# Telehealth Secure Solution to Provide Childhood Obesity Monitoring
## 6 Security Analysis
### Key Sensitivity Analysis  
The ability to produce two uncorrelated cryptograms using two highly similar secret keys must be performed in any encryption algorithm, which is called the avalanche effect. The use of chaotic maps helps in this ability of the encryption algorithm, since they are highly sensitive to the initial conditions and control parameters. In the decryption process, only the correct secret key must recover the original plain text. Table 5 depicts the key sensitivity results for the decryption process considering the secret keys in Table 3. The plain text is recovered correctly for users 2, 3, and 5 (since the same secret key of encryption is used), whereas users 1 and 4 cannot recover correctly the plain text (since secret keys are lightly different than the encryption keys).  
The numerical difference between the plain text of the five users and the corresponding decrypted text is calculated using the bit error rate (BER) with the \"_biter  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{x_{0}}\) & \(\mathbf{y_{0}}\) \\ \hline User 1 & 1.400112233445566 & 0.300112233445566 & 0.556677889900118 & 0.667788990011223 \\ User 2 & 1.400223344556677 & 0.300222344556677 & 0.6677889900112233 & 0.7788990011223345 \\ User 3 & 1.400334455667788 & 0.30022334456677 & 0.77889900112233445 & 0.889900112233445 \\ User 4 & 1.40045566778899 & 0.3003445566778899 & 0.8899011223344556 & 0.99011223344557 \\ User 5 & 1.400556677889900 & 0.300445566778899 & 0.990011223344556 & 0.001122334455667 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Secret key definition for the decryption process of the five users.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **User 1** & **User 2** & **User 3** & **User 4** & **User 5** \\ \hline Weekly steps (mean) & **4801** & 10,616 & 12,221 & **5416** & 14,504 \\ Weekly activity (mean min) & **11,734** & 134 & 154 & **44,277** & 184 \\ Distance week (km) & **818** & 30 & 38 & **33,968** & 44 \\ Calories burned (cal) & **32,953** & 700 & 1039 & **482** & 1023 \\ Sleep (mean min) & **56,196** & 480 & 562 & **19,737** & 546 \\ Deep sleep (mean min) & **36,955** & 217 & 220 & **61,087** & 146 \\ Light sleep (mean min) & **44,513** & 262 & 342 & **15,880** & 400 \\ Fell asleep at & **6900** & 326 & 630 & **56,226** & 651 \\ Woke up at & **22,135** & 447 & 472 & **12,731** & 481 \\ Awake time (mean min) & **23,445** & 0 & 0 & **58,027** & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Recovered plain text of five users, where bold numbers denote incorrect data.  
MATLAB [37]. The BER ratio is 47.91, 0, 0, 50.78, and 0, for users 1, 2, 3, 4, and 5, respectively. The results show that the decrypted text (for users 1 and 4) has close to 50% of different bits.  
Hence, the proposed telehealth monitoring scheme presents high sensitivity to the secret key in the decryption process, even when the last digit in the secret key is slightly different from the correct secret key. In comparison, the implementation in [22] found BER ratios with values of 0.5677 and 0.6543 when incorrect secret keys were used in the decryption process.