Local-Encoding-Preserving Secure Network Coding for Fixed Dimension

In the paradigm of network coding, information-theoretic security is considered in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this paper, we consider the problem of secure network coding for flexible pairs of information rate and security level with any fixed dimension (equal to the sum of rate and security level). We present a novel approach for designing a secure linear network code (SLNC) such that the same SLNC can be applied for all the rate and security-level pairs with the fixed dimension. We further develop a polynomial-time algorithm for efficient implementation and prove that there is no penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. Finally, by applying our approach as a crucial building block, we can construct a family of SLNCs that not only can be applied to all possible pairs of rate and security level but also share a common local encoding kernel at each intermediate node in the network.