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https://github.com/saymrwulf/anza-ed25519-verified.git
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mrwulf
7dc6bdc40d
Canonicity pass (the layer is now closed under its own preconditions): - sub_val_spec post carries the exact value equation (exists beta <= 1, scVal r + scVal b = scVal a + ell*beta, with the underflow guard beta = 1 -> scVal a < scVal b) - add/montgomery_reduce/mul/aggregate posts all carry scVal r < ell: canonical inputs give canonical outputs everywhere. Needed because from_bytes_wide (hash-to-scalar) feeds Montgomery outputs into add. Hash-to-scalar foundation (toward Scalar::from_hash / EdDSA verify): - extraction scope + from_bytes_wide (brings constants::R); regenerated gen - source repos carry a documented Aeneas-compat patch: the bare `hi[4] = words[7] >> 20` extracts ill-typed at pin bf13c42e; masked (semantic no-op, words[7] >> 20 < 2^44) - Proofs/ScalarWideSpec.lean: R constant lemmas (R = 2^260 mod ell, witness 2^260 = R + 255*ell) and montgomery_mul_spec, the single Montgomery round: [r]*2^260 = [a]*[b], canonical bounded output check-scalar.sh: 10 proof files, 11 kernel audits, all exactly [propext, Classical.choice, Quot.sound]. Button pressed fresh: green.
205 lines
10 KiB
Text
205 lines
10 KiB
Text
/- ──────────────────────────────────────────────────────────────────────────────
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Proofs/ScalarReduceSpec.lean — Scalar52 Montgomery reduction, the main walk.
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Rounds 0–4 of `montgomery_reduce` (the five exact-division `part1` rounds),
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composed with `mont_tail_spec` (rounds 5–8 + the `sub _ L`
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canonicalization) from ScalarMontSpec. Split across two files because the
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74-step monolith exceeds a single declaration's elaboration budget
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(METHOD 4); each half is a `mul_internal`-sized walk.
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────────────────────────────────────────────────────────────────────────────── -/
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import Proofs.ScalarMontSpec
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open Aeneas Aeneas.Std Result
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open curve25519
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set_option maxHeartbeats 16000000
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set_option linter.unusedSimpArgs false
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set_option exponentiation.threshold 600
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namespace ScalarProofs
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open Aeneas.Std.WP
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/-- The nonce value N = Σ n_k·2^52k of five 52-bit digits stays below R.
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Standalone so its `omega` sees exactly five hypotheses — inside the
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walk, the same goal drags ~110 hypotheses (five round equations with
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2^102-scale coefficients included) into one Presburger instance and
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never returns (measured: the single spiralling line of the file). -/
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theorem nonce_sum_bound {n0 n1 n2 n3 n4 : ℕ} (h0 : n0 < 2^52) (h1 : n1 < 2^52)
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(h2 : n2 < 2^52) (h3 : n3 < 2^52) (h4 : n4 < 2^52) :
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n0 + 2^52 * n1 + 2^104 * n2 + 2^156 * n3 + 2^208 * n4 < 2^260 := by
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omega
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/-- **Montgomery reduction is exact division by R = 2^260 mod ℓ.** For a
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9-limb double-width input Z = Σ z_k·2^52k with column-bounded limbs
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(z_k < 2^107, what `mul_internal` produces) and the standard
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Montgomery hypothesis Z < 2^260·ℓ, the transpiled `montgomery_reduce`
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returns a 52-bit-bounded scalar r with ⟦r⟧·2^260 = Z in ZMod ℓ.
