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mrwulf
b408df3765
Replicates dalek's gen merge: extract.sh co-extracts the Scalar52 backend and scalar::from_bytes_mod_order[_wide] into the CurveField model, pins RUSTFLAGS --cfg curve25519_dalek_backend="serial" (SIMD dispatch compiles out; get_selected_backend becomes a real definition), and upgrades unwrap_u8 to the documented Choice-model definition. ScalarDenote repoints by one import line; both buttons green over the merged gen. Prepares the verify-glue integration (CurveSig) on this fork. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
90 lines
4.1 KiB
Text
90 lines
4.1 KiB
Text
/- ──────────────────────────────────────────────────────────────────────────────
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Proofs/ScalarDenote.lean — SEMANTIC FOUNDATION for the scalar layer:
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from Scalar52 machine limbs to ℤ/ℓℤ, ℓ the ed25519 group order.
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WHAT THIS FILE PROVIDES
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* `Ell` — the group order ℓ = 2²⁵² + 27742317777372353535851937790883648493
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(a prime; the order of the ed25519 basepoint).
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* `Sc := Scalar52` — the transpiled type: 5 little-endian 52-bit-limbed u64s
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(`Array U64 5`; each limb < 2⁵² by the crate's representation invariant).
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* `scVal`/`scLimbs` — the exact ℕ value Σ lᵢ·2^(52i).
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* `ScBnd` — the limb-bound invariant (all limbs < 2⁵²).
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* `scDenote` (⟦·⟧) — the denotation Scalar52 → ZMod ℓ.
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* `L_val` — the transpiled `constants::L` denotes exactly ℓ (kernel-checked).
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RUST ANALOG: `src/backend/serial/u64/scalar.rs` — `pub struct Scalar52(pub [u64; 5])`,
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invariant "5 limbs of 52 bits each, value < ℓ after reduction".
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SCOPE NOTE. This file + the add/sub specs are the tractable core. The
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Montgomery multiplication path (`mul_internal` → `montgomery_reduce`) shares
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the 4×64-Montgomery big-coefficient structure that overflows the Lean kernel
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in the pasta field layer (documented in that repo's POSTMORTEM); it is built
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the same isolated-lemma way but is not yet complete.
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Imports: gen/CurveScalar (the transpiled Scalar52 arithmetic).
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────────────────────────────────────────────────────────────────────────────── -/
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import CurveField.Funs
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open Aeneas Aeneas.Std Result
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open curve25519_dalek
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namespace ScalarProofs
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/-- The ed25519 group order ℓ = 2²⁵² + 27742317777372353535851937790883648493. -/
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def Ell : ℕ := 7237005577332262213973186563042994240857116359379907606001950938285454250989
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/-- The transpiled Scalar52 element type (5 little-endian 52-bit limbs). -/
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abbrev Sc := backend.serial.u64.scalar.Scalar52
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/-- Exact ℕ value of five little-endian 52-bit limbs. -/
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def scLimbs (a0 a1 a2 a3 a4 : U64) : ℕ :=
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a0.val + 2^52 * a1.val + 2^104 * a2.val + 2^156 * a3.val + 2^208 * a4.val
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/-- Exact ℕ value of a `Scalar52`. -/
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def scVal (a : Sc) : ℕ :=
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match (↑a : List U64) with
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| [a0, a1, a2, a3, a4] => scLimbs a0 a1 a2 a3 a4
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| _ => 0
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/-- Every `Scalar52` IS five named u64 limbs. -/
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theorem Sc.exists_limbs (a : Sc) :
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∃ a0 a1 a2 a3 a4 : U64, (↑a : List U64) = [a0, a1, a2, a3, a4] := by
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obtain ⟨l, hl⟩ := a
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match l, hl with
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| [a0, a1, a2, a3, a4], _ => exact ⟨a0, a1, a2, a3, a4, rfl⟩
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@[simp]
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theorem scVal_eq (a : Sc) (a0 a1 a2 a3 a4 : U64)
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(h : (↑a : List U64) = [a0, a1, a2, a3, a4]) :
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scVal a = scLimbs a0 a1 a2 a3 a4 := by
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unfold scVal; rw [h]
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/-- The limb discipline: every limb below 2⁵² (52-bit limbs). -/
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def ScBnd (a : Sc) : Prop :=
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∃ a0 a1 a2 a3 a4 : U64, (↑a : List U64) = [a0, a1, a2, a3, a4] ∧
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a0.val < 2^52 ∧ a1.val < 2^52 ∧ a2.val < 2^52 ∧ a3.val < 2^52 ∧ a4.val < 2^52
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/-- The denotation: machine limbs ↦ ℤ/ℓℤ. -/
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def scDenote (a : Sc) : ZMod Ell := (scVal a : ZMod Ell)
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notation "⟦" a "⟧" => scDenote a
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/-- The transpiled `constants::L` as a limb list. -/
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theorem L_limbs :
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(↑backend.serial.u64.constants.L : List U64) =
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[671914833335277#u64, 3916664325105025#u64, 1367801#u64, 0#u64,
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17592186044416#u64] := by
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unfold backend.serial.u64.constants.L
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rfl
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/-- **The transpiled modulus constant denotes exactly the group order ℓ.**
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Kernel-checked literal arithmetic (no native_decide). -/
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theorem L_val : scVal backend.serial.u64.constants.L = Ell := by
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rw [scVal_eq _ _ _ _ _ _ L_limbs]
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unfold scLimbs Ell
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norm_num
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/-- The limbs of L are each below 2⁵² (needed by the add/sub reductions). -/
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theorem L_bnd : ScBnd backend.serial.u64.constants.L := by
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refine ⟨_, _, _, _, _, L_limbs, ?_, ?_, ?_, ?_, ?_⟩ <;> norm_num
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end ScalarProofs
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