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zeekay
9504b4c3ef
Sweep across the non-Tier-1 paper catalogue (Tier-1 owned by parallel
agent and excluded). All 113 processed papers already had luxcover
wired up; this commit fixes 14 build breakers and produces fresh
PDFs across the catalogue.
Fixes:
- lux-blockchain-scaling-laws: em-dash inside lstlisting -> --
- lux-corona-pq: stale \input pointing at 03-the-corona-scheme renamed
to 03-the-corona-scheme (file already renamed on disk)
- lux-lattice-cryptography-primer: same Corona->Corona rename
- lux-formal-verification-lean4: Lean source uses unicode (×, ∀, ∃, ∈,
∧, ∨, ∩, ≠, ⟨, ⟩, ·, ¬) inside lstlistings. Added a literate map
so listings render the math glyphs without UTF-8 fault.
- lux-id-did-specification: ✓/✗ in tabular replaced with \ding{51}
/ \ding{55} (pifont). Added pifont package.
- lux-ntt-simd-implementation: middle-dot in Go assembly listings;
added literate {·}{\textperiodcentered}.
- lux-ntt-transform: μ, ×, –, — in regular text (no T1 fontenc).
Replaced with $\mu$/$\times$/--/--- via unicode->latex sweep.
- lux-perpetuals-derivatives, lux-quantum-consensus,
lux-quasar-consensus, lux-tee-computing-mesh: same unicode->latex
sweep across body text. Also fixed broken math segment
`$L_{avg} = 10$\times$$` -> `$L_{avg} = 10\times$`.
- lux-zchain: emoji glyphs in roadmap table replaced with
parenthetical text.
- lux-triple-proof-consensus: dropped duplicate \usepackage{geometry}
that clashed with luxcover (geometry option clash).
- pq: defined missing \Rq macro.
PDFs rebuilt for the full Tier-2/3 set.
Excluded (Tier-1, parallel agent): lp-073-pulsar, lp-stricte2epq,
dag-evm-formal, lp-010-quasar-stm-{3,4}, lp-020-quasar-consensus,
lux-institutional-custody, lux-nist-mptc-submission, pqsafe-il5,
pq-rns, plus shared/ and lps/.
28 lines
1.4 KiB
TeX
28 lines
1.4 KiB
TeX
\section{Preliminaries}
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%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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\subsection{Shamir's Secret Sharing}
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\begin{definition}[$(t, n)$-Shamir Sharing]
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A dealer chooses a random polynomial $f(x) = s + a_1 x + a_2 x^2 + \cdots + a_{t-1} x^{t-1}$ over $\mathbb{Z}_q$ where $f(0) = s$ is the secret. Share $i$ is $s_i = f(i)$ for $i \in \{1, \ldots, n\}$. Any $t$ shares determine $f$ (and thus $s$) via Lagrange interpolation; any $t - 1$ shares reveal no information about $s$.
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\end{definition}
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\subsection{Linearity Property}
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The critical property enabling LSS is linearity:
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\begin{equation}
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f(x) + g(x) = (s_f + s_g) + (a_1^f + a_1^g)x + \cdots
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\end{equation}
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If parties hold shares of $f$ and independently generate shares of a polynomial $g$ with $g(0) = 0$, the sum produces a fresh sharing of the same secret $s_f$ under a new polynomial. This enables:
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\begin{itemize}[nosep]
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\item \textbf{Refresh}: Same $(t, n)$, new polynomial, same secret.
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\item \textbf{Reshare}: New $(t', n')$, new polynomial, same secret.
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\end{itemize}
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\subsection{Joint Verifiable Secret Sharing (JVSS)}
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JVSS~\cite{pedersen1991} adds verifiability: each party publishes Pedersen commitments $C_k = g^{a_k} h^{b_k}$ for each coefficient of their polynomial. Other parties verify their shares against these commitments. A complaint mechanism handles malicious dealers.
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%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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