Secure error-correcting network codes with side information leakage

Incorporating information security and error correction in network coding, which has various applications in communication theory, for example, secret key sharing through a network, is of special interest and has been widely studied. To make a more intensive analysis of secure error-correcting network codes, we investigate how to secure k source symbols transmission in a multicast network against an adversary that can obtain k₁ source symbols as side information, eavesdrop μ channels and contaminate d channels. We introduce relative network generalised Hamming weight (RNGHW) with network error correction (NEC), or briefly NEC-RNGHW, to measure the equivocation to the adversary. Network generalised singleton bound on NEC-RNGHW is obtained and code constructions achieving the bound are provided. By these constructions, the maximum rate of a secure linear multicast is n - 2d - μ - k₁, where n is the minimum value of the maxflows from a source node to sink nodes. We also characterise the equivocation by the relative profiles of a linear code and a subcode, and tighten the singleton bound on equivocation by the generalised Griesmer bound on relative profiles.