# Secure Physical Layer Network Coding versus Secure Network Coding +
## 1 Introduction
Unfortunately, secure PLNC has a completely different mathematical structure from the simple combination of secure NC and error correction. Hence, it is quite difficult to construct a general theory to compare them. Due to this reason, we address two typical network models in the area of secure NC, the butterfly network [22] and a network with three source nodes, which is a special network model studied in [23].Then, we make the above comparison numerically over these two networks. Indeed, many existing studies [7; 8; 9; 10; 11] for secure PLNC employed lattice codes. Only the reference [21] studied it with BPSK modulation. Notice that the QPSK modulation can be considered as twice the use of the BPSK modulation. For PLNC, references [24; 25; 26; 27] discussed CAF based on lattice codes. Indeed, \(2^{n}\)-phase shift keying (PSK) modulation works for practical systems such as conventional satellite communications with LDPC codes [28]. In addition, references [29; 30] demonstrated the efficiency of the CAF scheme composed of binary LDPC codes under the BPSK modulation. Reference [31] compared the BPSK modulation and the method based on lattice codes for CAF. Hence, to adopt the existing communication system, we focus on the BPSK modulation. Although, this paper is the journal version of the preceding conference paper [32], this paper is different from the conference version as follows. First, the conference version gave the secure NC protocol only when \(q\) is not a power of 2. This paper additionally gives the secure NC protocol when \(q\) is a power of 2 (not 2). This kind of extension enables us to consider the new protocol given in Section 3.3.2. Second, the conference version discussed only one type of secure NC protocol. This paper additionally considers another type of protocol in secure PLNC (See Sections 3.3.2 and 4.2.2). Totally, this paper discusses two types of protocols in secure PLNC. This additional protocol clarifies the merit of use of CAF. Third, the conference version compared the number of times slots only for two cases: secure NC without Gaussian MAC and secure PLNC with Gaussian MAC. Also, it did not considered the protocol in Sections 3.3.2 and 4.2.2. This paper additionally considers another case: secure NC with Gaussian MAC. Further, this version discussed the transmission time by considering the information transmission rate when the asymptotically best code is employed. To make this additional comparison, analytical discussions are newly made in this version by using the mutual information. Also, this version newly contains numerical graphs (Figures 3 and 5) for this comparison. Due to this additional comparison, we can compare the transmission time. The rest of this paper is organized as follows. First, Section 2 reviews the results in CAF and secure CAF, which is a typical example of secure PLNC. Next, Section 3 considers how secure communication can be implemented over the butterfly network based on secure PLNC. Finally, Section 4 discusses how secure communication can be implemented over a network with three source nodes based on secure PLNC. That is, Sections 3 and 4 are devoted to our contribution.