# A MILP Model for a Byzantine Fault Tolerant Blockchain Consensus
## 5 Conclusions and Future Works  
This work presents mixed-integer linear programming for exact adversarial model on a blockchain consensus. Blockchains are decentralized technologies that deal with many sorts of attacks, being both intentional (Byzantine) and non-intentional (general failures). We consider an application over the Neo Blockchain consensus, dBFT, which was inspired by the classical algorithm PBFT. Both PBFT and dBFT require that the number of Byzantine agents to be limited to \(f\), from a set of \(N=3f+1\) consensus nodes. Since Byzantine agents may also provoke delays on the decentralized peer-to-peer network, we also model these delays in minimization or maximization scenarios, namely P1-P7, considering distinct constraints over message reception. We propose a set of instances with different sizes, varying number of nodes, and maximum time allowed for the system to exchange messages, which leads to the discovery of practical limitations on the power of open-source and commercial solvers to deal with these MILP problems. Finally, we present graphical tools to analyze the scenarios, demonstrating how Byzantine agents can explore every _space_ of the MILP model, according to its own _intention_. To the best of our knowledge, this is the first work to successfully explore MILP exact models to find weakness on a blockchain consensus system, which we believe can help in the design of future consensus for large-scale public blockchain projects.  
Several extensions are possible from this work, such as finding better algorithms to resolve MILP models of medium and large sizes (due to solver capabilities, we were limited to cases with few nodes and small time intervals); exploring how every pack of constraints affects the problem-solving capabilities of the solvers (including internal heuristics); extending current modeling to other recent blockchain consensus systems (including newer generations of Neo Blockchain consensus); developing real-time integration of the drawing tools with dynamic consensus simulations under the described  
\begin{table}
\begin{tabular}{c c c c} \hline _Instance_ & \(N\) & \(t^{max}\) & **Optimal** \\ \hline dbft-2-4-5 & 4 & 5 & yes \\ dbft-2-4-10 & 4 & 10 & yes \\ dbft-2-4-15 & 4 & 15 & yes \\ dbft-2-7-5 & 7 & 5 & yes \\ dbft-2-7-10 & 7 & 10 & yes \\ dbft-2-7-15 & 7 & 15 & yes \\ dbft-2-10-5 & 10 & 5 & no \\ dbft-2-10-15 & 10 & 15 & no \\ dbft-2-10-25 & 10 & 25 & no \\ dbft-2-16-5 & 16 & 5 & no \\ dbft-2-16-25 & 16 & 25 & no \\ dbft-2-19-25 & 19 & 25 & no \\ \hline \end{tabular}
\end{table}
Table 3: Optimality of each instance according to time limit of 600 s, for all Scenarios P1–P7.  
adversarial conditions; and devising other exact adversarial conditions that may affect other families of consensus systems (e.g., purely asynchronous Byzantine agreements).  
V.N.C.: Conceptualization of this study, Methodology, Mathematical Formulation, and Writing--Original draft preparation. R.P.A.: Design of the graphs for the Mathematical Formulation, and Software, Writing--Revisions. H.G.S.: Reviews on the Mathematical Formulation, Software, and Writing--Revisions. W.Y.Q.: Reviews on the Mathematical Formulation, Methodology, and Writing--Revisions. I.M.C.: Advisor,Conceptualization of this study, Methodology, Mathematical Formulation, and Writing--Original draft preparation. All authors have read and agreed to the published version of the manuscript.  
This research was done in partnership with NeoResearch, a worldwide open-source community and support of Neo Foundation. [PERSON] was partially supported by the Brazilian funding agency CNPq (PQ-2 313777/2018-7).  
The authors declare no conflict of interest.