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Rounds 0–4 walk `part1_spec` here; rounds 5–8 and the `sub _ L`
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canonicalization are `mont_tail_spec`; the value accounting is
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`mont_head_telescope` + `mont_bound`. -/
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theorem montgomery_reduce_spec (limbs : Std.Array Std.U128 9#usize)
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(z0 z1 z2 z3 z4 z5 z6 z7 z8 : Std.U128)
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(hl : (↑limbs : List Std.U128) = [z0, z1, z2, z3, z4, z5, z6, z7, z8])
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(hzb : z0.val < 2^107 ∧ z1.val < 2^107 ∧ z2.val < 2^107 ∧ z3.val < 2^107 ∧
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z4.val < 2^107 ∧ z5.val < 2^107 ∧ z6.val < 2^107 ∧ z7.val < 2^107 ∧
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z8.val < 2^107)
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(hZ : z0.val + 2^52 * z1.val + 2^104 * z2.val + 2^156 * z3.val + 2^208 * z4.val + 2^260 * z5.val + 2^312 * z6.val + 2^364 * z7.val + 2^416 * z8.val < 2^260 * Ell) :
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backend.serial.u64.scalar.Scalar52.montgomery_reduce limbs
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⦃ r => (∃ s0 s1 s2 s3 s4 : U64, (↑r : List U64) = [s0, s1, s2, s3, s4] ∧
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s0.val < 2^52 ∧ s1.val < 2^52 ∧ s2.val < 2^52 ∧ s3.val < 2^52 ∧
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s4.val < 2^52) ∧
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scVal r < Ell ∧
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scDenote r * 2^260 = ((z0.val + 2^52 * z1.val + 2^104 * z2.val + 2^156 * z3.val + 2^208 * z4.val + 2^260 * z5.val + 2^312 * z6.val + 2^364 * z7.val + 2^416 * z8.val : ℕ) : ZMod Ell) ⦄ := by
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obtain ⟨hz0, hz1, hz2, hz3, hz4, hz5, hz6, hz7, hz8⟩ := hzb
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-- Hide the Montgomery bound behind an existential for the duration of
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-- the walk: a bare 2^416-coefficient inequality in the local context
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-- sends every step's side-condition automation into the huge literals
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-- (measured: the walk never finishes). It re-enters only for mont_bound.
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replace hZ : ∃ B : ℕ,
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(z0.val + 2^52 * z1.val + 2^104 * z2.val + 2^156 * z3.val
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+ 2^208 * z4.val + 2^260 * z5.val + 2^312 * z6.val + 2^364 * z7.val
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+ 2^416 * z8.val < B) ∧ B = 2^260 * Ell := ⟨_, hZ, rfl⟩
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unfold backend.serial.u64.scalar.Scalar52.montgomery_reduce
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backend.serial.u64.scalar.Scalar52.Insts.CoreOpsIndexIndexUsizeU64.index
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step as ⟨i, hi⟩
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simp [hl] at hi
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have hvi : i.val = z0.val := by rw [hi]
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have hbi : i.val < 2^124 := by clear hZ; omega
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step with (part1_spec i hbi) as ⟨carry, n0, hn0b, hE0⟩
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rw [hvi] at hE0
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have hcb0 : carry.val < 2^60 := by clear hZ; omega
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step as ⟨i1, hi1⟩
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simp [hl] at hi1
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have hvi1 : i1.val = z1.val := by rw [hi1]
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have hsi2 : carry.val + i1.val < 2^128 := by clear hZ; omega
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step as ⟨i2, hi2⟩
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have hvi2 : i2.val = carry.val + i1.val := by rw [hi2]
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have hbi2 : i2.val < 2^110 := by clear hZ; omega
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step as ⟨i3, hi3⟩
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try simp [L_limbs] at hi3
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have hvi3 : i3.val = 3916664325105025 := by rw [hi3]; rfl
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step with m_spec as ⟨i4, hi4⟩
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have hvi4 : i4.val = n0.val * 3916664325105025 := by rw [hi4, hvi3]
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have hbi4 : i4.val < 2^104 := by clear hZ; rw [hvi4]; omega
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have hsi5 : i2.val + i4.val < 2^128 := by clear hZ; omega
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step as ⟨i5, hi5⟩
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have hvi5 : i5.val = i2.val + i4.val := by rw [hi5]
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have hbi5 : i5.val < 2^112 := by clear hZ; omega
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have hbp1 : i5.val < 2^124 := by clear hZ; omega
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step with (part1_spec i5 hbp1) as ⟨carry1, n1, hn1b, hE1⟩
