## 4 Security and Performance Analyses
### Performance Comparison  
Table 3 presents the performance of the proposed scheme and other related schemes, where \(T_{H}\) denotes the time to execute the one-way hash function; \(T_{BH}\) the time to execute a biohashing; \(T_{S}\) denotes the time to execute symmetric en/decryption; \(T_{A}\) denotes the time to execute asymmetric en/decryption; \(T_{ECM}\) denotes the time to execute a scalar multiplication on elliptic curves; \(T_{FE}\) denotes the time to execute a fuzzy extractor function; and \(T_{C}\) denotes the time to execute extended Chebyshev chaotic maps. \(T_{FE}\) and \(T_{ECM}\) are considered to be the same by using the arguments presented in [35]. Table 4 presents the hardware/software specifications and the algorithms that were used in the simulation environment, including hash function-SHA256, symmetric en/decryption algorithm-AES, asymmetric en/decryption algorithm-RSA, scalar multiplication-elliptic curve, 256-bit strings. The schemes of [PERSON] et al. [3], [PERSON] and [PERSON] [5], and [PERSON] and [PERSON] [7] involve several symmetric en/decryptions and hash operations. Those schemes even require time-consuming asymmetric en/decryptions. The scheme of [PERSON] et al. [10] also involves several scalar multiplications on elliptic curves and fuzzy extractor operations, whose computational complexities are close to that of asymmetric en/decryptions. Although [PERSON] et al. [8], [PERSON] et al. [10], [PERSON] et al. [13] and [PERSON] et al. [15] provided efficient solutions for healthcare systems, their proposed schemes only consider authentication and key agreement, but their schemes do not consider PHI access and emergency exception. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Phases** & **Registration** & _PHI_ **Uploading** & _PHI_ **Access** & **Emergency-Exception** & **Password-Updating** \\ \hline [PERSON] et al. & \(6T_{A}+T_{S}\) & \(5T_{A}+T_{S}+T_{H}\) & \(4T_{A}+T_{S}\) & \(2T_{A}\) & - \\
[3] & \(2.254\,\mathrm{s}\) & \(1.8874\,\mathrm{s}\) & \(1.5198\,\mathrm{s}\) & \(0.7342\,\mathrm{s}\) & - \\ \hline Ray-Biswas & \(4T_{A}+T_{H}\) & \(5T_{A}+2T_{S}+2T_{H}\) & \(3T_{A}+2T_{S}\) & \(3T_{A}+T_{S}\) & - \\
[5] & \(1.4689\,\mathrm{s}\) & \(1.9939\,\mathrm{s}\) & \(1.2041\,\mathrm{s}\) & \(1.1527\,\mathrm{s}\) & \\ \hline Ray-Biswas [7] & \(4T_{A}+T_{H}\) & \(3T_{A}+4T_{S}+3T_{H}\) & \(T_{A}+4T_{S}+2T_{H}\) & \(T_{A}+4T_{S}+2T_{H}\) & - \\  & \(1.4689\,\mathrm{s}\) & \(1.3084\,\mathrm{s}\) & \(0.5737\,\mathrm{s}\) & \(0.5737\,\mathrm{s}\) & \\ \hline Proposed & \(T_{C}+T_{S}+T_{H}\) & \(10T_{C}+6T_{S}+13T_{H}\) & \(10T_{C}+8T_{S}+14T_{H}\) & \(3T_{C}+4T_{S}+4T_{H}\) & \(4T_{C}+3T_{S}+6T_{H}\) \\ scheme & \(0.0688\,\mathrm{s}\) & \(0.4839\,\mathrm{s}\) & \(0.5877\,\mathrm{s}\) & \(0.2583\,\mathrm{s}\) & \(0.2248\,\mathrm{s}\) \\ \hline
**Phases** & **Registration** & **Authentication and Key Agreement** & **Emergency Exception** & **Password Updating** \\ \hline [PERSON] et al. & \(5T_{H}+1T_{BH}\) & \(28T_{H}+1T_{BH}\) & & - & - \\
[8] & \(0.003\,\mathrm{s}\) & \(0.0145\,\mathrm{s}\) & & \(2T_{H}+2T_{FE}\) \\ \hline [PERSON] et al. & \(1T_{ECM}+1T_{H}+1T_{FE}\) & \(3T_{ECM}+8T_{H}+1T_{FE}\) & - & \(2T_{H}+2T_{FE}\) \\
[10] & \(0.3307\,\mathrm{s}\) & \(0.6644\) & & \(0.3312\,\mathrm{s}\) \\ \hline [PERSON] et al. [13] & \(5T_{H}\) & \(34T_{H}\) & & \(17T_{H}\) \\  & \(0.0025\,\mathrm{s}\) & \(0.0170\,\mathrm{s}\) & - & \(0.0085\,\mathrm{s}\) \\ \hline [PERSON] et al. [15] & \(5T_{H}\) & \(19T_{H}\) & & \(8T_{H}\) \\  & \(0.0025\,\mathrm{s}\) & \(0.0095\,\mathrm{s}\) & - & \(0.0040\,\mathrm{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison. Figure 7 illustrates the response-time statistics of the proposed scheme and other related schemes, which can fully provide the registration phase, PHI uploading phase, PHI access and emergency-exception-handling phase and include the schemes of [PERSON] et al. [3], [PERSON] [5] and [PERSON] [7].