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mrwulf
1029f0c251
The owner's requirement, now the book's explicit contract (new "ratchet
rule" box in chapter 1): every load-bearing idea is worked at least
twice - once at napkin scale (mod 13, inverses by scanning) and once at
REAL scale, the actual 77-digit Ed25519 constants printed in full with
no digits hidden and no artificial zeros; where raw size genuinely
exceeds paper (a 77-digit square root), the book says so and teaches
witness-auditing instead of pretending.
Socratic audit result: chapters 1-11 already honor the rule (verified
rung by rung - e.g. ch6 pairs the Z/12 clock with 19^-1 mod the real p;
ch7 pairs Pratt-for-97 with costing the real certificate; ch8 runs the
extracted model at the real envelope edge). The gaps were ch12's three
summit rungs, which had NO numeric examples at all. Filled:
- Group law: "running the addition law by hand - napkin curve, then the
real one". Doubling (2,4)->(10,11) mod 13 in full; then the real base
point with x1, y1 printed in 5-digit groups, the first machine step
certified as x1*y1 = q*p + u with the 77-digit witness q printed, and
the student auditing it by casting out nines AND elevens (both clocks
close: 3=3, 1=1). Lands on the real 2B coordinates.
- Scalars: the cycle felt on the napkin curve first (order 16, so
21P = 5P) before the real prime ell.
- Apex: "decompression, run twice". Encode/decode (10,11) as "(11,
even)" mod 13 - x^2 = 3/9 = 9, roots {3,10}, parity picks 10; then
the real compressed base point: all 32 bytes printed (58 66...66),
byte 31 = 0x66 sign-bit read, y_B printed in full, and the
no-shortcuts full-size hand verification 5*y_B - 4 = 4*p, both
78-digit sides printed for digit-by-digit comparison. The square
root honestly declared machine territory, with the witness-checked
certificate named.
- New paper exercise 12.4 (+pathway/solution): encode & decompress
3P = (6,10) solo - the wrong root lands on -3P, one bit doing real
cryptographic work.
Every printed constant machine-verified before typesetting (base point
on-curve, q*p+u exact, 5y-4 = 4p exact, 2B on-curve, toy order 16).
PDF rebuilt: 109 pages, zero errors. Honesty ledger records the
revision.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
132 lines
7.2 KiB
Markdown
132 lines
7.2 KiB
Markdown
# Verifying Cryptography with Lean 4
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**A hands-on curriculum for undergraduates with zero formal-verification
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background** — from `1 + 1 = 2` to reading (and extending) real,
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machine-checked proofs that production elliptic-curve code is correct.
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This is the educational companion to a family of verification projects in
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which complete Ed25519 proof pyramids (from
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[curve25519-dalek](https://github.com/dalek-cryptography/curve25519-dalek)
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and three production forks — field, group law, scalars, and the signature
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verifier itself) and the Pasta curves' field layer were machine-checked in
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Lean 4 against models extracted from the actual Rust sources:
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| Companion project | What is verified there |
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|---|---|
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| [dalek-ed25519-verified](https://github.com/saymrwulf/dalek-ed25519-verified) | the complete pyramid, upstream dalek: field 𝔽ₚ + Edwards group law + scalar arithmetic mod ℓ + the four-tier signature apex (accept ⇔ decompress(R) = [k](−A)+[s]B, hash opaque) |
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| [anza-ed25519-verified](https://github.com/saymrwulf/anza-ed25519-verified) | the complete pyramid, Solana's fork, its own extraction |
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| [risc0-ed25519-verified](https://github.com/saymrwulf/risc0-ed25519-verified) | the complete pyramid, RISC Zero's fork |
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| [betrusted-ed25519-verified](https://github.com/saymrwulf/betrusted-ed25519-verified) | the complete pyramid, Betrusted's fork |
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| [pasta-pallas-verified](https://github.com/saymrwulf/pasta-pallas-verified) | Pallas modulus primality (Lucas/Pratt), Montgomery foundations |
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| [formal-verification-control](https://github.com/saymrwulf/formal-verification-control) | the method: invariants, terrain map, failure map, tooling |
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## The book
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**[`main.pdf`](main.pdf)** — twelve chapters + interlude + three
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appendices, 106 pages, full color, built with LaTeX/TikZ from the sources
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in this repo. No prior Lean or formal methods assumed; high-school algebra
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and a little programming suffice.
