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saymrwulf
5861c73c22
- pen-and-paper worked examples in all 12 chapters, using the REAL constants throughout: 2^-64 waiting-time arithmetic, headroom budgets, hand type-checking, rfl traces, full goal-state boards, the column-sum audit at 2^54, inverting 19 mod p via Euclid, the x19 fold at real weights, denoting p itself (telescope), the 16p audit (8 fails by 151), the 254+11 inversion-chain bookkeeping, the substitution test, sizing the 28-vs-1000 extraction, cofactor/torsion arithmetic, and the full Bernstein-Lange completeness derivation - CORRECTNESS FIX: ch7 asserted a false factorization of p-1; replaced with the computationally verified p-1 = 2^2 * 3 * 65147 * Q (Q 71-digit prime), witness w=2 verified for all four Pratt conditions - every chapter's exercises now followed immediately by 'Solutions and pathways' (pathway first, then answer), incl. new exercises - NEW Interlude: a complete two-clause verification done entirely by hand, then mapped line-by-line onto the compiled Lean proof - NEW appendices: A pen-and-paper toolkit (8 recipe cards + drills + answers), B guided walkthroughs of every exercise-file hole, C tour of the real repositories; plus glossary, instructor notes, 13-week plan - preamble: worked-example box, solution macros, math-safe inline code Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
107 lines
5.5 KiB
Markdown
107 lines
5.5 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 the arithmetic core of Ed25519 (from
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[curve25519-dalek](https://github.com/dalek-cryptography/curve25519-dalek)
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and three production forks) and the Pasta curves' field layer were
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machine-checked in Lean 4 against models extracted from the actual Rust
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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) | field 𝔽ₚ + Edwards group law + scalar foundations, upstream dalek |
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| [anza-ed25519-verified](https://github.com/saymrwulf/anza-ed25519-verified) | same layers, Solana's fork, its own extraction |
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| [risc0-ed25519-verified](https://github.com/saymrwulf/risc0-ed25519-verified) | same layers, RISC Zero's fork |
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| [betrusted-ed25519-verified](https://github.com/saymrwulf/betrusted-ed25519-verified) | same layers, 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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- 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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