# A Medical Image Encryption Scheme for Secure Fingerprint-Based Authenticated Transmission
## 3 Materials and Methods
### Visually Secure Image Decryption Scheme  
The proposed decryption scheme, shown in Figure 2, starts with the visually meaningful encrypted image received. The first step is to extract the EHR previously embedded by the LSB algorithm. The least significant bits are retrieved from each image pixel and decoded to achieve the medical information contained in the EHR. Subsequently, the medical image and the physician's fingerprint feature have been retrieved. The 2D-DWT is performed on the visually meaningful encrypted image to obtain the four sub-bands \(LL^{\prime},\ LH^{\prime},\ HL^{\prime}\) according to Equation (11).  
\[LL^{\prime},\ LH^{\prime},\ HL^{\prime},\ HH^{\prime}=DWT2\left(I^{\prime},\ w\right) \tag{11}\]  
The wavelet \(\mathrm{w}\) is the same wavelet used in the encryption scheme in order to ensure the correct decomposition. The sub-band \(LH^{\prime}\) contains the values of matrix \(U\), the sub-band \(HL^{\prime}\) contains the values of matrix \(V^{H}\), and \(HH^{\prime}\) contains the \(c_{m}\) permuted vector in the position set by the permutation key \(k\). The permuted vector \(c_{m}\) is extracted from  
Figure 5: (**a**) Hybrid asymmetric encryption scheme; (**b**) Hybrid asymmetric decryption scheme.  
following the permutation key \(k\) according to Equation (12). \(k\) is obtained after the hybrid asymmetric decryption scheme described in the previous section.  
\[HH_{k_{ij+1},\;k_{ij+2}}\;\to c_{m_{k_{ij}}} \tag{12}\]  
Equation (12) is the inverse of the embedding process from Equation (6). The obtained vector \(c_{m}\) is the vector permuted, and it is sorted following the first column of \(k\) to achieve the original mapped vector. The original mapped vector contains the mapped vector \(s_{m}\) and the mapped feature vector \(f_{m}\) concatenated. After the de-concatenation process, the singular value vector \(s^{\prime}\) and fingerprint feature vector \(f^{\prime}\) are computed by applying the least-squares method (LSTSQ) to Equations (4) and (5). LSTSQ computes an approximate solution x to a linear matrix equation in the form \(a\;x=b\). So, the least-squares method is used to compute \(s^{\prime}\) and \(f^{\prime}\) that are approximate solutions of Equations (4) and (5), respectively. The coefficient matrixes \(Q\) and \(R\) to solve the Equations (4) and (5) are achieved through \(QR\) decomposition of sub-band \(LL^{\prime}\) as performed in the image encryption scheme.  
The fingerprint feature vector \(f^{\prime}\) is used to ensure the authenticity of the received image by matching \(f^{\prime}\) with the fingerprint feature of the physician sender contained in the hospital database. The vector \(s^{\prime}\) is used to retread the medical image by the SVD reconstruction process according to Equation (13). The vector s' is first converted into a diagonal matrix \(D=\left(d_{i,j}\right)\) of size \(N\times N\) and then the dot product with \(U\) and \(V^{H}\) has been computed.  
\[d_{i,j}=0\;i\
eq j\]  
\[d_{i,j}=s^{\prime}_{i}\;i=j\]  
\[\textit{medical\_image}=U\cdot\left(D\cdot\left(V^{H}\right)\right) \tag{13}\]  
The medical image