papers/ntt-transform/sections/10-security-analysis.tex

\section{Security Analysis}


\subsection{Quantum Attack Resistance}

\subsubsection{Grover's Algorithm}
Grover's algorithm provides quadratic speedup for brute-force search:
\begin{itemize}
\item Classical security: $2^n$ operations
\item Quantum security: $2^{n/2}$ operations
\item Mitigation: Double key sizes (256-bit → 512-bit)
\end{itemize}

\subsubsection{Shor's Algorithm}
Shor's algorithm breaks discrete log and factoring in polynomial time:
\begin{itemize}
\item Impact: Complete break of ECC and RSA
\item Timeline: 10-15 years for cryptographically relevant quantum computers
\item Mitigation: Lattice-based cryptography (immune to Shor)
\end{itemize}

\subsection{Classical Security}

The security of lattice-based cryptography relies on:

\begin{enumerate}
\item \textbf{Learning With Errors (LWE)}: Distinguishing noisy linear combinations
\item \textbf{Short Integer Solution (SIS)}: Finding short vectors in lattices
\item \textbf{NTRU assumption}: Specific to FALCON
\end{enumerate}

These problems have resisted classical attacks for decades and are believed quantum-hard.

\subsection{Implementation Security}

\subsubsection{Timing Attacks}
Our implementation uses constant-time operations:
\begin{itemize}
\item No data-dependent branches
\item Fixed iteration counts
\item Constant-time modular arithmetic
\end{itemize}

\subsubsection{Side-Channel Resistance}
\begin{itemize}
\item Power analysis: Mitigated by EVM abstraction
\item Cache timing: Not applicable to EVM
\item Electromagnetic emanation: Hardware-dependent
\end{itemize}