Secure Linear Network Code

According to secure network coding theory and information theoretic security to the network code, a linear network code with word length of (j+r), j-dimensional message and r-dimensional random key to be secured linearly, illegitimate users will obtain message of length lesser than j+r that is less information than a source message symbol can deliver to legitimate users.

For above network code, each network channel is to of unit capacity and the network code is linear. If a malicious user has access to r-channels then network code is r-secure network code. A network code is w-imperfectly secure if there r<w<j+r channels are accessible to malicious users. A network code is weakly secure if there is no randomness introduced to message and the malicious users get strictly less amount of message than j and the corresponding network code is efficient codes with optimal information rate. 

For linear and unit capacity network channels, entropy of a coded symbol is equivalent to j+r and that eavesdropped by malicious user is with w-channel tapped is equivalently to w, where w is in between r and j+r.