papers/lux-consensus/sections/06-parameterization.tex

\section{Parameterization}

\label{sec:params}

The three primary parameters $k$, $\alpha$, and $\beta$ govern the security-performance
trade-off of Lux Consensus.

\subsection{Sample Size $k$}

The sample size $k$ controls the probability that a biased sample creates a false
supermajority. From the Chernoff bound (Equation~\ref{eq:chernoff}), the probability of
safety failure per round decreases as $e^{-kD(\alpha \| 1-\alpha)}$. Setting a target
safety parameter $\lambda$ (e.g., $\lambda = 2^{-128}$):
\begin{equation}
k \geq \frac{\lambda \ln 2}{D(\alpha \| 1-\alpha)}
\label{eq:k_lower_bound}
\end{equation}

For $\alpha = 0.8$: $D(0.8 \| 0.2) \approx 0.832$.
Setting $\lambda = 128$: $k \geq 128 \ln 2 / 0.832 \approx 107$.

In practice, $k = 20$ provides $\approx 60$ bits of security per round, which compounds
over the $\beta$ finalization rounds to yield overall security of $60\beta \approx 1200$
bits for $\beta = 20$.

\subsection{Supermajority Threshold $\alpha$}

The threshold $\alpha \in (1/2, 1)$ trades off:
\begin{itemize}
\item \textbf{Higher $\alpha$}: Stronger safety (Byzantine minority cannot manufacture
supermajority), slower convergence (needs larger honest majority to achieve threshold).
\item \textbf{Lower $\alpha$}: Faster convergence, weaker safety guarantee.
\end{itemize}

The critical constraint is $\alpha > 2/3$: this ensures that even if all $f < n/3$
Byzantine validators coordinate, they cannot alone achieve a supermajority. We recommend
$\alpha = 0.8$ for production networks.

\subsection{Finalization Threshold $\beta$}

The finalization threshold $\beta$ sets the number of consecutive supermajority rounds
required. The safety failure probability decreases as $e^{-\beta k D(\alpha \| 1-\alpha)}$.
For $k=20$, $\alpha=0.8$, $\beta=20$:
\begin{equation}
P(\text{safety failure}) \leq e^{-20 \cdot 20 \cdot 0.832} = e^{-333} \approx 10^{-144}
\end{equation}

The expected finalization time is $\beta / f_{\text{round}}$. For 50ms round-trips,
finalization takes $\approx 1\text{s}$. With pipelining, effective finality is 650ms.

\subsection{Joint Parameter Optimization}

We define the security margin as $\mathcal{S}(k, \alpha, \beta) = \beta \cdot k \cdot D(\alpha \| 1-\alpha)$
and the expected finality as $\mathcal{F}(k, \alpha, \beta) = \beta \cdot \Delta_{\text{RTT}} + O(\log n) \cdot \Delta_{\text{RTT}}$.
The optimization problem is:
\begin{equation}
\min_{k, \alpha, \beta} \mathcal{F}(k, \alpha, \beta) \quad \text{s.t.} \quad \mathcal{S}(k, \alpha, \beta) \geq \lambda_{\text{target}}
\end{equation}

For $\lambda_{\text{target}} = 128$ (128-bit security), the Pareto-optimal configurations
include $(k=20, \alpha=0.8, \beta=20)$ yielding $\mathcal{S} = 333$ bits with $\mathcal{F} \approx 650$ms, and
$(k=40, \alpha=0.75, \beta=10)$ yielding $\mathcal{S} = 225$ bits with $\mathcal{F} \approx 400$ms.

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