# Analysis and Research on Secondary LT Coding Anti-Eavesdropping Scheme Based on LT Code Degree-1
## 3 Program Performance Analysis
### Decoding Complexity  
(1) Double LT fountain code decoding complexity  
The fountain code uses the \(\mu(d)\) distribution to obtain the coding matrix and the XOR computation to obtain the coded symbols, and the number of XOR calculations determines the size of the complexity of the compiled code. A computational approach to coding complexity is suggested in the literature [22]. The decoding complexity \(E\) is therefore correlated with the average degree \(\overline{d}\) and the number of decoding symbols \(m_{1}\). The following is the mathematical expression:  
\[E=m_{1}\cdot(\overline{d}-1) \tag{11}\]  
Due to the low decoding overhead of the fountain code, the number of decoded symbols \(m_{1}=k+e\), and \(e\) can be an arbitrarily small number greater than zero. Then, from Equation (11), the decoding complexity \(E_{LT}\) of LT code is obtained:  
\[E_{LT}=(k+e)\Big{(}\overline{d_{LT}}-1\Big{)}\approx k\bigg{(}o\left(\ln\frac {k}{\delta}\right)-1\bigg{)} \tag{12}\]  
The \(E_{double\_LT}\) complexity in the double LT approach is divided into two sections, LT-a code and LT-b code:  
\[E_{double\_LT}=E_{LT-a}+E_{LT-b} \tag{13}\]  
LT-a, the first LT encoding process needs to complete decoding for \(k\) symbols, and the decoding complexity \(E_{LT-a}\) can be obtained from Equation (12):  
\[E_{LT-a}=k(\overline{d_{LT}}-1) \tag{14}\]  
LT-b is the secondary LT encoding for the \(k\cdot u(1)\) degree-1 symbols in LT-a after extraction, and the decoding complexity \(E_{LT-b}\) can be obtained from Equation (12):  
\[E_{LT-b}=k\cdot u(1)(\overline{d_{LT-b}}-1) \tag{15}\]  
where \(\overline{d_{LT-b}}\) denotes the average degree of participation of \(k\cdot u(1)\) symbols in the LT-b code and \(u(1)\) is the probability of the degree-1 symbols. In summary, substituting (14) and (15) into Equation (13) yields the decoding complexity \(E_{double\_LT}\):  
\[ \begin{array}{ll}E_{double\_LT}&=E_{LT-a}+E_{LT-b}\\ &=k\Big{(}\overline{d_{LT_{(k)}}}-1\Big{)}+k\cdot u(1)\Big{(}\overline{d_{LT_{ (k\cdot u(1))}}}-1\Big{)}\\ &=k\Big{(}o\left(\ln\frac{k}{\delta}\right)-1\Big{)}+k\cdot u(1)\Big{(}o\left( \ln\Big{(}\frac{k\cdot u(1)}{\delta}\Big{)}\right)-1\Big{)}\end{array}  \tag{16}\]  
Obviously, from Equations (12) and (16), it can be seen that the decoding complexity of the double LT method is increased by \(o\Big{(}k\cdot u(1)\ln\Big{(}\frac{k\cdot u(1)}{\delta}\Big{)}\Big{)}-k\cdot u (1)\) compared to the LT code, and increases with the increase in the \(k\) value. (2) Comparison of decoding complexity of double LT fountain code with two other fountain codesFor the DEMR-LT fountain code [21], a twice LT code cascade coding was performed, and the decoding complexity is expressed as follows:  
\[ \begin{array}{ll}E_{DEMR\_LT}&=E_{LT}\cdot E_{LT}\cdot\frac{k}{(1-\frac{k}{P_{ AB}})}\\ &=k\left(\overline{d_{LT}}-1\right)\cdot k\left(\overline{d_{LT}}-1\right) \end{array}  \tag{17}\]  
Equation (17) shows that \(E_{DEMR\_LT}\) increases with the increase in \(P_{AB}\). When \(P_{AB}=0\), Equation (17) shows that:  
\[E_{DEMR\_LT}=\left(k\cdot\left(\overline{d_{LT}}-1\right)\right)^{2} \tag{18}\]  
In Equation (18), \(k\left(\overline{d_{LT}}-1\right)\) denotes the LT coding complexity, i.e., the number of XOR calculations during the coding process, \(k\) is chosen to be large in LT coding, and the number of XOR calculations is much larger than 2, which can be obtained from (18). \[E_{DEMR\_LT}\gg 2k\left(\overline{d_{LT}}-1\right) \tag{19}\]  
In Equation (16), \(u(1)\ll 1\), then, \(k\cdot u(1)\left(\overline{d_{LT}}_{(k\cdot u(1))}-1\right)\ll k\left( \overline{d_{LT}}_{(k)}-1\right)\). \[ \begin{array}{ll}E_{double\_LT}&=k\left(\overline{d_{LT}}_{(k)}-1\right)+k \cdot u(1)\left(\overline{d_{LT}}_{(k\cdot u(1))}-1\right)\\ &\ll 2\left(\overline{d_{LT}}_{(k)}-1\right)\end{array}  \tag{20}\]  
Comparing Equations (19) and (20), it is clear that \(E_{double\_LT}\ll E_{DEMR\_LT}\) and the double LT fountain code is much smaller than the decoding complexity of the DEMR-LT fountain code.