## 3 Proposed Scheme
### Decryption Algorithm  
Suppose that the communication channel between the sender and receiver is free of any noise or interference, and the receiver obtains the ciphertext \(C\) of \(n\) characters without any errors or distortion. The decryption process involves the following steps. The initial step is to convert each character of the ciphertext \(C\), which is a string of \(n\) characters, into its corresponding ASCII code; we denote it as \(a_{1}^{\prime},a_{2}^{\prime},\ldots,a_{n}^{\prime}\). This conversion facilitates the reconstruction of the original signal by using the ASCII codes, the decryption keys \(q\) and \(B\), and secret key \(s\) described in Section 3.1. Figure 2: An illustrative example of the proposed encryption algorithm. The next step is to reconstruct the approximation and detail signals from the ASCII codes using Equations (3) and (4), respectively. These equations enable the calculation of the values of the approximation signal \(w_{1}(i)\) and the detail signal \(v_{1,1}(i)\):  
\[w_{1}(i)=94q_{i}+(a^{\prime}_{i}-32), \tag{3}\]  
\[v_{1,1}(i)=(94q_{i+\lceil\frac{n}{2}\rceil}+(a^{\prime}_{i+\lceil\frac{n}{2} \rceil}-32))(-1)^{B_{i}}, \tag{4}\]  
where \(i=1,2,\ldots,\lceil\frac{n}{2}\rceil\). Since we generate the ciphertext using the Type IVa MP wavelets, in the decryption process, we apply the inverse transform process to obtain the original signal. To do so, we use the synthesis process, which starts from the highest-frequency level and works its way down to the original signal. We use Equations (5) and (6) to compute the inverse approximation and detail signal, respectively [14]:  
\[w_{\ell}(c_{\ell+1}i)=w_{\ell+1}(i)-(v_{\ell+1,1}(i)\oplus 0), \tag{5}\]  
\[w_{\ell}(c_{\ell+1}i+j)=v_{\ell+1,j}(i)\otimes w_{\ell}(c_{\ell+1}i+j-1), \tag{6}\]  
where \(\ell\geq 0\), \(c_{\ell}\) represents the channel of a signal at level \(\ell\), \(i=1,2,\ldots,\lceil\frac{n}{2}\rceil\), \(j=1,2,\ldots,c_{\ell+1}-1\), and we use the symbols \(\oplus\) and \(\otimes\) to denote the operations of max-plus addition and max-plus multiplication, respectively. Next, using a MEC \(E_{p,b}\) and a fixed point \(G\) on the curve, we generate a sequence of points \(2G,3G,\ldots,(m-1)G\) on the curve \(E_{p,b}\) using the group law operations. We assume that for each point \(kG\), where \(1\leq k<m\), the \(y\)-coordinate of the point is denoted by \(y_{k}\). Our goal is to find a particular value of \(k_{i}\) for each \(w_{0}(i)\) such that \(k_{i}G=(x_{k_{i}},w_{0}(i))\), where \(w_{0}(i)\) is the resultant signal obtained after applying the modulo operation with respect to \(p\) to the output received during the synthesis process. In other words, we are looking for a value of \(k\) that generates a point on the curve with the same \(x\)-coordinate as \(G\) but with a \(y\)-coordinate equal to \(w_{0}(i)\). Since \(w_{0}(i)\) exists only once on the MEC, there will be only one corresponding value of \(k_{i}\) for each \(w_{0}(i)\). Finally, we subtract \(s\) from each value \(k_{i}\) to obtain the original ASCII codes \(a_{i}\). The original ASCII codes \(a_{i}\) obtained from the previous step can be converted back into their corresponding plaintext characters using the ASCII table. After performing all these steps, we obtain the original plaintext. Figure 3 illustrates an example of the decryption process of the proposed scheme. The decryption process for the ciphertext \"$FOU-O5 aJbSTI3Q2\" involves the following steps. Initially, each character of the ciphertext is converted into its corresponding ASCII value. Subsequently, the approximation signal \(w_{1}(i)\) and the detail signal \(v_{1,1}(i)\) are reconstructed using Equations (3) and (4), respectively, where \(1\leq i\leq 16\). The inverse transform process, also known as the synthesis process, is then applied to obtain the original signal \(w_{0}(i)\) by utilizing Equations (5) and (6).