# Constellation Encryption Design Based on Chaotic Sequence and the RSA Algorithm
## 4 Experimental Simulation 7
### Constellation Diagram Comparison before and after Encryption  
Attack methods such as modulation identification for digital modulation will cause security risks in the communication process. By changing the shape of the constellation diagram, the purpose of the physical layer encryption is achieved, thereby ensuring the security of the system.  
Common constellation encryption methods, such as phase rotation, amplitude adjustment, or a combination of the two methods, are used here to conduct simulation experiments with Binary Phase Shift Keying (BPSK) as an example. The time domain expression of the BPSK signal is:  
\[e_{2 PSK}(t)=\left\{\begin{array}{cc}A\cos w_{c}t,&\text{The probability is P}\\ -A\cos w_{c}t,&\text{The probability is }(1-\text{P})\end{array}\right. \tag{4}\]  
In the simulation, the chaotic sequence is used as the random phase key and random amplitude key, and the shape of the constellation diagram is affected by encrypting the original sequence. Figure 8 is the simulation result of only encrypting the BPSK constellation diagram by using the random phase key. It can be seen from the diagram that the constellation points are all distributed on a circle with a radius of one. Under the condition of the same signal-to-noise ratio, the size of the selected phase interval is related to the bit error performance of the system.  
Figure 9 is the simulation result of encrypting the BPSK constellation diagram using the random phase key and the random amplitude key at the same time. It can be seen that the constellation points of the encrypted constellation diagram are evenly distributed in a circle with a radius of about one, because the constellation the shape of the graph is further unrecognizable, so the illegal interceptor cannot obtain communication parameters through attack methods, such as constellation diagram clustering identification, which greatly improves the security of the communication process.  
Figure 8: BPSK constellation encryption algorithm. (**a**) Phase rotation and (**b**) rotation and amplitude adjustment.  
The constellation map encryption method, based on the random transformation of phase and amplitude, makes the distribution of constellation points random, and the cracking cost of the digital modulation link at the illegal receiver is greatly increased, so the security of the communication process is greatly improved. However, it also has some defects. This method cannot guarantee the judgment performance of the legitimate communication receiver, and even has a negative impact. The fundamental reason is that the closer the constellation point is to the center of the circle, the easier it is to be interfered with by noise in the process of passing through the transmission channel, which will lead to difficulty in the receiver's decision, and then will increase the bit error rate of the decision.  
The encryption algorithm proposed in this paper uses the secret key sequence to map the real constellation points, so that the encrypted constellation points no longer have the meaning of traditional constellation points. For the collaborator, the encrypted constellation point can be decrypted using the same chaotic sequence as the sender and the encryption rules of this algorithm, so that it is in the real position, and then the signal is subsequently processed.  
As can be seen from Figure 9, there is no difference in the shape of the constellation diagram before and after encryption. The advantage of this is that the communication interceptor can determine the modulation method used by the sender through the constellation diagram identification or even machine learning, butabt they could not observe any trace of encryption from the constellation map.