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1. **Why Verify?** — the carry bug testing cannot find
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2. **Meet Lean** — programs, types, inductive data
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3. **Propositions as Types** — Curry–Howard: proofs *are* programs
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4. **Tactics** — proving as a dialogue with the goal state
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5. **Numbers and Automation** — `omega`, `ring`, `norm_num`, `decide`, and the `simp` discipline
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6. **Modular Arithmetic** — clock worlds, fields, why 2²⁵⁵ − 19
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7. **Primality Certificates** — convincing a paranoid kernel a 77-digit number is prime
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8. **From Rust to Lean** — the Charon/Aeneas extraction pipeline
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9. **The Denotation Bridge** — the commuting square at the heart of it all
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— ***Interlude*** — a complete verification, entirely by hand, then re-enacted in Lean line by line
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10. **Verifying a Field** — the full campaign, told honestly (including the crash)
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11. **Honesty and Axioms** — `#print axioms`, hollow certificates, trusted bases
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12. **The Pyramid** — group law, scalars, signatures, and where you come in
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Appendices: **A** — the pen-and-paper toolkit (recipe cards with drills);
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**B** — guided walkthroughs of every exercise-file hole; **C** — a tour of
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the real repositories. Plus a glossary and a thirteen-week course plan.
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The didactic machinery, deliberately heavy:
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- **Pen-and-paper worked examples in every chapter** — computations with
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the *real* constants (2²⁵⁵−19, radix 2⁵¹, the fold constant 19, the
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actual 254-squaring inversion chain, the true Pratt tree
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p−1 = 2²·3·65147·Q), because the real numbers carry the real arguments.
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Highlights: inverting 19 modulo the 77-digit prime in five lines of
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Euclid; a fully hand-checked primality certificate for 97; the ×19 fold
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derived at the real weights; the 16p subtraction constant audited to the
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bit (8 fails by 151); the complete Bernstein–Lange completeness chain.
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- **Solutions immediately after every exercise set** — each one leads with
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the *pathway* (how a person finds the answer) before the answer itself.
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- Boxed **Big idea / Try it / Pitfall / Aha / Checkpoint** elements, TikZ
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figures throughout.
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Everything the book claims about the companion projects reflects their
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actual, auditable state — including open frontiers.
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## The exercises (they run!)
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`exercises/ChNN.lean` are working files with `sorry` holes;
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`solutions/ChNN.lean` are complete. **Every solution file compiles with
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zero errors** against the pinned toolchain (Lean `v4.30.0-rc2`, Mathlib
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`5450b53e`); solutions to proof exercises contain no `sorry`.
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Setup (one-time, ~5 min + Mathlib cache download):
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```bash
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# 1. install elan (Lean version manager) if you haven't:
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curl https://elan.lean-lang.org/elan-init.sh -sSf | sh
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# 2. fetch the Mathlib build cache (do NOT build Mathlib yourself):
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cd verifying-crypto-with-lean
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lake exe cache get
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# 3. open the folder in VS Code with the "Lean 4" extension, or:
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lake build Solutions # compiles all solution files as a check
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```
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Chapters 2–4 need no Mathlib at all — you can start them with any Lean 4
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install while the cache downloads.
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## Building the book
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Any TeX Live ≥ 2023 with `tikz`, `tcolorbox`, `listings`, `lmodern`:
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```bash
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pdflatex main.tex && pdflatex main.tex # twice for the TOC
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```
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## Honesty ledger
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In the spirit of Chapter 11:
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- All `solutions/*.lean` were compiled (and their `#eval` outputs checked
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against their comments) at authoring time with the pinned versions above.
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- Exercise templates compile with `sorry` warnings only.
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- The book's claims about the companion projects (what is proven, what is
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frontier) mirror those repos' own READMEs and TRUSTED-BASE ledgers at the
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time of writing; the repos, not this book, are the source of truth.
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Re-audited 2026-07-06 after the signature apex reached its final
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four-tier form (coherence pass 4): chapter 12's status diagram, apex
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section, and audit-drill solution, chapter 11's boundary example,
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chapter 8's extraction notes, the repo tour, and this table were
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brought up to the proven state.
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- Didactic revision (2026-07-06, same day): the book now states and keeps
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a "ratchet rule" (chapter 1) — every load-bearing idea worked at napkin
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scale AND at real scale with the full 77-digit constants printed,
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nothing elided. Chapter 12 gained the missing rungs: the addition law
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run by hand on a mod-13 curve and then on the real base point (with a
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machine-supplied quotient witness audited by casting out nines and
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elevens), the scalar cycle felt on the napkin curve, decompression run
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twice (mod-13 sign-bit walk, then the real compressed base point:
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byte-31 sign bit, and the full-size hand verification 5·y_B − 4 = 4·p,
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every digit printed), plus a new paper exercise (12.4). Every printed
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constant was machine-verified before typesetting; the PDF (109 pages)
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is rebuilt from these sources.
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- The PDF in the repo is built from the committed sources by the command
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above; rebuild it yourself if you don't trust binaries (good instinct).
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- The three named solution certificates were kernel-audited
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(coherence pass 2, 2026-07-03): `Ch09.add_spec` depends on
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`[propext, Classical.choice, Quot.sound]`; `Ch09.mulVal_spec` and
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`Ch12.addFixed_spec` on `[propext, Quot.sound]` only. The Interlude's
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"compiled and axiom-audited" phrase shipped one pass before its audit
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had actually been run — caught by the verification projects' own
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coherence process and made true; recorded here in the spirit of
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Chapter 11.
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