## 7 Safety Analysis  
The necessary definitions for proving the security of the message hash chain are given below. **Definition 1**. : _If there is always a \(\mu_{0}\) for all \(e\) such that \(\varepsilon(\mu)<\frac{1}{p^{\mu}}\) when \(\mu>\mu_{0}\), then \(\varepsilon(\mu)\) is said to be a negligible value with \(\mu\) as the parameter._  
**Definition 2**. : _Note that \(H\) is the set of all hash functions, and \(h\) is a hash function. If \(h\) can find \(a\), \(b\), \(a\
eq b\), \(h(a)=h(b)\) in polynomial time, then it is considered that \(h\) will have a hash collision. For \(\forall h\in H\), if the probability of hash collision in \(h\) is equal to \(\varepsilon(\mu)\), i.e., the probability of hash collision in \(h\) is negligible, then \(H\) is a non-collision hash function set._  
**Definition 3**. : _Denote a digital signature scheme triple \((Gen,Sig,Ver)\), which satisfies:_  
* _Gen represents the asymmetric key generation algorithm. For the key pair_ \((ek,dk)\)_,_ \(ek\) _is the private key of the signature, and_ \(dk\) _is the public key of the signature._
* _Sig is the signature algorithm of the digital signature scheme. For the communication transmission sequence_ \(p_{1},p_{2},p_{3},\ldots\)_, there is_ \(Sig(ek,HC_{i},HC_{i},HC_{i+1},\ldots,HC_{i+q})=s_{i},s_{i+1},\ldots\)_,_ \(s_{i+q}\) _on a certain segment of data transmitted, where_ \(q\) _is a positive integer, and_ \(HC_{i}\)_,_ \(HC_{i+1},\ldots\)_,_ \(HC_{i+q}\) _come from_ \(p_{i},p_{i+1},\ldots,p_{i+q}\)_._
* _Ver is the verification algorithm of the digital signature scheme. For the digital signature_ \(s_{i},s_{i+1},\ldots,s_{i+q}\) _of a certain segment of data and the_ \((ek,dk)\) _generated by Gen, there is always_ \(Ver(dk,s_{i},s_{i+1},\ldots,s_{i+q})=1\)_._  
**Definition 4**. : _For the digital signature scheme \((Gen,Sig,Ver)\), if only the \(dk\) cannot forge the \(ek\) of the scheme in polynomial time, then the scheme is secure._  
**Definition 5**. : _The message hash chain verification scheme \((MHC,Gen,Sig,Ver)\) takes the digital signature scheme \((Gen,Sig,Ver)\) as an option. On the basis of \((Gen,Sig,Ver)\), it also satisfies:_\(MHC\) _is the construction algorithm of the message hash chain. For the transmission sequence_ \(p_{i},p_{i+1},\ldots,p_{i+q}\)_, there is_ \(MHC(m_{i},m_{i+1},\ldots,m_{i+q})=HC_{i},HC_{i+1},\ldots,HC_{i+q}\)_, where_ \(m_{i},m_{i+1},\ldots,m_{i+q}\) _come from_ \(p_{i}\)_,_ \(p_{i+1},\ldots,p_{i+q}\)_, respectively,._ _After receiving the sequence_ \(p_{i},p_{i+1},\ldots,p_{i+q}\) _and the_ \(HC_{i},HC_{i+1},\ldots,HC_{i+q}\) _encapsulated in it, the receiver also constructs a message hash chain node_ \(MHC(m_{i},m_{i+1},\ldots,m_{i+q})\)__\(=HC_{i}^{\prime},HC_{i+1}^{\prime},\ldots,HC_{i+q}^{\prime}\) _for the received sequence through_ \(MHC\)_, and there is_ \(\forall\delta\)_,_ \(\delta\in(i+1,\ldots,i+q)\)_,_ \(HC_{\delta}=HC_{\delta}^{\prime}\)_._  
Theorem 3.1: _The messages between two messages authenticated by chain signature also have authenticity and non-repudiation._  
\((Gen,Sig,Ver)\) is a secure digital signature scheme, \(h\) is a known hash function, and the probability of hash collision at \(h\) is less than \(\varepsilon(\mu)\), i.e., \(h\) is a non-collision hash function. In this case, if the digital signatures of \(p_{\alpha}\) and \(p_{\beta}\) can be verified successfully and satisfy \(\alpha<\alpha+1<\beta\), then \(\forall\delta\), \(\alpha<\delta<\beta\), \(p_{\delta}\) can verify their authenticity and non-repudiation through message hash chain verification. Proof of Theorem 3.1: It is assumed that the message hash chain verification scheme \((MHC,Gen,Sig,Ver)\) is insecure.