1}{16}\right)^{64}\approx 8.64\times 10^{-78}. \tag{10}\]  
This means that, theoretically, you expect to find a collision every \(\approx\)1.16 \(\times 10^{77}\) tests.  
\begin{table}
\begin{tabular}{c c c} \hline
**Specs\textbackslash{}Model** & **Moto G** & **Nexus 5X** \\ \hline \multirow{3}{*}{Processor} & 1.2 GHz Qualcomm & 1.8 GHz Qualcomm \\  & Snapdragon 400, & Snapdragon 800, \\  & Quad-core & Hexa-core \\ \hline OS & Android 5.1 & Android 7.0 \\ \hline Memory & 1 GB RAM & 2 GB RAM \\ \hline \end{tabular}
\end{table}
Table 1: Characteristics of the mobile devices used for the tests.  
\begin{table}
\begin{tabular}{c c} \hline
**Duration (s)** & **Number of Recordings per Device** \\ \hline
0–10 & 10 \\
10–20 & 10 \\
20–40 & 10 \\
40–80 & 10 \\
80–160 & 10 \\
160–320 & 10 \\
320–640 & 10 \\ \hline \end{tabular}
\end{table}
Table 2: Time ranges of the recordings acquired by each device.  
To verify the presence of collisions, the Hamming distance (\(HD\)) is calculated between each pair of codes of the test signals. The percentage of coordinates that differ from each other (\(HD\)) is given by Equation (11).  
\[d_{st}=(\#(x_{sj}\
eq x_{tj}/n)). \tag{11}\]  
If the result obtained in \(HD\) is 0, it implies that the two hash codes are identical, whereas, if the obtained value is 1, it means that the hash codes are completely different.  
In a complementary way, each pair of evaluated codes are compared to each other using the Pearson Correlation Coefficient (\(PCC\)) given by Equation (12), in order to determine which is the linear dependence between two codes. The goal here is to evaluate the degree in which a pair of hash codes are linearly related, in order to evaluate the similarity between them.  
\[PCC=\frac{\sum_{m}\sum_{n}\left(A_{mn}-\bar{A}\right)(B_{mn}-\bar{B})}{\sqrt{ \left(\sum_{m}\sum_{n}\ \left(A_{mn}-\bar{A}\right)^{2}\right)\left(\sum_{m}\sum_{n} \left(B_{mn}-\bar{B}\right)^{2}\right)}}, \tag{12}\]  
where \(A\) and \(B\) are matrices or vectors of the same size. The result of the correlation can vary between \(-1\) and \(1\); if the absolute \(PCC\) value is 1, it indicates a perfect linear relationship, whereas a \(PCC\) close to 0 means that there is no linear relationship between the data. The sign indicates the direction of the relationship, i.e., if both variables tend to increase or decrease at the same time, the coefficient will be positive, whereas if one variable tends to increase and the other to decrease, the coefficient will be negative. In practice, when evaluating the collision resistance in binary hash codes by means of the correlation coefficient, \(PCC\
eq 1\) is expected.