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mrwulf
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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.
99 lines
5.4 KiB
Text
99 lines
5.4 KiB
Text
/- ──────────────────────────────────────────────────────────────────────────────
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Proofs/ScalarMain.lean — the scalar-layer aggregate certificate.
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Clean-interface corollaries of the assembly proofs, stated through
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`ScBnd` (52-bit limb representation) and `scDenote` (⟦·⟧ : Scalar52 →
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ZMod ℓ), plus the single bundled certificate `scalarImplementation`:
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· add: canonical inputs → ⟦add a b⟧ = ⟦a⟧ + ⟦b⟧
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· sub: canonical inputs → ⟦sub a b⟧ = ⟦a⟧ − ⟦b⟧
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· mul: Montgomery input bound → ⟦mul a b⟧ = ⟦a⟧ · ⟦b⟧
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(canonical inputs satisfy it: ℓ·ℓ < 2^260·ℓ)
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Every output is both ScBnd (52-bit limbs) and canonical (scVal < ℓ):
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the layer is closed under its own preconditions.
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Audit: `#print axioms ScalarProofs.scalarImplementation` must report
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exactly [propext, Classical.choice, Quot.sound].
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────────────────────────────────────────────────────────────────────────────── -/
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import Proofs.ScalarFullMulSpec
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import Proofs.ScalarAddSpec
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open Aeneas Aeneas.Std Result
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open curve25519
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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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/-- Addition, clean interface. -/
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theorem scalar_add_correct (a b : Sc) (ha : ScBnd a) (hb : ScBnd b)
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(hca : scVal a < Ell) (hcb : scVal b < Ell) :
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backend.serial.u64.scalar.Scalar52.add a b
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⦃ r => ScBnd r ∧ scVal r < Ell ∧ scDenote r = scDenote a + scDenote b ⦄ := by
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obtain ⟨a0, a1, a2, a3, a4, hal, hA0, hA1, hA2, hA3, hA4⟩ := ha
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obtain ⟨b0, b1, b2, b3, b4, hbl, hB0, hB1, hB2, hB3, hB4⟩ := hb
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apply spec_mono (add_val_spec a b a0 a1 a2 a3 a4 b0 b1 b2 b3 b4 hal hbl
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⟨hA0, hA1, hA2, hA3, hA4⟩ ⟨hB0, hB1, hB2, hB3, hB4⟩ hca hcb)
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intro r hr
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exact ⟨hr.1, hr.2.1, hr.2.2⟩
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/-- Subtraction, clean interface: canonical inputs give a canonical output
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(β = 0: r = a − b < ℓ; β = 1: the guard says a < b, so r = a − b + ℓ < ℓ). -/
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theorem scalar_sub_correct (a b : Sc) (ha : ScBnd a) (hb : ScBnd b)
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(hca : scVal a < Ell) (hcb : scVal b ≤ Ell) :
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backend.serial.u64.scalar.Scalar52.sub a b
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⦃ r => ScBnd r ∧ scVal r < Ell ∧ scDenote r = scDenote a - scDenote b ⦄ := by
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obtain ⟨a0, a1, a2, a3, a4, hal, hA0, hA1, hA2, hA3, hA4⟩ := ha
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obtain ⟨b0, b1, b2, b3, b4, hbl, hB0, hB1, hB2, hB3, hB4⟩ := hb
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apply spec_mono (sub_val_spec a b a0 a1 a2 a3 a4 b0 b1 b2 b3 b4 hal hbl
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⟨hA0, hA1, hA2, hA3, hA4⟩ ⟨hB0, hB1, hB2, hB3, hB4⟩ hcb)
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intro r hr
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obtain ⟨hbnds, ⟨β, hβle, heq, hguard⟩, hden⟩ := hr
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refine ⟨hbnds, ?_, hden⟩
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rcases Nat.le_one_iff_eq_zero_or_eq_one.mp hβle with h0 | h1
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· subst h0; omega
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· subst h1; have := hguard rfl; omega
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/-- Multiplication, clean interface. The Montgomery hypothesis
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scVal a · scVal b < 2^260·ℓ holds in particular for canonical inputs. -/
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theorem scalar_mul_correct (a b : Sc) (ha : ScBnd a) (hb : ScBnd b)
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(hm : scVal a * scVal b < 2^260 * Ell) :
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backend.serial.u64.scalar.Scalar52.mul a b
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⦃ r => ScBnd r ∧ scVal r < Ell ∧ scDenote r = scDenote a * scDenote b ⦄ := by
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obtain ⟨a0, a1, a2, a3, a4, hal, hA0, hA1, hA2, hA3, hA4⟩ := ha
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obtain ⟨b0, b1, b2, b3, b4, hbl, hB0, hB1, hB2, hB3, hB4⟩ := hb
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apply spec_mono (mul_spec a b a0 a1 a2 a3 a4 b0 b1 b2 b3 b4 hal hbl
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⟨hA0, hA1, hA2, hA3, hA4⟩ ⟨hB0, hB1, hB2, hB3, hB4⟩ hm)
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intro r hr
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exact ⟨hr.1, hr.2.1, hr.2.2⟩
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/-- Canonical inputs always satisfy the Montgomery multiplication bound. -/
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theorem canonical_mul_bound {a b : Sc} (hca : scVal a < Ell) (hcb : scVal b < Ell) :
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scVal a * scVal b < 2^260 * Ell := by
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have h1 : scVal a * scVal b < Ell * Ell := Nat.mul_lt_mul'' hca hcb
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have h2 : Ell * Ell ≤ 2^260 * Ell :=
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Nat.mul_le_mul_right Ell (by unfold Ell; norm_num)
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exact lt_of_lt_of_le h1 h2
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/-- **The scalar-layer certificate**: the transpiled `Scalar52` add, sub
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and mul all denote the ring operations of ZMod ℓ on canonical inputs,
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and every output is again 52-bit-bounded AND canonical — the layer is
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closed under its own preconditions. One theorem, one axiom audit. -/
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theorem scalarImplementation :
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(∀ a b : Sc, ScBnd a → ScBnd b → scVal a < Ell → scVal b < Ell →
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backend.serial.u64.scalar.Scalar52.add a b
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⦃ r => ScBnd r ∧ scVal r < Ell ∧ scDenote r = scDenote a + scDenote b ⦄) ∧
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(∀ a b : Sc, ScBnd a → ScBnd b → scVal a < Ell → scVal b ≤ Ell →
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backend.serial.u64.scalar.Scalar52.sub a b
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⦃ r => ScBnd r ∧ scVal r < Ell ∧ scDenote r = scDenote a - scDenote b ⦄) ∧
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(∀ a b : Sc, ScBnd a → ScBnd b → scVal a < Ell → scVal b < Ell →
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backend.serial.u64.scalar.Scalar52.mul a b
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⦃ r => ScBnd r ∧ scVal r < Ell ∧ scDenote r = scDenote a * scDenote b ⦄) :=
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⟨fun a b ha hb hca hcb => scalar_add_correct a b ha hb hca hcb,
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fun a b ha hb hca hcb => scalar_sub_correct a b ha hb hca hcb,
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fun a b ha hb hca hcb =>
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scalar_mul_correct a b ha hb (canonical_mul_bound hca hcb)⟩
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end ScalarProofs
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