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New Chapter 13, 'The Second Summit: A Hash-Based Pyramid' — SLH-DSA
(FIPS 205) as the transfer experiment for the whole method:
- opens on leaf 18 as the anomaly; correctness-vs-security across the
quantum divide ('a correct implementation of a broken lock is still a
broken lock')
- Lamport -> Winternitz chains with the checksum see-saw run twice on
real w=16 numbers, including a concrete failed forgery (480 -> 479,
digit 14 -> 13)
- FORS worked at napkin scale (k=2, a=2, one reuse = one forgery) and
real scale (28 of 57,344, exponent 14)
- the virtual hypertree: digest split 21/7/2 to the bit, the 54-bit
meter peeled 9 bits per layer, verification priced exactly (254 fixed
oracle calls; the see-saw itself caps a layer at 510, so worst case
3,824 — the naive 525*35 bound is unreachable, and the chapter says
why); ~2^72 to build vs ~2^12 to check
- the eleven certificates, the loop-to-fold bridges, the honest
'visible, not correct' boundary (no second semantics — and why the
natural move fails), the cone-growth table, the t_l/t_len naming
inversion told as the war story it was, the apex as an audit
invitation with the verbatim theorem named
- 'The leaf, live': leaf-vs-head precision ('plausible, and wrong
twice'), the three-clause self-reference ledger (attested machinery /
attested scheme / honest gap), one-command tryit
- six exercises with pathway'd solutions; checkpoint hands the
who-checks-them question to the finale
Structural: attestation renamed ch14 and now carries the book's single
ending (where-to-go, further reading, final reframe, prospective
checkpoint — moved from ch12); its two interior checkpoints demoted to
bigidea/tryit so the terminal checkpoint stands alone; opening now
receives ch13's baton. ch12 ends as a chapter. Front matter: three-summit
arc, fourteen-week plan, honest discussion-exercise count; ch01 promise
ladder extended to Chapters 13/14; glossary +5 entries (and the
pre-existing Hasse-bound misordering fixed); README fourteen chapters +
build.sh recipe.
Every constant verified against fips205-slhdsa-verified and
lean-transparency-log by four adversarial checkers; arithmetic
independently recomputed; didactic panel scored the chapter 9/8 —
the book's high-water mark. Build: 128 pages, zero unresolved refs.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
188 lines
8.6 KiB
TeX
188 lines
8.6 KiB
TeX
\chapter*{Glossary}
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\markboth{Glossary}{}
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\addcontentsline{toc}{chapter}{Glossary}
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\newcommand{\gloss}[1]{\par\smallskip\noindent{\bfseries #1.}\ }
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\gloss{Axiom-clean} Of a theorem: \lean{\#print axioms} reports exactly
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Lean's standard trio \lean{[propext, Classical.choice, Quot.sound]} and
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nothing else. The gold standard for shipped certificates
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(Chapter~\ref{ch:honesty}).
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\gloss{Bounds invariant} A predicate limiting how large limbs may grow
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(e.g.\ every limb $< 2^{54}$), maintained across operations so that
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machine arithmetic never overflows. One of the two clauses of every
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operation spec (Chapters~\ref{ch:rust}--\ref{ch:denotation}).
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\gloss{Carry} Value moved from one limb position to the next when a
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limb exceeds its radix. Delayed (``lazy'') carries are the central
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performance trick of fast field arithmetic and the habitat of its
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characteristic bugs (Chapter~\ref{ch:why}; Interlude).
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\gloss{Certificate} Data that makes a fact cheap to \emph{check}
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regardless of how expensive it was to \emph{find}: a Pratt witness
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tree for primality, a proof object for a theorem
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(Chapter~\ref{ch:prime}).
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\gloss{Charon / Aeneas} The two-stage extraction pipeline: Charon
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compiles Rust to the LLBC intermediate representation; Aeneas
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translates LLBC into pure Lean definitions
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(Chapter~\ref{ch:rust}).