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rw [hvi5, hvi2, hvi1, hvi4] at hE1
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have hcb1 : carry1.val < 2^60 := by clear hZ; omega
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step as ⟨i6, hi6⟩
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simp [hl] at hi6
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have hvi6 : i6.val = z2.val := by rw [hi6]
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have hsi7 : carry1.val + i6.val < 2^128 := by clear hZ; omega
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step as ⟨i7, hi7⟩
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have hvi7 : i7.val = carry1.val + i6.val := by rw [hi7]
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have hbi7 : i7.val < 2^110 := by clear hZ; omega
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step as ⟨i8, hi8⟩
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try simp [L_limbs] at hi8
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have hvi8 : i8.val = 1367801 := by rw [hi8]; rfl
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step with m_spec as ⟨i9, hi9⟩
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have hvi9 : i9.val = n0.val * 1367801 := by rw [hi9, hvi8]
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have hbi9 : i9.val < 2^104 := by clear hZ; rw [hvi9]; omega
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have hsi10 : i7.val + i9.val < 2^128 := by clear hZ; omega
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step as ⟨i10, hi10⟩
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have hvi10 : i10.val = i7.val + i9.val := by rw [hi10]
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have hbi10 : i10.val < 2^112 := by clear hZ; omega
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step with m_spec as ⟨i11, hi11⟩
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have hvi11 : i11.val = n1.val * 3916664325105025 := by rw [hi11, hvi3]
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have hbi11 : i11.val < 2^104 := by clear hZ; rw [hvi11]; omega
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have hsi12 : i10.val + i11.val < 2^128 := by clear hZ; omega
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step as ⟨i12, hi12⟩
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have hvi12 : i12.val = i10.val + i11.val := by rw [hi12]
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have hbi12 : i12.val < 2^113 := by clear hZ; omega
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have hbp2 : i12.val < 2^124 := by clear hZ; omega
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step with (part1_spec i12 hbp2) as ⟨carry2, n2, hn2b, hE2⟩
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rw [hvi12, hvi10, hvi7, hvi6, hvi9, hvi11] at hE2
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have hcb2 : carry2.val < 2^62 := by clear hZ; omega
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step as ⟨i13, hi13⟩
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simp [hl] at hi13
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have hvi13 : i13.val = z3.val := by rw [hi13]
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have hsi14 : carry2.val + i13.val < 2^128 := by clear hZ; omega
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step as ⟨i14, hi14⟩
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have hvi14 : i14.val = carry2.val + i13.val := by rw [hi14]
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have hbi14 : i14.val < 2^110 := by clear hZ; omega
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step with m_spec as ⟨i15, hi15⟩
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have hvi15 : i15.val = n1.val * 1367801 := by rw [hi15, hvi8]
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have hbi15 : i15.val < 2^104 := by clear hZ; rw [hvi15]; omega
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have hsi16 : i14.val + i15.val < 2^128 := by clear hZ; omega
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step as ⟨i16, hi16⟩
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have hvi16 : i16.val = i14.val + i15.val := by rw [hi16]
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have hbi16 : i16.val < 2^112 := by clear hZ; omega
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step with m_spec as ⟨i17, hi17⟩
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have hvi17 : i17.val = n2.val * 3916664325105025 := by rw [hi17, hvi3]
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have hbi17 : i17.val < 2^104 := by clear hZ; rw [hvi17]; omega
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have hsi18 : i16.val + i17.val < 2^128 := by clear hZ; omega
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step as ⟨i18, hi18⟩
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have hvi18 : i18.val = i16.val + i17.val := by rw [hi18]
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have hbi18 : i18.val < 2^113 := by clear hZ; omega
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have hbp3 : i18.val < 2^124 := by clear hZ; omega
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step with (part1_spec i18 hbp3) as ⟨carry3, n3, hn3b, hE3⟩
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rw [hvi18, hvi16, hvi14, hvi13, hvi15, hvi17] at hE3
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have hcb3 : carry3.val < 2^62 := by clear hZ; omega
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step as ⟨i19, hi19⟩
