papers/lux-verkle-trees/sections/11-security-analysis.tex

\section{Security Analysis}


\subsection{Binding Property}

\begin{theorem}[Strong Binding]
Under the $q$-SDH assumption, no PPT adversary can find $(f, f', z, y, y', \pi)$ such that:
\begin{itemize}
    \item $f \neq f'$
    \item $\mathsf{Commit}(f) = \mathsf{Commit}(f')$
    \item $\mathsf{Verify}(\mathsf{Commit}(f), z, y, \pi) = 1$
    \item $\mathsf{Verify}(\mathsf{Commit}(f'), z, y', \pi) = 1$
\end{itemize}
\end{theorem}

\subsection{Hiding Property}

Optional hiding through randomization:
\begin{equation}
    C_{hiding} = [f(\tau)]_1 \cdot [r \cdot \gamma(\tau)]_1
\end{equation}
where $\gamma(X)$ is a random polynomial.

\subsection{Trusted Setup Security}

The ceremony security relies on:
\begin{enumerate}
    \item At least one honest participant
    \item Secure deletion of toxic waste $\tau$
    \item Public verifiability of contributions
\end{enumerate}

Probability of compromise with $n$ participants and $t$ corrupt:
\begin{equation}
    P_{compromise} = \left(\frac{t}{n}\right)^n < 2^{-128} \text{ for } n > 1000, t < 0.99n
\end{equation}

\subsection{Quantum Resistance}

Verkle trees are vulnerable to quantum attacks:
\begin{itemize}
    \item Shor's algorithm breaks discrete log
    \item Pairing inversion becomes feasible
    \item Migration to post-quantum required
\end{itemize}

Post-quantum alternatives under research:
\begin{itemize}
    \item Lattice-based commitments
    \item Hash-based accumulators
    \item STARK-based proofs
\end{itemize}