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62 lines
2.3 KiB
TeX
62 lines
2.3 KiB
TeX
\section{FROST: Threshold Schnorr and BLS Signatures}
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\label{sec:frost}
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\subsection{Protocol Overview}
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FROST (Flexible Round-Optimized Schnorr Threshold)~\cite{komlo2020frost} is a
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two-round threshold Schnorr signature protocol. Unlike CGGMP21, FROST operates
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over groups where Schnorr signatures are native, making it suitable for:
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\begin{itemize}[leftmargin=2em]
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\item Ed25519 (Solana, Cosmos, Polkadot)
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\item BLS12-381 (Ethereum consensus, Lux consensus)
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\item BIP-340 Taproot (Bitcoin)
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\end{itemize}
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\subsection{Two-Round Protocol}
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\begin{algorithm}[ht]
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\caption{FROST Signing.}
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\label{alg:frost}
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\begin{algorithmic}[1]
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\Require Message $m$; key shares $sk_i$ for $i \in S$; $|S| = t$
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\Ensure Schnorr signature $(\tilde{R}, z)$
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\State \textbf{Round 1 (commitment):}
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\ForAll{$P_i \in S$}
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\State Sample $(d_i, e_i) \leftarrow \mathbb{Z}_q^2$
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\State Broadcast $(D_i, E_i) = (d_i \cdot G, e_i \cdot G)$
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\EndFor
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\State \textbf{Round 2 (response):}
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\State $B \leftarrow \{(i, D_i, E_i)\}_{i \in S}$
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\State $\rho_i \leftarrow H(\text{``binding''}, i, m, B)$ for each $i$
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\State $\tilde{R} \leftarrow \sum_{i \in S} (D_i + \rho_i \cdot E_i)$
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\State $c \leftarrow H(\tilde{R}, pk, m)$ \Comment{Fiat-Shamir challenge}
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\ForAll{$P_i \in S$}
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\State $z_i \leftarrow d_i + \rho_i \cdot e_i + \lambda_i \cdot sk_i \cdot c$
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\State Broadcast $z_i$
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\EndFor
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\State $z \leftarrow \sum_{i \in S} z_i$
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\State \Return $(\tilde{R}, z)$
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\end{algorithmic}
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\end{algorithm}
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\subsection{FROST for BLS}
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FROST extends naturally to BLS signatures because BLS signing is a scalar
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multiplication $\sigma = sk \cdot H(m)$ in $\mathbb{G}_1$. Threshold BLS:
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\begin{enumerate}[leftmargin=2em]
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\item Each party computes partial signature $\sigma_i = sk_i \cdot H(m)$.
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\item The coordinator reconstructs $\sigma = \sum_{i \in S} \lambda_i \cdot \sigma_i$.
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\item Verification: $e(\sigma, G_2) = e(H(m), pk)$ where $pk \in \mathbb{G}_2$.
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\end{enumerate}
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BLS threshold signing requires only \emph{one} round (each party independently
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computes their partial signature). No precomputation or commitment phase is needed.
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This makes BLS threshold signing significantly faster than ECDSA threshold signing.
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%% ========================================================================
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%% 6. KEY RESHARING
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%% ========================================================================
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