papers/lux-quantum-consensus/sections/03-quantum-consensus-protocol.tex

\section{Quantum Consensus Protocol}


\subsection{Architecture Overview}

\begin{figure}[h]
\centering
\begin{verbatim}
+-------------------------------------------------+
|          Lux Quantum Consensus Layer            |
+-------------------------------------------------+
|  +--------------+  +--------------+            |
|  | ML-DSA    |  | SLH-DSA (FIPS 205, formerly SPHINCS+)     |            |
|  | Signatures   |  | Checkpoints  |            |
|  +--------------+  +--------------+            |
+-------------------------------------------------+
|  +--------------------------------------+      |
|  |  Lattice Threshold Signatures        |      |
|  |  (n-of-m multi-sig, quantum-safe)    |      |
|  +--------------------------------------+      |
+-------------------------------------------------+
|  +--------------+  +--------------+            |
|  | ML-KEM KEM    |  | zk-STARKs    |            |
|  | (Key Exch)   |  | (Privacy)    |            |
|  +--------------+  +--------------+            |
+-------------------------------------------------+
         |                    |
         v                    v
+-----------------+  +-----------------+
|  Wave Family    |  |  Hybrid Mode    |
|  (Metastable)   |  |  (ECDSA+Dilith) |
+-----------------+  +-----------------+
\end{verbatim}
\caption{Lux Quantum Consensus architecture}
\end{figure}

\subsection{Validator Key Management}

**Classical blockchain** (ECDSA):
\begin{equation}
\text{Sig}_{sk}(m) = \text{ECDSA}(sk, m) \quad \text{(32 bytes)}
\end{equation}

**Quantum consensus** (ML-DSA):
\begin{equation}
\text{Sig}_{sk}(m) = \text{ML-DSA}(sk, m) \quad \text{(3,293 bytes)}
\end{equation}

**Challenge**: 100$\times$ larger signatures impact network bandwidth.

**Solution**: Aggregation via BLS-like lattice schemes.

\subsubsection{Lattice-Based Threshold Signatures}

We adapt FROST \cite{komlo2020frost} to lattice setting:

\begin{algorithm}
\caption{Quantum-Safe Threshold Signature}
\begin{algorithmic}
\STATE \textbf{Setup Phase}:
\STATE Validators generate secret shares: $s_i \in \mathbb{Z}_q$
\STATE Commitment: $c_i = \mathbf{A} \cdot s_i + e_i$
\STATE Public key: $pk = \sum_{i=1}^n c_i$

\STATE \textbf{Signing Phase} (message $m$):
\FOR{each validator $i$ in quorum}
    \STATE Compute partial sig: $\sigma_i = \text{Dilithium-Partial}(s_i, m)$
    \STATE Broadcast $\sigma_i$ to coordinator
\ENDFOR

\STATE \textbf{Aggregation}:
\STATE Coordinator: $\sigma = \text{Aggregate}(\{\sigma_i\})$
\STATE Verify: $\text{Dilithium-Verify}(pk, m, \sigma)$
\RETURN $\sigma$ (3,293 bytes)
\end{algorithmic}
\end{algorithm}

**Properties**:
\begin{itemize}
    \item Threshold: Requires $t$ of $n$ validators ($t \geq 2n/3$)
    \item Quantum-safe: MLWE hardness
    \item Aggregate size: Same as single ML-DSA signature (constant)
\end{itemize}

\subsection{Consensus Flow}

\begin{enumerate}
    \item \textbf{Propose}: Leader creates block, signs with ML-DSA
    \item \textbf{Vote}: Validators vote using threshold signatures
    \item \textbf{Finalize}: Once $t$ votes collected, block is final
    \item \textbf{Checkpoint}: Every 1000 blocks, SLH-DSA signature for long-term security
\end{enumerate}

\begin{equation}
\text{Finality Time} = T_{\text{network}} + T_{\text{crypto}} + T_{\text{consensus}}
\end{equation}

**Benchmarks**:
\begin{itemize}
    \item $T_{\text{network}}$: 200ms (global gossip)
    \item $T_{\text{crypto}}$: 50ms (ML-DSA sign + verify)
    \item $T_{\text{consensus}}$: 1,500ms (Wave metastability)
    \item \textbf{Total}: 1,750ms (< 2 seconds)
\end{itemize}

\subsection{Quantum-Safe Finality Gadget}

We prove finality via quantum-resistant proof:

\begin{equation}
\text{Proof}_{\text{final}}(B) = \text{zk-STARK}(\exists \sigma : \text{Dilithium-Verify}(pk, B, \sigma) \land |\sigma| \geq 2n/3)
\end{equation}

**Properties**:
\begin{itemize}
    \item Zero-knowledge: Doesn't reveal validator identities
    \item Post-quantum: Hash-based (SHA3-256)
    \item Succinct: $O(\log^2 n)$ proof size
    \item Fast verification: $O(\log n)$ time
\end{itemize}