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Network coding provides the advantage of maximizing the usage of networkresources, and has great application prospects in future networkcommunications. However, the properties of network coding also make thepollution attack more serious. In this paper, we give an unconditional secureauthentication scheme for network coding based on a linear code $C$.Safavi-Naini and Wang gave an authentication code for multi-receivers andmultiple messages. We notice that the scheme of Safavi-Naini and Wang isessentially constructed with Reed-Solomon codes. And we modify theirconstruction slightly to make it serve for authenticating subspace codes overlinear network. Also, we generalize the construction with linear codes. Thegeneralization to linear codes has the similar advantages as generalizingShamir's secret sharing scheme to linear secret sharing sceme based on linearcodes. One advantage of this generalization is that for a fixed message space,our scheme allows arbitrarily many receivers to check the integrity of theirown messages, while the scheme with Reed-Solomon codes has a constraint on thenumber of verifying receivers. Another advantage is that we introduce accessstructure in the generalized scheme. Massey characterized the access structureof linear secret sharing scheme by minimal codewords in the dual code whosefirst component is 1. We slightly modify the definition of minimal codewords.Let $C$ be a $[V,k]$ linear code. For any coordinate $i\in \{1,2,\cdots,V\}$, acodeword $\vec{c}$ 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 othercodeword 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 ischaracterized by the minimal codewords respect to $i$ in the dual code$C^\bot$.

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