papers/zoo-confidential-defi/sections/07-threshold-settlement.tex

\section{Threshold Settlement}

\label{sec:settlement}
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\subsection{T-Chain Coordination}

Settlement is the only phase where decryption occurs. \TChain{} coordinates:

\begin{enumerate}
    \item The matching engine submits matched pairs $(\hat{O}_b, \hat{O}_a, \hat{q}_f)$ to \TChain{}.
    \item \TChain{} validators verify the match proof (the encrypted comparison result $\hat{r} = \hat{1}$).
    \item Each validator computes a partial decryption of $\hat{q}_f$ using its key share $sk_i$.
    \item Once $t = 67$ partial decryptions are collected, the fill quantity $q_f$ is reconstructed.
    \item The settlement contract transfers $q_f$ tokens from seller to buyer and $q_f \cdot p_{\textit{match}}$ payment from buyer to seller.
\end{enumerate}

\subsection{Partial Revelation}

The settlement reveals only the information necessary for execution:

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
\textbf{Information} & \textbf{Matched Orders} & \textbf{Unmatched Orders} \\
\midrule
Existence & Revealed & Revealed \\
Side (buy/sell) & Revealed & Revealed \\
Price & \textbf{Match price only} & \textbf{Never revealed} \\
Quantity & \textbf{Fill quantity only} & \textbf{Never revealed} \\
Submitter identity & Revealed to counterparty & Never revealed \\
\bottomrule
\end{tabular}
\caption{Information revelation under Confidential DeFi}
\label{tab:revelation}
\end{table}

Unmatched orders are \emph{never} decrypted. They expire silently, revealing nothing about the submitter's valuation.

\subsection{Atomic Settlement}

Settlement is atomic: either both legs of the trade execute or neither does. This is enforced by the settlement contract on \Zoo{} EVM:

\begin{verbatim}
function settle(
    bytes32 match_id,
    uint256 fill_qty,
    uint256 match_price,
    bytes   threshold_proof
) external {
    require(TChain.verify(threshold_proof, match_id));

    token_base.transferFrom(seller, buyer, fill_qty);
    token_quote.transferFrom(buyer, seller,
                             fill_qty * match_price);

    emit Settlement(match_id, fill_qty, match_price);
}
\end{verbatim}

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