papers/zoo-threshold-signatures/sections/07-security-analysis.tex

\section{Security Analysis}

\label{sec:security}

\subsection{Open DKG Safety}

The DKG executes over an open network where validators may be adversarial. Safety relies on two mechanisms:

\begin{enumerate}
\item \textbf{Feldman commitments}: Each participant commits to their polynomial coefficients. Share recipients verify consistency, detecting any deviation.
\item \textbf{Zero-knowledge proofs}: Each participant proves knowledge of their secret coefficient $a_{i,0}$ without revealing it, preventing a rogue-key attack where an adversary biases the group public key.
\end{enumerate}

\begin{proposition}[DKG Soundness]
If fewer than $t$ of the $n$ DKG participants are corrupted, the resulting group public key $Y$ is uniformly distributed and the adversary learns no information about honest parties' shares.
\end{proposition}

\subsection{Consensus-Coupled Threshold Guarantees}

In \TChain{}, the MPC threshold $t$ is set to match the consensus fault tolerance. For $n$ validators with total stake $W$, the consensus safety threshold requires honest control of $\geq \frac{2}{3}W$. We set $t = \lceil \frac{n}{3} \rceil + 1$, ensuring:

\begin{equation}
\text{Forge signature} \implies \text{Corrupt } \geq t \text{ validators} \implies \text{Break consensus}
\end{equation}

An adversary who can forge a threshold signature can also break consensus, and vice versa. The two security boundaries collapse into one.

\subsection{Identifiable Abort}

Both FROST and CGGMP21 support identifiable abort. If a participant submits an invalid round message:

\begin{enumerate}
\item Other participants detect the inconsistency via commitment verification or range proof checking.
\item The session aborts and the offending validator's identity is recorded on-chain.
\item The validator is slashed according to the staking parameters defined in \cite{zoo-tokenomics}.
\item A new session is initiated excluding the slashed validator.
\end{enumerate}

This mechanism converts MPC liveness failures into economic penalties, aligning cryptographic and economic security.

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