# Analysis and Research on Secondary LT Coding Anti-Eavesdropping Scheme Based on LT Code Degree-1
## 3 Program Performance Analysis
### Security Capacity  
The number of symbols sent by the source is \(k\), and the channel deletion probability of the main channel and the wiretap channel are \(P_{AB}\) and \(P_{AE}\), respectively, which are transmitted through a binary memoryless deletion channel. The channel security capacity \(C_{s}\) is described as maximizing the information transmission rate while ensuring security and is denoted as \(C_{s}=\max R\).  
As can be seen from 2.3, the double LT scheme is divided into two decoding stages. The first phase is the decoding of degree-1 symbols, in which [PERSON] encodes degree-1 symbols to create the LT-b code (U-sequence) and sends it to [PERSON]. [PERSON] then receives the source symbols to obtain \(U_{Bob}\). According to [PERSON]'s Information Theory [1], [PERSON]'s information transmission rate \(R_{U\_Bob}\) is:  
\[R_{U\_Bob}=I(U;U_{Bob})=1-P_{AB} \tag{6}\]  
In the second stage, [PERSON] encodes \(k\) symbols to form an LT-a code (\(S_{U}\) sequence) and sends it to [PERSON], and by utilizing the degree-1 symbol sequences that have been decoded in the first stage to be decoded together, [PERSON]'s information transmission rate \(R_{S\_[PERSON]}\) is satisfied:  
\[\begin{array}{ll}R_{S\_[PERSON]}&=I(S_{U};S_{U\_[PERSON]})\times R_{U\_[PERSON]}\\ &=I(S_{U};S_{U\_[PERSON]})\times(1-P_{AB})\\ &=(1-P_{AB})^{2}\end{array} \tag{7}\]  
Similarly, the eavesdropper eavesdrops through the eavesdropping channel to get [PERSON]'s information transmission rate \(R_{S\_[PERSON]}\) as:  
\[R_{S\_[PERSON]}=(1-P_{AE})^{2} \tag{8}\]  
Setting parameter \(S_{r}=(1-P_{AE})/(1-P_{AB})\), the safe rate \(R\) of the system is satisfied:  
\[\begin{array}{ll}C_{S\_{double\_LT}}&=\max R=R_{Bob}-R_{Eve}\\ &=(1-P_{AB})^{2}-(1-P_{AE})^{2}\\ &=(1-P_{AB})^{2}-S_{r}^{2}(1-P_{AB})^{2}\\ &=(1-P_{AB})^{2}(1-S_{r}^{2})\end{array} \tag{9}\]  
Due to \((1-P_{AB})^{2}>0\), achieving \(C_{s}>0\) merely requires completing \(S_{r}{}^{2}<1\). This ensures that the security capacity is greater than zero, thus guaranteeing the secure transmission of information. Then, to satisfy \(S_{r}{}^{2}<1\), that is, \(S_{r}<1\), that is, \(1-P_{AE}<1-P_{AB}\), yields \(P_{AB}<P_{AE}\). Therefore, in summary, we only need to satisfy that \(P_{AB}<P_{AE}\), the legal channel environment is superior to the eavesdropping channel, and then necessarily \(C_{s}>0\), that is, the security capacity exists and is higher.  
Next, the security capacity of the double LT code is simplified:  
\[\begin{array}{ll}C_{S\_{double\_LT}}&=(1-P_{AB})^{2}-(1-P_{AE})^{2}\\ &=((1-P_{AB})+(1-P_{AE}))\times(1-P_{AB}-(1-P_{AE}))\\ &=(P_{AE}-P_{AB})(2-P_{AB}-P_{AE})\end{array} \tag{10}\]  
Since better channel conditions are taken into account in this paper's practical applications, both \(P_{AB}\) and \(P_{AE}\) take values less than 0.5; then, \(2-P_{AB}-P_{AE}>1\), which results in \(C_{S\_{double\_LT}}>C_{S\_{UT}}\). Even though the channel state condition of the eavesdropping channel is better than that of the legitimate channel, it still satisfies the larger security capacity ofthe double LT method, which can theoretically be shown to have a larger security capacity than the traditional LT code.