papers/lp-073-pulsar/sections/08-lifecycle.tex

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\section{Pulsar Key-Era Lifecycle}
\label{sec:lifecycle}
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The static Pulsar signing primitive presupposes a one-shot key
generation followed by a fixed signing committee. Permissionless chains
do not have that luxury: validator sets rotate every epoch. Pulsar
extends the static primitive with a \emph{key-era lifecycle} that supports
validator rotation without reconstructing the master secret and without
requiring a trusted dealer --- not after genesis, and (with the dealerless
Mithril RSS keygen of \S\ref{sec:protocol:bcc:keydkg}) not at genesis
either.

The metaphor is load-bearing, not flavor. A pulsar (the cosmic object)
is a rotating neutron star: the body persists, the beam direction
rotates, observed pulses appear as the beam sweeps. In Pulsar:
\begin{itemize}[leftmargin=2em,nosep]
  \item \textbf{Persistent body}: the group public key
        $(\mathbf{A}, \tilde{\mathbf{b}})$ and the hidden master
        secret $\mathbf{s}$ persist within a key era.
  \item \textbf{Rotating beam}: the share distribution
        $\{\mathbf{s}_i\}_{i \in V}$ rotates each epoch with
        validator-set changes.
  \item \textbf{Observed pulse}: each Quasar bundle emits one
        threshold certificate (a Pulsar \emph{pulse}) signed under the
        unchanged group public key.
\end{itemize}

\subsection{Three lineage fields, kept distinct}
\label{sec:lifecycle:lineage}

Every Pulsar share state carries three lineage scalars that must not be
collapsed:

\begin{itemize}[leftmargin=2em,nosep]
  \item \textbf{KeyEraID} ($\eta$). The group-key lineage. Bumps
        \emph{only} at Reanchor, when a new group public key is
        established.
  \item \textbf{Generation} ($g$). The LSS resharing version within a
        key era. Bumps every Refresh or Reshare under the same
        $(\mathbf{A}, \tilde{\mathbf{b}})$.
  \item \textbf{RollbackFrom} ($r$). Audit lineage. $r=0$ on
        ordinary forward transitions; $r > 0$ records the prior
        generation reverted from when the chain rolls back.
\end{itemize}

A trajectory across one key era and one Reanchor:
\begin{verbatim}
KeyEraID=7, Gen=0, Rb=0     genesis state of era 7
KeyEraID=7, Gen=1, Rb=0     reshared to a new validator set
KeyEraID=7, Gen=2, Rb=0     refreshed (same set; HJKY zero-poly)
KeyEraID=7, Gen=3, Rb=0     reshared again
KeyEraID=7, Gen=4, Rb=2     activation @gen 3 failed; reverted to gen 2
KeyEraID=8, Gen=0, Rb=0     reanchor: fresh group key, new era
\end{verbatim}

\subsection{Three layers, one shipping path}

\begin{tabular}{@{}p{0.18\textwidth}p{0.32\textwidth}p{0.42\textwidth}@{}}
  \toprule
  \textbf{Layer} & \textbf{Operation} & \textbf{Trust} \\
  \midrule
  Bootstrap & Dealerless Mithril RSS keygen (\S\ref{sec:protocol:bcc:keydkg}) for the production FIPS-204 scheme; Pedersen DKG over $R_q$ (\S\ref{sec:lifecycle:dkg}) for the shared kernel & No trusted dealer; group key signs under the stock FIPS-204 verifier. \\
  Reshare   & Old qualified subset $Q \subseteq O$, $|Q| \geq t_{\text{old}}$, runs $g_i(x)$ with $g_i(0) = s_i$; new party $j$ computes $s'_j = \sum_{i \in Q} \lambda^Q_i \cdot g_i(\beta_j)$ \cite{desmedtjajodia97} & No new trust. Master secret preserved structurally. \\
  Reanchor  & Fresh ceremony with new $\eta$ & Same as Bootstrap; rare governance event. \\
  \bottomrule
\end{tabular}

\subsection{Refresh vs Reshare: distinct primitives}

Two primitives, never collapsed:

\paragraph{Refresh (HJKY97).}
Same committee, same threshold. Each party $i$ samples a degree-$(t-1)$
zero-polynomial $z_i(x)$ with $z_i(0) = 0$, distributes $z_i(\alpha_j)$,
and updates $s'_j = s_j + \sum_i z_i(\alpha_j)$. Defends against
mobile-adversary share accumulation across time \cite{hjky97}.

