Secure Network Coding For Wiretap Networks

This thesis studies secure network coding for wiretap networks, in which linear network codes should be considerably designed to obtain perfect secrecy against the wiretapper who can eavesdrop some arbitrarily selected edges.Firstly, we consider four kinds of wiretap networks, including wiretap net-works of typeâ…¡, erroneous wiretap networks, wiretap networks with side in-formation to the wiretapper and multi-source wiretap networks. In the first three network models, we study the security performances of the linear network codes, derive the necessary and sufficient conditions for the perfect secrecy, and give construction methods of secure linear network codes. For the multi-source wiretap network, we obtain a general necessary and sufficient secure condition for the network to be perfect secure.Secondly, in network coding settings, we generalize the classic definitions of generalized Hamming weight and relative generalized Hamming weight to the network generalized Hamming weight (NGHW) and relative network gen-eralized Hamming weight (RNGHW), respectively. NGHW completely charac-terizes the security performance of the linear network codes in wiretap networks of typeâ…¡, while RNGHW completely characterizes the security performance of the linear network codes in erroneous wiretap networks with nested coset cod-ing schemes and wiretap networks with side information to the wiretapper. The basic properties and generalized Singleton bounds of the two concepts can help to construct optimal linear network codes with perfect secrecy.Finally, we investigate the intrinsic relations between secure network cod-ing for wiretap networks and coherent network error correction. We prove that the first NGHW of linear network codes equals the minimum distance of linear network error correction codes in a specific setting. By this equality and the ex- isting coding bounds for linear network error correction codes, we can readily get the Hamming bound, Sphere-packing bound and Gilbert-Varshamov bound for secure network codes.