papers/lux-fraud-proofs/sections/09-security-analysis.tex

\section{Security Analysis}


We analyze the security properties of the fraud proof system under various threat models.

\subsection{Safety Property}

The system maintains safety if no invalid state is finalized:

\begin{theorem}[Safety]
Given at least one honest party with sufficient resources to challenge, no invalid state transition will be finalized with probability:
\begin{equation}
P(\text{safety}) \geq 1 - e^{-\lambda \tau}
\end{equation}
where $\lambda$ is the rate of honest monitoring and $\tau$ is the challenge period.
\end{theorem}

\subsection{Liveness Property}

The system maintains liveness if valid states eventually finalize:

\begin{theorem}[Liveness]
Valid state transitions finalize within time $T$ with probability:
\begin{equation}
P(\text{finality} \leq T) = 1 - e^{-\mu(T-\tau)}
\end{equation}
where $\mu$ is the rate of honest assertion and $\tau$ is the challenge period.
\end{theorem}

\subsection{Censorship Resistance}

Censorship attacks attempting to prevent challenges fail if:

\begin{equation}
P(\text{censorship}) < 1 - (1 - p)^n
\end{equation}

where $p$ is the probability of successful transaction inclusion and $n$ is the number of attempts within the challenge period.

\subsection{Delay Attack Analysis}

Adversaries attempting to delay resolution face exponentially increasing costs:

\begin{equation}
\text{Cost}_{\text{delay}}(t) = B \times (1 + r)^{t/\tau_{\text{round}}}
\end{equation}

where $r$ is the interest rate and $\tau_{\text{round}}$ is the round timeout.