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Peer-to-Peer Networking and Applications

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07-01-2021

A certificateless linearly homomorphic signature scheme for network coding and its application in the IoT

Authors: Bin Wu, Caifen Wang, Hailong Yao

Published in: Peer-to-Peer Networking and Applications | Issue 2/2021

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Abstract

Network coding is an effective method to optimize network throughput and improve routing reliability, and has been widely used in a decentralized Internet of Things system. However, the packet-mixing property of network coding renders transmission susceptible to pollution attacks, which may prevent the reconstruction of the original file. A homomorphic signature scheme is a powerful tool that enables network coding to combat pollution attacks. Although a series of homomorphic signature schemes already exists, no construction has been proposed to support both homomorphic network coding signatures and the certificateless characteristic. In this paper, we construct a certificateless linearly homomorphic signature scheme for network coding, thus avoiding the disadvantages of certificate management and key escrow problems. We then prove the security of the scheme in a random oracle model against an adaptively chosen dataset attack under two types of adversaries. Moreover, performance analysis results show that our scheme has a lower communication overhead and enjoys a comparable computation cost with related schemes.

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Since we embed the hard problem in the term 1 of Eq. 1, that is, Q_ID = g^b, P_pub = g^a. In order to successfully answer the signing query, our idea is to eliminate item 1 by carefully setting the values of T_i(i ∈ [N = n + m]) and U while ensuring that the values of T_i and U are random (\( T_{i}=(\frac {g^{u_{i}}}{Q_{ID}} )^{r^{-1}}\), \( U=P^{r}_{pub}=(g^{a})^{r} \)). The item 2^′ in (2) is further arranged to obtain items 21′ and 22′ in (3). It is not difficult to find that item 21′ can eliminate item 1, because item 21′ and item 1 are inverses of each other in group \(\mathbb { G}_{2} \).

In this case, ID ≠ ID_k, then \( Q_{ID}=g^{w_{ID}} \), where w_ID is a known random number, so the required values generated in the process of various queries can be directly brought into the signature algorithm of the proposed scheme to obtain the signature.

Here, the expression of hash value \( T^{*}_{i} \) is different from that of hash value T_i in signing queries. This is because if an adversary outputs a type 1 forgery, the identifier τ^∗ never appears in the signing query, so the hash value \( T^{*}_{i} \) corresponding to the identifier τ^∗ come from H₁ queries.

In detail, \( \boldsymbol {\alpha }\cdot \boldsymbol {v}_{i}= (\alpha _{1}, \cdots , \alpha _{n}, \beta _{1}, \cdots , \beta _{m})\cdot (v_{i1}, \cdots , v_{in}, \underbrace { 0,\cdots , 1}_{i} , \cdots , 0)= \alpha _{1}v_{i1}+ \cdots + \alpha _{n}v_{in}+\beta _{i} = \alpha _{1}v_{i1}+ \cdots + \alpha _{n}v_{in}+\left (-\sum \limits ^{n}_{j=1}\alpha _{j}v_{ij}\right )=0. \) In particular, since we set \( (T_{1}, \cdots , T_{n}, T_{n+1}, \cdots , T_{n+m})=((g^{a} )^{\alpha _{1}}, (g^{a} )^{\alpha _{n}}, (g^{a} )^{\beta _{1}}, \cdots , (g^{a} )^{\beta _{m}} ) \), we have \( \underset {j\in {[N]}}{\prod }T_{j}^{v_{ij}}= \underset {j\in {[n]}}{\prod }(g^{a})^{\alpha _{j}v_{ij}}\cdot \underset {j\in {[m]}}{\prod }(g^{a})^{\beta _{j}v_{i,(n+j)}}=(g^{a})^{\boldsymbol {\alpha }\cdot \boldsymbol {v}_{i}}=1 \)

Since we embed the hard problem in the term

https://static-content.springer.com/image/art%3A10.1007%2Fs12083-020-01028-8/MediaObjects/12083_2020_1028_Fige_HTML.gif

of Eq. 6. In order to successfully answer the signing query, our idea is to carefully set the value of T_i(i ∈ [N = n + m]) such that \((\underset {j\in {[N]}}{\prod }T_{j}^{v_{ij}})^{br}=1 \), while ensuring that the values of T_i are random. As we know from the previous, the vector α formed by the exponents of T_i(i ∈ [N = n + m]) satisfies α ∈ V^⊥, so term

https://static-content.springer.com/image/art%3A10.1007%2Fs12083-020-01028-8/MediaObjects/12083_2020_1028_Figf_HTML.gif

in Eq. 6 is equal to 1, that is, \( (g^{ab})^{(\boldsymbol {\alpha }\cdot \boldsymbol {v}_{i})r}=1 \).

Since we embed the hard problem in the term 1 of Eq. 6, that is, T = (g^b)^t, PK_ID = g^a. In order to successfully answer the signing queries, our idea is to eliminate item 1 by carefully setting the values of T_j(j ∈ [N = n + m]) and U while ensuring that the values of T_j and U are random (\( T_{j}=(\frac {g^{u_{i}}}{T} )^{r^{-1}}\), \( U=PK^{r}_{ID}=(g^{a})^{r} \)). The item 2^′ in (11) is further arranged to obtain items 21′ and 22′ in (12). It is not difficult to find that item 22′ can eliminate item 1, because item 22′ and item 1 are inverses of each other in group \(\mathbb { G}_{2} \).

In this case, ID ≠ ID_k, then \( PK_{ID}=g^{x_{ID}} \), where x_ID is a known random number, so the required values generated in the process of various queries can be directly brought into the signature algorithm of the proposed scheme to obtain the signature.

In detail, \( \boldsymbol {\alpha }\cdot \boldsymbol {v}_{i}= (\alpha _{1}, \cdots , \alpha _{n}, \beta _{1}, \cdots , \beta _{m})\cdot (v_{i1}, \cdots , v_{in}, \underbrace { 0,\cdots , 1}_{i} , \cdots , 0)= \alpha _{1}v_{i1}+ \cdots + \alpha _{n}v_{in}+\beta _{i} = \alpha _{1}v_{i1}+ \cdots + \alpha _{n}v_{in}+(-\sum \limits ^{n}_{j=1}\alpha _{j}v_{ij})=0. \) In particular, since we set \( (T^{\prime }_{1}, \cdots , T^{\prime }_{n}, T^{\prime }_{n+1}, \cdots , T^{\prime }_{n+m})=((g^{b} )^{\alpha _{1}}, (g^{b} )^{\alpha _{n}}, (g^{b} )^{\beta _{1}}, \cdots , (g^{b} )^{\beta _{m}} ) \), we have \( \underset {j\in {[N]}}{\prod }(T^{\prime }_{j})^{v_{ij}}= \underset {j\in {[n]}}{\prod }(g^{b})^{\alpha _{j}v_{ij}}\cdot \underset {j\in {[m]}}{\prod }(g^{b})^{\beta _{j}v_{i,(n+j)}}=(g^{b})^{\boldsymbol {\alpha }\cdot \boldsymbol {v}_{i}}=1 \)

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Title: A certificateless linearly homomorphic signature scheme for network coding and its application in the IoT
Authors: Bin Wu
Caifen Wang
Hailong Yao
Publication date: 07-01-2021
Publisher: Springer US
Published in: Peer-to-Peer Networking and Applications / Issue 2/2021
Print ISSN: 1936-6442
Electronic ISSN: 1936-6450
DOI: https://doi.org/10.1007/s12083-020-01028-8

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