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\section{Practical Applications}
\label{sec:applications}
\subsection{Encrypted Orderbook (Dark Pool)}
A dark pool is a trading venue where orders are not visible to other participants
until matched. On a transparent blockchain, orderbook entries are publicly visible,
enabling front-running and sandwich attacks. FHE eliminates this:
\begin{enumerate}[leftmargin=2em]
\item \textbf{Order submission}: Trader encrypts $(price, quantity, side)$ under
the matching engine's public key and submits the ciphertext on-chain.
\item \textbf{Matching}: The smart contract performs encrypted comparisons:
$\textsf{FHE.GTE}(\textsf{ct}_{\text{bid}}, \textsf{ct}_{\text{ask}})$.
Matched orders produce an encrypted execution price.
\item \textbf{Settlement}: Only matched counterparties receive decryption keys
for their specific trades. Unmatched orders remain encrypted and invisible.
\end{enumerate}
This construction satisfies Regulation ATS requirements for fair access and
confidential order handling, as the matching engine operates on encrypted data
without ever observing plaintext order flow.
\subsection{Private Governance Voting}
On-chain governance typically uses transparent voting, which enables:
\begin{itemize}[leftmargin=2em]
\item Vote buying (verifiable votes can be sold)
\item Strategic last-minute voting (seeing the tally before voting)
\item Social pressure (publicly visible individual votes)
\end{itemize}
FHE voting:
\begin{enumerate}[leftmargin=2em]
\item Each voter encrypts their vote $(0 \text{ or } 1)$ under the governance
contract's public key.
\item The contract homomorphically adds all encrypted votes:
$\textsf{ct}_{\text{tally}} = \sum_i \textsf{ct}_{\text{vote}_i}$.
\item At the end of the voting period, a threshold decryption (requiring $t$-of-$n$
governance committee members) reveals only the final tally, not individual
votes.
\end{enumerate}
\subsection{Sealed-Bid Auctions}
Sealed-bid auctions (first-price, second-price/Vickrey) require that bids remain
secret until the reveal phase. FHE enables this without a trusted auctioneer:
\begin{enumerate}[leftmargin=2em]
\item Bidders encrypt their bids under the auction contract's public key.
\item The contract computes $\textsf{FHE.Max}(\textsf{ct}_1, \ldots, \textsf{ct}_n)$
using a sorting network of encrypted comparisons.
\item For Vickrey auctions, the contract computes the second-highest bid
homomorphically.
\item The winner is notified via threshold decryption of a per-bidder indicator
ciphertext.
\end{enumerate}
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%% 7. PERFORMANCE
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