papers/lux-mchain-mpc/sections/09-security-considerations.tex

\section{Security Considerations}


\subsection{Threat Model}

\textbf{Adversary Capabilities:}
\begin{itemize}
\item Can corrupt up to $f < n/3$ validators (Byzantine)
\item Can delay network messages by up to $\Delta_{max}$
\item Cannot break cryptographic assumptions (discrete log, lattice hardness)
\end{itemize}

\subsection{Byzantine Fault Tolerance}

\begin{theorem}[Threshold Security]
With $t = \lceil \frac{2n}{3} \rceil$ and $f < \frac{n}{3}$ Byzantine validators, an adversary cannot forge a valid threshold signature.
\end{theorem}

\begin{proof}
A valid signature requires $t$ shares. With $f < n/3$ Byzantine nodes:
\begin{align}
\text{Honest shares available} &= n - f > n - \frac{n}{3} = \frac{2n}{3} \\
\text{Required threshold} &= t = \lceil \frac{2n}{3} \rceil
\end{align}
Thus, $n - f > t$, ensuring honest validators always have enough shares. Byzantine nodes alone cannot reach threshold $t$.
\end{proof}

\subsection{Quantum Security Timeline}

\begin{table}[h]
\centering
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{Phase} & \textbf{Timeline} \\
\midrule
Phase 0 (Classical) & 2024-2027 \\
Phase 1 (Transition) & 2027-2030 \\
Phase 2 (Dual-sig required) & 2030-2035 \\
Phase 3 (Quantum primary) & 2035+ \\
\bottomrule
\end{tabular}

\label{tab:quantum-timeline}
\end{table}