papers/lux-flare-protocol/sections/05-luring-attack-analysis.tex

\section{Luring Attack Analysis}

\label{sec:attacks}

\subsection{Attack Strategy}

A \emph{luring attack} attempts to split the honest validators into two groups, each
building conviction for a different color. The adversary equivocates: responding with
color 1 to validators in group $H_1$ and color 0 to validators in group $H_0$.

\begin{definition}[Luring Attack]
The adversary partitions its responses such that:
\begin{itemize}[nosep]
\item When queried by $v \in H_1$: respond with color 1 (reinforcing their conviction).
\item When queried by $v \in H_0$: respond with color 0.
\end{itemize}
The goal is to have both groups simultaneously reach $cnt = \beta_2$.
\end{definition}

\begin{theorem}[Luring Attack Failure]
\label{thm:luring}
For any luring attack strategy and any initial partition $H_1 \cup H_0 = H$, the
probability that both groups reach $cnt = \beta_2$ simultaneously is bounded by:
\[
\Pr[\text{luring success}] \leq \left(\frac{f}{n - f}\right)^{\beta_2} \cdot \binom{n}{k}^{-\beta_2}
\]
which is negligible for standard parameters.
\end{theorem}

\begin{proof}
For the luring attack to succeed, consider group $H_1$ with $|H_1|$ honest validators.
When a validator $v \in H_1$ samples $k$ validators, the expected number of honest
validators from $H_0$ in the sample is $k \cdot |H_0|/n$. These validators respond with
color 0, working against $v$'s conviction for color 1.

For $v$ to observe $\geq \alpha k$ responses for color 1, we need the $f$ Byzantine
validators (responding with color 1) plus the $|H_1|$-fraction of honest validators
to collectively provide $\alpha k$ responses. This requires:
\[
\frac{f + |H_1|}{n} \geq \alpha
\]
Since $f < n/3$ and $|H_1| < 2n/3$ (because $|H_0| \geq 1$), the maximum supportable
fraction is $f/n + |H_1|/n < 1/3 + |H_1|/n$.

For a balanced split ($|H_1| = |H_0| = |H|/2 \approx n/3$), we get support fraction
$1/3 + 1/3 = 2/3 < \alpha = 0.8$. This means a balanced luring attack cannot produce
supermajority for \emph{either} group, making success probability zero.

For unbalanced splits where $|H_1| > |H_0|$, the larger group may achieve supermajority,
but the smaller group cannot. The attack reduces to a standard Byzantine attack where
the adversary supports one color, which our CSE theorem already covers.
\end{proof}

\subsection{Adaptive Equivocation}

\begin{lemma}[Equivocation Detection]
\label{lem:equivocation}
If the adversary equivocates on $> f$ total responses across all honest validators'
queries in a round, at least one honest validator detects the equivocation with
probability $\geq 1 - (1 - f/n)^k$.
\end{lemma}

\begin{proof}
If an honest validator $v$ queries Byzantine validator $b$, and another honest validator
$u$ also queries $b$ in the same round, $v$ and $u$ can compare $b$'s responses through
a post-round gossip protocol. The probability that at least two honest validators query
the same Byzantine validator is $1 - \prod_{i=0}^{k-1}(1 - i/n) \geq 1 - (1-k/n)^k$
by the birthday bound. For $k = 20$ and $n = 500$, this exceeds 0.33 per round, making
equivocation detectable within a few rounds.
\end{proof}

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