Secure error‐correcting network codes with side information leakage

Abstract

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 k 1 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 − k 1, 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.