\paragraph{Reshare (Desmedt--Jajodia).}
New committee. Old qualified subset $Q$ samples $g_i(x)$ of degree
$t_{\text{new}}-1$ with $g_i(0) = s_i$. Each new party $j$ aggregates
contributions weighted by Lagrange coefficients evaluated at $0$ for
$Q$. New polynomial $G(x) = \sum_{i \in Q} \lambda^Q_i \cdot g_i(x)$
satisfies $G(0) = \sum_{i \in Q} \lambda^Q_i \cdot s_i = s$ by Lagrange
identity \cite{desmedtjajodia97}. Verifiability per
Wong--Wang--Wing \cite{wongwangwing02}.

Both primitives leave the public key unchanged: the lattice
Pedersen-style commits $\mathbf{C}_k = \mathbf{A}\cdot\mathrm{NTT}(c_k)
+ \mathbf{B}\cdot\mathrm{NTT}(r_k)$ bind shares to the polynomial
without revealing $\mathbf{s}$.

\subsection{Activation certificate: the consensus circuit-breaker}
\label{sec:lifecycle:activation}

After the resharing math finishes, the chain does \emph{not} accept the
new generation on faith. The new committee threshold-signs an
activation message under the \emph{unchanged} group public key:

\begin{verbatim}
QUASAR-PULSAR-ACTIVATE-v1 ||
    chain_id || network_id ||
    key_era_id || group_id ||
    old_epoch || new_epoch ||
    old_validator_set_hash || new_validator_set_hash ||
    old_threshold || new_threshold ||
    group_public_key_hash ||
    reshare_transcript_hash ||
    pairwise_material_commitment_hash ||
    implementation_version
\end{verbatim}

If the threshold-aggregated signature verifies under
$(\mathbf{A}, \tilde{\mathbf{b}})$, the new shares actually reconstruct
$\mathbf{s}$ and the chain commits the new state. If verification
fails, the chain stays at the old generation; the LSS rollback
mechanism (\S\ref{sec:lifecycle:rollback}) restores the prior
snapshot.

\paragraph{What activation proves.}
Successful new signing capability under the unchanged GroupKey. That
is all.

\paragraph{What activation does \emph{not} prove.}
Which old party sent a bad contribution; which new party lied; whether
all contributions were polynomial-consistent; whether a malicious old
subset caused a failed activation. Those are responsibilities of the
complaint and disqualification pipeline (commit / share verification
under lattice Pedersen) and the slashing-evidence pipeline. Activation
is necessary but \emph{not sufficient} for full Byzantine attribution.

\subsection{Rollback and the no-slashing failure ladder}
\label{sec:lifecycle:rollback}

Pulsar's failure response is graded and \emph{non-punitive}. There is
no slashing dependency:

\begin{enumerate}[leftmargin=2em,nosep]
  \item Coordinator timeout / unavailable
        $\rightarrow$ next coordinator picked deterministically.
  \item Resharing round fails (commit mismatch, bad share, etc.)
        $\rightarrow$ emit signed evidence; retry under fresh
        transcript binding next epoch.
  \item Activation cert fails to verify under unchanged GroupKey
        $\rightarrow$ \texttt{Rollback(targetGeneration = current - 1)};
        chain stays at the previous generation; signing continues.
  \item Multiple consecutive activation failures
        $\rightarrow$ governance Reanchor: new $\eta$, fresh ceremony.
\end{enumerate}

This ladder works for chains, bridges, enterprise custody, and
validator committees alike, because none of the steps require
identifying a malicious actor. Slashing evidence is \emph{collected}
during steps 2--3 (the framework persists signed messages from the
failed run for forensic review), but the chain's liveness and safety
do not depend on attribution.

