Complex Modified Projective Difference Function Synchronization of Coupled Complex Chaotic Systems for Secure Communication in WSNs  
[PERSON]  
1 School of Control Science and Engineering, Shandong University, Jinan 250061, China; [EMAIL_ADDRESS]  
[PERSON]  
1 School of Control Science and Engineering, Shandong University, Jinan 250061, China; [EMAIL_ADDRESS]  
[PERSON]  
2 Department of Electrical Engineering and Automation, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China; [EMAIL_ADDRESS]  
[PERSON]  
3 School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China; jianqce  
and block ciphers using chaotic systems [11; 12; 13]. In traditional chaotic masking communication, chaotic signals are employed as carriers produced by the master systems to obscure the message signals. Then, the message signals are retrieved through chaos synchronization, for example, through complete synchronization (CS) between the master system at the transmitting end and the slave system at the receiving end. Hence, the type of synchronization is critical. New types of synchronization for CVCSs have attracted extensive increasing attention recently [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31], including phase synchronization (PHS) [14], complete synchronization (CS) [15], anti-synchronization (AS) [16; 17], lag synchronization (LS) [18], anti-lag synchronization [19], modified function projective synchronization (MFPS) [20], full-state hybrid synchronization [21], modified projective phase synchronization (MPPS) [22], hybrid MFPS [23], complex modified projective synchronization (CMPS) [24], time-delay chaotic system [25; 26], complex function projective synchronization (MFPS) [27; 28], complex anti-synchronization (CAS) [29], difference function synchronization (DFS) [30; 31], etc. Synchronization for CVCSs involves designing a controller that synchronizes the states variables of two chaotic systems synchronized, which is called a synchronization controller. More accurately, synchronization is generally for a chaotic system while control involves any system. Over the past twenty years, the field of the synchronization and control of CVCSs has grown rapidly, and so have the communication schemes for CVCSs. In 2014, [PERSON] and [PERSON] [6] developed complex function projective synchronization (CFPS) along with a new chaos communication scheme. In 2015, [PERSON] et al. applied a self-delay synchronization scheme [25] and the CS of coupled multiple time-delay CVCSs [26] to communication systems. In 2018, [PERSON] and [PERSON] [29] presented CAS and discussed the corresponding communication schemes. It is worth noting that as the amplitude of the signal generated by the sender approaches zero, the denominator in CFPS also approaches zero and will, therefore, affect synchronization and message recovery. [PERSON] et al. proposed fractional difference function synchronization in 2019 [30]. The type of synchronization in these two papers is modified difference function synchronization (MDFS). However, these schemes have an amplitude limitation of chaos communication, which limits its application in general chaos communication. In 2021, [PERSON] et al. proposed modified fractional projective difference function synchronization (MFPDFS) for a time-delay fractional complex chaotic system [31]. Inspired by MFPDFS, we propose a complex modified projective difference function synchronization (CMPDFS) for common complex chaotic system and address issues originating from the amplitude limitation. Complete synchronization (CS), phase synchronization (PHS), projective synchronization (PS), modified projective synchronization (MPS), and modified difference function synchronization (MDFS) are all special cases of CMPDFS. CMPDFS combines the advantages from CMPS and MDFS, because CMPS can adjust the amplitude of chaotic signals according to plaintext information to be transmitted confidentially while MDFS avoids the denominator being zero.