papers/lux-teleport-protocol/sections/04-cross-ecosystem-light-client.tex

\section{Cross-Ecosystem Light Client}
\label{sec:lightclient}

\subsection{Lux Light Client on Ethereum}

For Lux-to-Ethereum transfers, an on-chain light client on Ethereum tracks Lux validator set changes and verifies finalized block headers:

\begin{definition}[Lux Light Client Contract]
The light client contract on Ethereum maintains:
\begin{itemize}
    \item Current Lux epoch number $e$
    \item Current validator set commitment $\mathsf{VRoot}_e$
    \item Total validator stake $W_e$
    \item Latest verified block height $h_{\text{latest}}$
\end{itemize}
\end{definition}

\begin{algorithm}[H]
\caption{Light Client Block Header Verification}
\label{alg:lightclient}
\begin{algorithmic}[1]
\Procedure{VerifyHeader}{blockHeader $B$, proof $\Pi$}
    \State Parse $\Pi = (\sigma_{\text{agg}}, \mathsf{bitmap}, \mathsf{valProofs})$
    \State Reconstruct signer set from $\mathsf{bitmap}$ and $\mathsf{VRoot}_e$
    \State Verify Merkle proofs in $\mathsf{valProofs}$ for each signer
    \State $W_{\text{signed}} \gets \sum_{\text{signers}} w_i$
    \State \textbf{require} $W_{\text{signed}} \geq 2W_e/3$
    \State $d \gets \mathsf{SHA256}(B.\text{serialize}())$
    \State \textbf{require} BLS12-381 pairing check on $(\sigma_{\text{agg}}, d)$ passes
    \State $h_{\text{latest}} \gets B.\mathsf{height}$
    \State Store $B.\mathsf{stateRoot}$
\EndProcedure
\end{algorithmic}
\end{algorithm}

\subsection{State Proof Verification}

After a block header is verified, state proofs (Merkle Patricia Trie proofs) verify specific state entries:

\begin{algorithm}[H]
\caption{State Proof Verification on Ethereum}
\label{alg:state_proof}
\begin{algorithmic}[1]
\Procedure{VerifyState}{stateRoot, key, value, merkleProof}
    \State $\mathsf{computedRoot} \gets \mathsf{VerifyMPT}(\mathsf{key}, \mathsf{value}, \mathsf{merkleProof})$
    \State \textbf{require} $\mathsf{computedRoot} = \mathsf{stateRoot}$
    \State \Return true
\EndProcedure
\end{algorithmic}
\end{algorithm}

\subsection{Gas Cost Analysis}

\begin{table}[H]
\centering
\caption{On-chain light client gas costs on Ethereum.}
\label{tab:lc_gas}
\begin{tabular}{lrr}
\toprule
\textbf{Operation} & \textbf{Gas Cost} & \textbf{Notes} \\
\midrule
Header verification (BLS) & 250,000 & BLS12-381 via EIP-2537 \\
Validator set update & 400,000 & Epoch transition \\
MPT state proof & 50,000--150,000 & Depth-dependent \\
Token mint (ERC-20) & 50,000 & Standard transfer \\
\midrule
\textbf{Total per transfer} & \textbf{350,000--450,000} & First transfer per header \\
\textbf{Amortized (batched)} & \textbf{100,000--200,000} & Shared header verification \\
\bottomrule
\end{tabular}
\end{table}

Header verification can be amortized across multiple transfers in the same block, reducing per-transfer cost.

\subsection{Ethereum Light Client on Lux}

For Ethereum-to-Lux transfers, a Lux C-Chain contract tracks Ethereum's beacon chain via the sync committee:

\begin{definition}[Ethereum Sync Committee Light Client]
The contract tracks:
\begin{itemize}
    \item Current sync committee public keys (512 validators)
    \item Next sync committee commitment
    \item Latest finalized beacon block root
\end{itemize}
Updates occur every sync committee period ($\sim$27 hours) with BLS aggregate signature verification.
\end{definition}

\begin{table}[H]
\centering
\caption{Ethereum light client costs on Lux C-Chain.}
\label{tab:eth_lc}
\begin{tabular}{lrr}
\toprule
\textbf{Operation} & \textbf{Gas (Lux C-Chain)} & \textbf{Notes} \\
\midrule
Sync committee update & 300,000 & Every $\sim$27 hours \\
Finality proof & 200,000 & Per Ethereum block \\
Storage proof (MPT) & 80,000 & Per key verified \\
Token release & 50,000 & ERC-20 transfer \\
\midrule
\textbf{Total per transfer} & \textbf{330,000} & Amortized header cost \\
\bottomrule
\end{tabular}
\end{table}