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\gloss{Cofactor} The factor $8$ in the Ed25519 group order $8\ell$;
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multiplying by it annihilates the small-torsion component of any point,
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which is why \emph{cofactored} verifiers (the ZIP-215 lineage) check
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$8sB = 8R + 8kA$. The verified dalek-lineage path checks the stricter
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\emph{canonical} uncofactored equation byte-exactly
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(Chapter~\ref{ch:pyramid}).
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\gloss{Commuting square} The diagram --- machine operation along the
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top, ideal operation along the bottom, denotation down the sides ---
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whose closure \emph{is} implementation correctness
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(Chapter~\ref{ch:denotation}).
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\gloss{Complete (addition law)} An addition formula with no exceptional
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cases: valid for every pair of points, including doubling and identity.
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Ed25519's Edwards law is complete because $d$ is a non-square
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(Chapter~\ref{ch:pyramid}).
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\gloss{Decision procedure} An algorithm that settles \emph{every}
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statement in a defined logical fragment --- \lean{omega} for linear
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arithmetic, \lean{decide} for finite computations, \lean{ring} for
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ring identities. Failure on an in-fragment goal means the goal is
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false (Chapter~\ref{ch:automation}).
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\gloss{Definitional equality} Two terms being identical after the
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kernel computes (unfolds definitions, reduces recursion). What
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\lean{rfl} checks; blocked by opaque variables in recursion position
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(Chapter~\ref{ch:pat}).
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\gloss{Denotation} The function $\denote{\cdot}$ mapping a machine
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representation (limb array) to the mathematical value it \emph{means}
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(an element of $\Fp$). The bridge on which all correctness statements
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stand (Chapter~\ref{ch:denotation}).
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\gloss{Euler's criterion} $a^{(p-1)/2} \equiv \pm 1 \pmod p$ decides
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whether $a$ is a square modulo the odd prime $p$ ($+1$: square; $-1$:
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non-square). Settles both completeness facts of
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Chapter~\ref{ch:pyramid} (toolkit Card~7).
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\gloss{Extraction} Mechanical translation of source code (Rust) into a
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proof assistant's language via Charon/Aeneas, producing the \emph{model}
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--- the artifact actually verified, never hand-edited
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(Chapter~\ref{ch:rust}).
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\gloss{Fermat's little theorem} $a^{p-1} \equiv 1 \pmod p$ for prime
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$p$ and $a \not\equiv 0$; hence $a^{p-2} = a^{-1}$, the identity behind
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the verified inversion chain (Chapters~\ref{ch:modular},
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\ref{ch:field}).
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\gloss{Find/check asymmetry} The gap between the cost of discovering a
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fact and the cost of verifying a certificate for it --- the engine of
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Pratt certificates, proof kernels, and (in disguise) the P-vs-NP
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question (Chapter~\ref{ch:prime}).
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\gloss{Fold} Reducing an overflow of the representation (weight
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$2^{255}$ and above) back into range using the modulus identity
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$2^{255} \equiv 19$; costs exactly one multiple of $p$ per unit folded
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(Chapter~\ref{ch:denotation}; Interlude).
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\gloss{FORS} Forest Of Random Subsets: SLH-DSA's few-time signature ---
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$k$ small Merkle trees, one secret revealed per tree, all recomputed
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roots compressed and certified by the hypertree; reuse degrades
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gracefully instead of breaking, which is what buys statelessness
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(Chapter~\ref{ch:secondsummit}).
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\gloss{Goal state} The proof assistant's board: hypotheses above the
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turnstile $\vdash$, obligation below. Reading it is the core tactic
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skill (Chapter~\ref{ch:tactics}).
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\gloss{Hash oracle} A hash function entering a proof as an \emph{axiom}
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with assumed functional behavior and no proven properties; the five
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SLH-DSA verify-path oracles are the standing example, and the audit's
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cone table shows exactly which certificate leans on which
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(Chapter~\ref{ch:secondsummit}).
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\gloss{Hasse bound} An elliptic curve over $\Fp$ has $p + 1 - t$ points
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with $|t| \le 2\sqrt{p}$; the thirty-second sanity check for any
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claimed group order (Chapter~\ref{ch:pyramid}).
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\gloss{Headroom} Bits of slack between a limb's payload (e.g.\ 51 bits)
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and its machine word (64 bits); the budget lazy carries spend
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(Chapter~\ref{ch:why}).
