papers/lux-verkle-trees/sections/07-lux-implementation.tex

\section{Lux Implementation}


\subsection{Integration Architecture}

Lux's multi-chain architecture requires Verkle tree integration at three levels:

\begin{enumerate}
    \item \textbf{P-Chain}: Platform chain managing validators and appchains
    \item \textbf{C-Chain}: EVM-compatible contract chain
    \item \textbf{X-Chain}: UTXO-based exchange chain
\end{enumerate}

\subsection{P-Chain State Model}

The P-Chain maintains:
\begin{itemize}
    \item Validator sets: $\mathcal{V} = \{(pk_i, stake_i, end_i)\}$
    \item Delegations: $\mathcal{D} = \{(del_i, val_i, amount_i)\}$
    \item Appchain configurations: $\mathcal{S} = \{(id_i, params_i)\}$
\end{itemize}

Verkle commitment:
\begin{equation}
    C_{P} = \mathsf{Commit}(\mathcal{V} || \mathcal{D} || \mathcal{S})
\end{equation}

\subsection{C-Chain EVM Integration}

EVM state comprises:
\begin{itemize}
    \item Account states: nonce, balance, storageRoot, codeHash
    \item Contract storage: key-value mappings
    \item Contract code: bytecode blobs
\end{itemize}

Account encoding in Verkle tree:
\begin{equation}
    \mathsf{Account}(addr) = \{nonce || balance || code_{hash} || storage_{root}\}
\end{equation}

Storage slot mapping:
\begin{equation}
    \mathsf{StorageKey}(addr, slot) = \mathsf{Hash}(addr || slot)
\end{equation}

\subsection{State Transition Proofs}

For transaction $tx$ modifying state from $S$ to $S'$:

\begin{equation}
    \pi_{transition} = (\pi_{pre}, \pi_{post}, \pi_{witnesses})
\end{equation}

where:
\begin{itemize}
    \item $\pi_{pre}$: Proofs for pre-state values
    \item $\pi_{post}$: Proofs for post-state values
    \item $\pi_{witnesses}$: Auxiliary data (e.g., code proofs)
\end{itemize}

\subsection{Cross-Chain Verification}

Appchain validators verify cross-chain transfers using Verkle proofs:

\begin{algorithmic}[1]
\STATE \textbf{function} $\mathsf{VerifyCrossChain}(proof, sourceRoot, targetRoot)$
\STATE $valid_{source} \gets \mathsf{VerifyProof}(proof.source, sourceRoot)$
\STATE $valid_{target} \gets \mathsf{VerifyProof}(proof.target, targetRoot)$
\STATE \textbf{return} $valid_{source} \land valid_{target}$
\end{algorithmic}