papers/zoo-fhe-ai/sections/02-background.tex

\section{Background}


\subsection{Fully Homomorphic Encryption}

\begin{definition}[Fully Homomorphic Encryption]
An \FHE{} scheme $\mathcal{E} = (\text{KeyGen}, \text{Enc}, \text{Dec}, \text{Eval})$ satisfies:
\begin{enumerate}
    \item \textbf{Correctness}: $\text{Dec}(sk, \text{Eval}(pk, f, \text{Enc}(pk, x))) = f(x)$
    \item \textbf{Compactness}: $|\text{Eval}(pk, f, c)| = \text{poly}(\lambda)$ independent of $|f|$
    \item \textbf{Semantic Security}: $\text{Enc}(pk, x_0) \approx_c \text{Enc}(pk, x_1)$
\end{enumerate}
\end{definition}

\subsection{TFHE}

\TFHE{}~\cite{chillotti2020tfhe} operates on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ and supports fast bootstrapping for boolean and small-integer operations:

\begin{equation}
\text{Bootstrap}(\text{ct}) : O(n \cdot \log n) \quad \text{in } \approx 10\text{ms}
\end{equation}

\TFHE{} is ideal for comparison operations, relu activations, and control flow in neural networks.

\subsection{CKKS}

\CKKS{}~\cite{cheon2017ckks} supports approximate fixed-point arithmetic on packed vectors:

\begin{equation}
\text{Enc}(\mathbf{x}) \otimes \text{Enc}(\mathbf{y}) = \text{Enc}(\mathbf{x} \cdot \mathbf{y} + e), \quad |e| < \epsilon
\end{equation}

where $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{N/2}$ are packed into $N$-slot ciphertexts. \CKKS{} is ideal for matrix multiplications and convolutions in neural networks.

\subsection{Lux FHE Infrastructure}

The Lux Network provides a native \FHE{} precompile at address \texttt{0x0700}~\cite{lux-tfhe,lux-fhevm} supporting:

\begin{itemize}
    \item \texttt{fheAdd}, \texttt{fheSub}, \texttt{fheMul}: Arithmetic on encrypted integers
    \item \texttt{fheLt}, \texttt{fheGt}, \texttt{fheEq}: Comparison operations
    \item \texttt{fheMax}, \texttt{fheMin}: Conditional selection
    \item \texttt{fheRand}: Encrypted random number generation
    \item \texttt{decrypt}: Threshold decryption request
\end{itemize}

Zoo Network inherits this precompile identically, enabling \FHE{} operations within Zoo EVM smart contracts.