# Secure Physical Layer Network Coding versus Secure Network Coding +
## 1 Introduction  
Wireless communication networks with relay nodes have a risk for information leakage to untrusted relays. To resolve this problem, several studies [1, 2, 3, 4, 5, 6] considered the relay terminals untrustworthy based on the result of secure computation-and-forward (CAF) tests [7, 8, 9, 10, 11], which is the main topic of the secure extension of physical layer network coding (PLNC), in short, secure PLNC. However, this type of security can be realized by the secure extension of network coding (NC), in short, secure NC, which is an upper-layered protocol to securely transmit a message via a noiseless network when a part of the edges and/or a part of the intermediate (untrusted) nodes are eavesdropped [12, 13, 14, 15, 16, 17]. Since a wireless channel is disturbed by noise, an error correction needs to be applied to the channel. Then, secure NC is applied to noiseless channels virtually implemented by an error correction. In other words, the error correction and secure NC are separately performed in the different layers under the above scenario. In contrast, since secure PLNC combines both parts, it can be considered as a cross-layer protocol. In order to clarify the advantage of this cross-layer protocol, it is needed to compare secure PLNC with a simple combination of secure NC and error correction over wireless channels, and this comparison has not been studied yet. That is, this type of comparison is strongly required in the viewpoint of wireless communication networks. Secure PLNC is based on PLNC [18, 19, 20], which efficiently transmits the modulo sum of two transmitters' messages via a Gaussian channel. To guarantee the security, the preceding studies [7, 8, 9, 10, 11, 21] invented a secure extension of PLNC, i.e., secure PLNC, which is a schemeto securely transmit a message by a combination of coding operations on nodes when the network is given as a set of noisy channels. Secure PLNC can be classified into two types. In the first case, secure NC is applied to the noiseless CAF process realized by PLNC. This method can be considered as a simple combination of secure NC and PLNC. The other type is a direct method to realize security in the PLNC. The typical example is secure CAF. The code of the latter type cannot be made by such a simple combination. All existing studies [7; 8; 9; 10; 11; 21] belong to the latter case and address only a two-hop relay scheme or its simple extension, the multi-hop relay scheme, which are based on secure CAF to securely transmit the modulo sum of two input message when the channel is a noisy multiple access channel (MAC). Indeed, a secure NC can guarantee the secrecy for the eavesdropper that eavesdrops the channels. Several typical secure NCs cannot guarantee the secrecy when one of the intermediate (untrusted) nodes is eavesdropped. In this way, secure PLNC has an advantage under attacks on intermediate (untrusted) nodes. However, the network models studied in secure NC are more advanced and more complicated, and no study discussed secure PLNC over such typical network models in secure NC. That is, the network models studied in secure PLNC is too limited and too primitive in the comparison with typical network models in secure NC. In other words, no prior study investigated the application of secure PLNC to such typical network models. In order for secure PLNC to overcome secure NC, we need to demonstrate that secure PLNC can be used in more advanced network models. At least, it is needed to study secure PLNC over typical network models in secure NC. Since no existing paper has made the comparison between secure PLNC and the simple combination of secure NC and error correction, this paper aims to make this type of comparison under typical network models in secure NC. That is, this paper is the first study for secure PLNC over typical network models in secure NC, the butterfly network model and the network model composed of three source nodes under certain assumptions for the attack.