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simp [hl] at hi19
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have hvi19 : i19.val = z4.val := by rw [hi19]
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have hsi20 : carry3.val + i19.val < 2^128 := by clear hZ; omega
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step as ⟨i20, hi20⟩
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have hvi20 : i20.val = carry3.val + i19.val := by rw [hi20]
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have hbi20 : i20.val < 2^110 := by clear hZ; omega
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step as ⟨i21, hi21⟩
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try simp [L_limbs] at hi21
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have hvi21 : i21.val = 17592186044416 := by rw [hi21]; rfl
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step with m_spec as ⟨i22, hi22⟩
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have hvi22 : i22.val = n0.val * 17592186044416 := by rw [hi22, hvi21]
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have hbi22 : i22.val < 2^104 := by clear hZ; rw [hvi22]; omega
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have hsi23 : i20.val + i22.val < 2^128 := by clear hZ; omega
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step as ⟨i23, hi23⟩
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have hvi23 : i23.val = i20.val + i22.val := by rw [hi23]
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have hbi23 : i23.val < 2^112 := by clear hZ; omega
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step with m_spec as ⟨i24, hi24⟩
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have hvi24 : i24.val = n2.val * 1367801 := by rw [hi24, hvi8]
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have hbi24 : i24.val < 2^104 := by clear hZ; rw [hvi24]; omega
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have hsi25 : i23.val + i24.val < 2^128 := by clear hZ; omega
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step as ⟨i25, hi25⟩
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have hvi25 : i25.val = i23.val + i24.val := by rw [hi25]
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have hbi25 : i25.val < 2^113 := by clear hZ; omega
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step with m_spec as ⟨i26, hi26⟩
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have hvi26 : i26.val = n3.val * 3916664325105025 := by rw [hi26, hvi3]
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have hbi26 : i26.val < 2^104 := by clear hZ; rw [hvi26]; omega
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have hsi27 : i25.val + i26.val < 2^128 := by clear hZ; omega
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step as ⟨i27, hi27⟩
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have hvi27 : i27.val = i25.val + i26.val := by rw [hi27]
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have hbi27 : i27.val < 2^114 := by clear hZ; omega
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have hbp4 : i27.val < 2^124 := by clear hZ; omega
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step with (part1_spec i27 hbp4) as ⟨carry4, n4, hn4b, hE4⟩
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rw [hvi27, hvi25, hvi23, hvi20, hvi19, hvi22, hvi24, hvi26] at hE4
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have hcb4 : carry4.val < 2^62 := by clear hZ; omega
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-- the head telescope gives X′·2^260 = Z + N·ℓ; mont_bound gives X′ < 2ℓ
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have hHT := mont_head_telescope z0.val z1.val z2.val z3.val z4.val z5.val
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z6.val z7.val z8.val n0.val n1.val n2.val n3.val n4.val
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carry.val carry1.val carry2.val carry3.val carry4.val
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hE0 hE1 hE2 hE3 hE4
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have hNb := nonce_sum_bound hn0b hn1b hn2b hn3b hn4b
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obtain ⟨B, hZlt, hBeq⟩ := hZ
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subst hBeq
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have hXb := mont_bound _ _ _ hHT hZlt hNb
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-- rounds 5–8 + canonicalization
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apply spec_mono (mont_tail_spec limbs z0 z1 z2 z3 z4 z5 z6 z7 z8 carry4
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n1 n2 n3 n4 i3 i8 i21 hl hvi3 hvi8 hvi21 hcb4
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⟨hn1b, hn2b, hn3b, hn4b⟩ ⟨hz5, hz6, hz7, hz8⟩ hXb)
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intro r hr
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refine ⟨hr.1, hr.2.1, ?_⟩
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rw [hr.2.2]
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have hEz : (Ell : ZMod Ell) = 0 := ZMod.natCast_self Ell
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have hc := congrArg (Nat.cast (R := ZMod Ell)) hHT
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push_cast at hc
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rw [hEz] at hc
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push_cast
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linear_combination hc
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end ScalarProofs
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