Multi-receiver Authentication Scheme for Multiple Messages Based on Linear Codes

Abstract

In this paper, we construct an authentication scheme for multi-receivers and multiple messages based on a linear code C. This construction can be regarded as a generalization of the authentication scheme given by Safavi-Naini and Wang [1]. Actually, we notice that the scheme of Safavi-Naini and Wang is constructed with Reed-Solomon 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 [2–6]. For a fixed message base field ${\mathbb F}_q$, our scheme allows arbitrarily many receivers to check the integrity of their own messages, while the scheme of Safavi-Naini and Wang has a constraint on the number of verifying receivers $V\leqslant q$. We further introduce access structure in our scheme. Massey [4] 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 in [4]. Let C be a [V,k] linear code. For any coordinate i ∈ {1,2, ⋯ ,V}, a codeword c in C is called minimal respect to i if the codeword 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 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 $C^\bot$. Finally, we illustrate our authentication scheme based on the elliptic curve codes, a special class of algebraic geometry codes. We use the group of rational points on the elliptic curve to determine all the malicious groups that can successfully make a substitution attack to any fixed receiver.

This paper is supported by the National Key Basic Research Program of China (973 Program Grant No. 2013CB834204), and the National Natural Science Foundation of China (No. 61171082).