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\gloss{Hypertree} SLH-DSA's tower of $d$ Merkle-tree layers, each tree's
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root signed by a one-time key of the layer above --- a virtual structure
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of $2^h$ keys that is never materialized: any path can be recomputed
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from a seed, and one root pins it all (Chapter~\ref{ch:secondsummit}).
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\gloss{Inductive type} A type defined by listing its constructors
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exhaustively (\lean{Nat}: \lean{zero} and \lean{succ}). Grants both
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pattern matching and the induction principle (Chapters~\ref{ch:lean},
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\ref{ch:tactics}).
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\gloss{Kernel} The small, paranoid core of a proof assistant that
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re-checks every proof object against a fixed rule set; the only
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component whose correctness soundness depends on
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(Chapter~\ref{ch:why}).
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\gloss{Limb} One machine word of a multi-word big-number
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representation; Ed25519 field elements use five 51-bit limbs in 64-bit
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words (Chapters~\ref{ch:why}, \ref{ch:denotation}).
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\gloss{Model} The extracted Lean rendition of the source code, living
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in \code{gen/}; the object theorems quantify over
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(Chapter~\ref{ch:rust}).
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\gloss{Montgomery form} Representing $x$ as $x \cdot R \bmod p$
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(typically $R = 2^{256}$) to make post-multiplication reduction cheap;
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absorbed by adjusting the denotation (Chapter~\ref{ch:denotation}).
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\gloss{Pratt witness} An element $w$ with $w^{p-1} \equiv 1$ and
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$w^{(p-1)/q} \not\equiv 1$ for every prime $q \mid p-1$; its existence
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certifies $p$ prime, given certificates for the $q$'s
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(Chapter~\ref{ch:prime}).
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\gloss{Radix} The base of a limb representation ($2^{51}$ for the
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dalek field, $4$ for this book's toy system).
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\gloss{SLH-DSA} The stateless hash-based digital signature algorithm of
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FIPS~205 (descended from SPHINCS\textsuperscript{+}): FORS under a
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hypertree of Winternitz chains, built from hash functions and nothing
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else --- no structure for Shor's algorithm to attack
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(Chapter~\ref{ch:secondsummit}).
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\gloss{Specification (spec)} The precise statement a program is proven
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to satisfy. The two-clause shape for arithmetic: bounds propagation
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plus value equation. A proof is only as good as its spec
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(Chapters~\ref{ch:denotation}, \ref{ch:honesty}).
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\gloss{Substitution test} Auditing a spec by substituting an
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adversarial implementation and checking whether the statement notices;
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detects trivial specs no tool can flag (Chapter~\ref{ch:honesty}).
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\gloss{Tactic} A command in Lean's interactive proof mode that
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transforms the goal state (\lean{intro}, \lean{rw}, \lean{induction},
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\lean{omega}, \dots), assembling a proof object behind the scenes
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(Chapter~\ref{ch:tactics}).
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\gloss{Torsion} The small-order component of a curve point (order
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dividing the cofactor); killed by multiplying by $8$, hence invisible
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to cofactored verification (Chapter~\ref{ch:pyramid}).
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\gloss{Trusted base} Everything a verification result assumes rather
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than proves: the kernel, the extraction tool, declared axioms
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(SHA-512, untranslatable backends). Honest projects keep it small,
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documented, and machine-visible (Chapters~\ref{ch:rust},
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\ref{ch:honesty}).
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\gloss{Two-clause spec} This book's name for the standard operation
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theorem: \emph{(1)} the operation succeeds and its output satisfies the
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(possibly widened) bounds invariant; \emph{(2)} the output's denotation
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equals the ideal result (Chapter~\ref{ch:denotation}; Interlude).
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\gloss{Winternitz chain (WOTS\textsuperscript{+})} A hash chain
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$c_0, F(c_0), F(F(c_0)), \dots$ signing one digit by revealing the
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chain value at the digit's position; the verifier walks the remaining
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steps to the published end. A checksum makes forward-walking forgeries
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self-defeating (Chapter~\ref{ch:secondsummit}).
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