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An Authentication Scheme for Subspace Codes over Network Based on Linear Codes

Network coding provides the advantage of maximizing the usage of network resources, and has great application prospects in future network communications. However, the properties of network coding also make the pollution attack more serious. In this paper, we give an unconditional secure authentication scheme for network coding based on a linear code $C$. Safavi-Naini and Wang gave an authentication code for multi-receivers and multiple messages. We notice that the scheme of Safavi-Naini and Wang is essentially constructed with Reed-Solomon codes. And we modify their construction slightly to make it serve for authenticating subspace codes over linear network. Also, we generalize the construction with linear codes. The generalization to linear codes has the similar advantages as generalizing Shamir's secret sharing scheme to linear secret sharing sceme based on linear codes. One advantage of this generalization is that for a fixed message space, our scheme allows arbitrarily many receivers to check the integrity of their own messages, while the scheme with Reed-Solomon codes has a constraint on the number of verifying receivers. Another advantage is that we introduce access structure in the generalized scheme. Massey characterized the access structure of linear secret sharing scheme by minimal codewords in the dual code whose first component is 1. We slightly modify the definition of minimal codewords. Let $C$ be a $[V,k]$ linear code. For any coordinate $i\in \{1,2,\cdots,V\}$, a codeword $\vec{c}$ in $C$ is called minimal respect to $i$ if the codeword $\vec{c}$ has component 1 at the $i$-th coordinate and there is no other codeword whose $i$-th component is 1 with support strictly contained in that of $\vec{c}$. Then the security of receiver $R_i$ in our authentication scheme is characterized by the minimal codewords respect to $i$ in the dual code $C^\bot$.