papers/zoo-confidential-defi/sections/06-private-amm.tex

\section{Private AMM}

\label{sec:amm}
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\subsection{Encrypted Constant-Product Invariant}

A standard constant-product AMM maintains the invariant $x \cdot y = k$ where $x$ and $y$ are reserve amounts. In the private AMM, reserves are encrypted:

\begin{equation}
\text{CKKS.Mul}(\hat{x}, \hat{y}) = \hat{k}
\end{equation}

A swap of encrypted amount $\hat{\Delta x}$ for token $X$ yields:

\begin{equation}
\hat{\Delta y} = \hat{y} - \frac{\hat{k}}{\hat{x} + \hat{\Delta x}}
\end{equation}

This computation is evaluated entirely in \CKKS{}, producing an encrypted output amount $\hat{\Delta y}$ that is decrypted only for the swapper via \TChain{}.

\subsection{Liquidity Provider View}

Liquidity providers cannot see individual swap amounts. They observe only:

\begin{itemize}
    \item Aggregate pool reserves (decrypted periodically, e.g., every 100 blocks)
    \item Total fees earned (decrypted for LP withdrawal)
    \item Pool utilization metrics (computed homomorphically)
\end{itemize}

This protects swappers from LP-side information extraction while giving LPs sufficient information for position management.

\subsection{Concentrated Liquidity}

Encrypted concentrated liquidity (a la Uniswap V3) requires range-bound position management:

\begin{enumerate}
    \item LPs define price ranges in plaintext (the range is public; the amount is private).
    \item Deposit amounts are encrypted: $\hat{L} = \text{CKKS.Encrypt}(pk, L)$.
    \item The AMM uses \TFHE{} to check whether the current price falls within each LP's range.
    \item Active positions contribute encrypted liquidity; out-of-range positions contribute nothing.
\end{enumerate}

\subsection{Impermanent Loss Protection}

Because swap amounts are encrypted, LPs cannot be targeted by toxic flow specifically designed to maximize impermanent loss. An attacker cannot observe pool state in real-time and craft adversarial trades.

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