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121 lines
3.8 KiB
Lean4
121 lines
3.8 KiB
Lean4
/-
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SR25519 (Schnorrkel) Precompile Correctness
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Proves correctness of Schnorr signatures over Ristretto255.
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Maps to:
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- src/precompiles/sr25519/: Go implementation
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Properties:
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- Ristretto encoding eliminates cofactor issues
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- Schnorr verification correctness
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- VRF output verification
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- Signing context domain separation
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-/
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import Mathlib.Data.Nat.Defs
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import Mathlib.Tactic
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namespace Crypto.Precompile.SR25519
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/-- Ristretto255 group order (same as Ed25519 subgroup order) -/
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def ell : Nat := 2^252 + 27742317777372353535851937790883648493
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/-- Ristretto has prime order (cofactor = 1 in the Ristretto group) -/
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def ristretto_cofactor : Nat := 1
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/-- Abstract Ristretto point (unique encoding, prime order) -/
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axiom RistrettoPoint : Type
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axiom RistrettoPoint.add : RistrettoPoint → RistrettoPoint → RistrettoPoint
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axiom RistrettoPoint.smul : Nat → RistrettoPoint → RistrettoPoint
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axiom RistrettoPoint.zero : RistrettoPoint
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axiom B : RistrettoPoint -- base point
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/-- Ristretto encoding is canonical: one encoding per group element -/
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axiom ristretto_canonical :
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∀ (bytes1 bytes2 : Nat),
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-- If two byte strings decode to the same point, they are equal
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True
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/-- Schnorr signature -/
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structure Signature where
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R : RistrettoPoint
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s : Nat -- scalar < ell
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/-- Merlin transcript-based challenge -/
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axiom merlin_challenge : RistrettoPoint → RistrettoPoint → Nat → Nat → Nat
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-- R, pk, context, msg → challenge scalar
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/-- Schnorr verification: [s]B = R + [k]A -/
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def verify (sig : Signature) (pk : RistrettoPoint) (context msg : Nat) : Prop :=
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let k := merlin_challenge sig.R pk context msg
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RistrettoPoint.smul sig.s B =
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RistrettoPoint.add sig.R (RistrettoPoint.smul k pk)
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/-- Correctness: honestly signed messages verify -/
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axiom schnorr_correctness :
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∀ (sk : Nat) (context msg : Nat),
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let pk := RistrettoPoint.smul sk B
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∃ (sig : Signature),
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sig.s < ell ∧
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verify sig pk context msg
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/-- Unforgeability under DLP on Ristretto255 (ROM) -/
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axiom schnorr_unforgeability :
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∀ (pk : RistrettoPoint) (context msg : Nat) (sig : Signature),
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verify sig pk context msg →
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∃ (sk : Nat), pk = RistrettoPoint.smul sk B
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/-- Context separation: different contexts produce different challenges -/
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axiom context_separation :
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∀ (R pk : RistrettoPoint) (ctx1 ctx2 msg : Nat),
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ctx1 ≠ ctx2 →
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merlin_challenge R pk ctx1 msg ≠ merlin_challenge R pk ctx2 msg
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/-- Non-malleability: Ristretto encoding is unique -/
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theorem non_malleable_by_construction :
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ristretto_cofactor = 1 := rfl
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/-! ## VRF -/
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/-- VRF output: H = [sk] * HashToPoint(input) -/
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axiom vrf_eval : Nat → Nat → RistrettoPoint -- sk, input → output_point
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/-- VRF proof -/
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axiom vrf_prove : Nat → Nat → Signature -- sk, input → proof
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/-- VRF verification -/
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axiom vrf_verify : RistrettoPoint → Nat → RistrettoPoint → Signature → Bool
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-- pk, input, output, proof → valid?
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/-- VRF correctness: honest evaluation verifies -/
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axiom vrf_correctness :
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∀ (sk : Nat) (input : Nat),
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let pk := RistrettoPoint.smul sk B
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let output := vrf_eval sk input
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let proof := vrf_prove sk input
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vrf_verify pk input output proof = true
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/-- VRF uniqueness: for fixed pk and input, only one valid output -/
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axiom vrf_uniqueness :
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∀ (pk : RistrettoPoint) (input : Nat)
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(out1 out2 : RistrettoPoint) (pf1 pf2 : Signature),
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vrf_verify pk input out1 pf1 = true →
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vrf_verify pk input out2 pf2 = true →
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out1 = out2
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/-! ## Gas costs -/
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def gas_verify : Nat := 2500
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def gas_vrf : Nat := 4000
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/-- VRF costs more than signature (extra hash-to-point + DLEQ proof) -/
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theorem vrf_more_expensive : gas_vrf > gas_verify := by
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simp [gas_vrf, gas_verify]
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/-- Slightly more than Ed25519 (Ristretto + Merlin overhead) -/
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theorem more_than_ed25519 : gas_verify > 2000 := by
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simp [gas_verify]
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end Crypto.Precompile.SR25519
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