# A P2P Scheme for Debating and Voting with Unconditional Flexibility
## 4 Analysis
### Security Requirements
#### 4.3.1 Privacy  
In e-voting systems, any traceability between the user and their vote must be removed. Clearly, if the rest of the users reveal their vote, the latter would be exposed. When this is the only case in which a vote can be associated with its user, the highest level of privacy is achieved, called'maximum privacy'. However, this scheme protects messages without constraints until they are published. Messages could provide information about users in different ways (style of text, content of messages, alias profile, Italian attacks, etc.). Depending on the context, this data leakage is possible or not. In a open debate users can include revealing details in their opinion or proposals. Nevertheless, a simple yes/no vote cannot reveal anything about its author except the case of maximum privacy. Therefore, the following property is stated.  
Under the conditions of the security model and as long as the content of the messages does not reveal anything about the users, any two of them are indistinguishable in phase 2.  
In this context, all the information that adversaries can accumulate is the following: a list of blind messages \((m_{ij}=Blind(a_{i},r_{ij},K_{j}))\) that are linked with their users, a list of blind signatures, a list of signed aliases \((a_{i})\), and the contributions (proposals, votes) signed by each alias. Since the premise of the proposition discards the information obtained from the contributions, only the information related to the alias itself remains. Distinguishing one user from another in phase 2 means distinguishing their aliases. However, considering all the information available to adversaries, there is only one element in which the user and alias are involved, which is the blinded message. If adversaries manage to distinguish two users, \(U_{1}\) and \(U_{2}\), then they are able to guess the relationship between user and alias, which implies that they know two sets of values, \(r_{1j}\) and \(r_{2j}\), such that:  
\[m_{1j} = Blind(a_{1},r_{1j},K_{j})=a_{1}\cdot r_{1j}^{K_{j}}(mod\;\;n_{j})\] \[m_{2j} = Blind(a_{2},r_{2j},K_{j})=a_{2}\cdot r_{2j}^{K_{j}}(mod\;\;n_{j})\]  
The other option is that they are able to show that there is no set of values \(s_{1j}\) capable of satisfying \(m_{1j}=Blind(a_{2},s_{1j},K_{j})\). However, given \(r_{ij}\), users could have chosen \(s_{ij}\), defined as \(s_{ij}=\left(a_{h}^{-1}\cdot a_{i}\right)^{K_{j}^{-1}}\cdot r_{ij}\) for any other alias \(h\). Specifically, if the values 1 and 2 are chosen for \(i\) and \(h\):  
\[Blind(a_{2},s_{1j},K_{j})=a_{2}\cdot s_{1j}^{K_{j}}=a_{2}\cdot((a _{2}^{-1}\cdot a_{1})^{K_{j}^{-1}}\cdot r_{1j})^{K_{j}}(mod\;\;n_{j})=Blind(a _{1},r_{1j},K_{j})=m_{1j}\] \[Blind(a_{1},s_{2j},K_{j})=a_{1}\cdot s_{2j}^{K_{j}}=a_{1}\cdot( (a_{1}^{-1}\cdot a_{2})^{K_{j}^{-1}}\cdot r_{2j})^{K_{j}}(mod\;\;n_{j})=Blind(a _{2},r_{2j},K_{j})=m_{2j}\]  
In other words, depending on what values users \(U_{1}\) and \(U_{2}\) would have chosen, they could very well be the authors of either alias \(a_{1}\) or alias \(a_{2}\). There is no unique mathematical link between blinded messages and aliases. Consequently, adversaries have no way to distinguish one user from another with the information they have.  
As a result of this proposition, if content of messages do not reveal its authorship (e.g., yes/no votes) maximum privacy is preserved.