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https://github.com/saymrwulf/betrusted-ed25519-verified.git
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92 lines
4.7 KiB
Text
92 lines
4.7 KiB
Text
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/- ──────────────────────────────────────────────────────────────────────────────
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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⟧, ScBnd out
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· sub: canonical subtrahend → ⟦sub a b⟧ = ⟦a⟧ − ⟦b⟧, ScBnd out
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· mul: Montgomery input bound → ⟦mul a b⟧ = ⟦a⟧ · ⟦b⟧, ScBnd out
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(canonical inputs satisfy it: ℓ·ℓ < 2^260·ℓ)
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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_dalek
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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 ∧ 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⟩
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/-- Subtraction, clean interface. -/
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theorem scalar_sub_correct (a b : Sc) (ha : ScBnd a) (hb : ScBnd b)
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(hcb : scVal b ≤ Ell) :
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backend.serial.u64.scalar.Scalar52.sub a b
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⦃ r => ScBnd r ∧ 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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exact ⟨hr.1, hr.2⟩
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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 ∧ 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⟩
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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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with 52-bit-bounded limb output. 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 ∧ scDenote r = scDenote a + scDenote b ⦄) ∧
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(∀ a b : Sc, ScBnd a → ScBnd b → scVal b ≤ Ell →
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backend.serial.u64.scalar.Scalar52.sub a b
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⦃ r => ScBnd r ∧ 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 ∧ 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 hcb => scalar_sub_correct a b ha hb 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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