\subsection{Pedersen DKG over $R_q$: the shared-kernel dealerless path}
\label{sec:lifecycle:dkg}

For the \emph{production} FIPS-204 byte-equal scheme, the default Bootstrap
path is the dealerless Mithril RSS keygen of
\S\ref{sec:protocol:bcc:keydkg} --- no trusted dealer, no toxic-waste
assumption, and the group key signs under the stock FIPS-204 verifier. The
construction below is the dealerless keygen for the \emph{shared kernel}
(the $(c,z,\Delta)$ scheme shared with Corona): a Pedersen-style DKG over
$R_q$ (\texttt{pulsar/dkg2/}) that likewise requires no dealer:

\begin{itemize}[leftmargin=2em,nosep]
  \item Two uniform matrices $\mathbf{A}, \mathbf{B}$ derived
        deterministically from public seeds.
  \item Each party samples Gaussian polynomials $f_i, g_i$ and
        broadcasts commits $\mathbf{C}_{i,k} = \mathbf{A}\cdot
        \mathrm{NTT}(c_{i,k}) + \mathbf{B}\cdot\mathrm{NTT}(r_{i,k})$.
  \item Hiding under MLWE on $\mathbf{B}$; binding under MSIS on
        $[\mathbf{A}\,|\,\mathbf{B}]$.
\end{itemize}

This avoids the upstream Pulsar Feldman-style DKG, which is
pseudoinverse-recoverable: $(\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T
\cdot \mathbf{C}_k = c_k$ when commits omit the noise term. The
Pedersen construction restores hiding by adding the
$\mathbf{B}\cdot\mathrm{NTT}(r_k)$ mask.

The Pedersen-DKG path solves an open distributed-construction problem:
jointly sampling $\tilde{\mathbf{b}} = \lfloor \mathbf{A}\cdot
\mathbf{s} + \mathbf{e} \rceil_\Xi$ with the correct small-noise
distribution, without any party learning $\mathbf{s}$ or $\mathbf{e}$,
and producing a publicly verifiable proof of public-key
well-formedness. Threshold Raccoon \cite{thresholdraccoon} and Hermine
\cite{hermine} are the closest related constructions for the shared
kernel. For the production FIPS-204 scheme this question is settled by
Mithril RSS \cite{mithril2026} (\S\ref{sec:protocol:bcc:keydkg}): the
joint key is dealerless, its group key signs under the stock FIPS-204
verifier (compatibility proven), and the only residual is full KeyGen
distribution-equivalence. Neither path requires a trusted dealer or a
toxic-waste assumption; the foundation-ceremony bootstrap of earlier
drafts is retired.

\subsection{What is preserved, what changes}

\begin{tabular}{@{}p{0.30\textwidth}p{0.65\textwidth}@{}}
  \toprule
  \textbf{Quantity} & \textbf{Status across resharing within an era} \\
  \midrule
  $\mathbf{A}$ (public matrix) & PRESERVED \\
  $\mathbf{s}$ (master secret) & PRESERVED structurally; share distribution rotates \\
  $\tilde{\mathbf{b}}$ (rounded public key) & PRESERVED \\
  GroupKey pointer (byte-identical) & PRESERVED \\
  $\mathbf{e}$ (LWE error) & NOT preserved; used at genesis to form $\tilde{\mathbf{b}}$, then erased with dealer state. Signers only need shares of $\mathbf{s}$. \\
  Share distribution $\{\mathbf{s}_i\}$ & ROTATED \\
  Pairwise PRF/MAC material & REGENERATED for the new committee via authenticated KEX (X25519 baseline; ML-KEM-768 hybrid recommended for PQ-aware deployments). \\
  $\eta$ (KeyEraID) & PRESERVED until Reanchor \\
  $g$ (Generation) & INCREMENTED \\
  $r$ (RollbackFrom) & 0 on forward; $>0$ on rollback \\
  \bottomrule
\end{tabular}