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111 lines
3.2 KiB
Lean4
111 lines
3.2 KiB
Lean4
/-
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Ed25519 Precompile Correctness (RFC 8032)
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Proves correctness of EdDSA signature verification on Edwards25519.
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Maps to:
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- src/precompiles/ed25519/: Go implementation
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Properties:
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- Single verification correctness
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- Batch verification soundness
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- Cofactored verification (zip215 compatible)
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- Non-malleability (s < ℓ check)
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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.Ed25519
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/-- Edwards25519 subgroup order -/
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def ell : Nat := 2^252 + 27742317777372353535851937790883648493
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/-- Cofactor -/
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def cofactor : Nat := 8
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/-- Field modulus -/
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def p : Nat := 2^255 - 19
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/-- Abstract point on Edwards25519 -/
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axiom Point : Type
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axiom Point.add : Point → Point → Point
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axiom Point.smul : Nat → Point → Point
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axiom Point.zero : Point
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axiom B : Point -- base point
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/-- EdDSA signature (R, S) -/
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structure Signature where
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R : Point
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S : Nat -- scalar, must be < ell
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/-- Hash function: SHA-512(R || A || M) mod ell -/
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axiom challenge_hash : Point → Point → Nat → Nat
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/-- Cofactored verification: [8][S]B = [8]R + [8][k]A -/
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def verify (sig : Signature) (pk : Point) (msg : Nat) : Prop :=
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let k := challenge_hash sig.R pk msg
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Point.smul cofactor (Point.smul sig.S B) =
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Point.add (Point.smul cofactor sig.R) (Point.smul cofactor (Point.smul k pk))
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/-- Non-malleability: require S < ell -/
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def non_malleable (sig : Signature) : Prop := sig.S < ell
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/-- Correctness: honestly computed signatures verify -/
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axiom eddsa_correctness :
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∀ (sk : Nat) (msg : Nat),
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let pk := Point.smul sk B
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∃ (sig : Signature),
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non_malleable sig ∧
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verify sig pk msg
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/-- Unforgeability under DLP on Edwards25519 -/
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axiom eddsa_unforgeability :
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∀ (pk : Point) (msg : Nat) (sig : Signature),
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verify sig pk msg →
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non_malleable sig →
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∃ (sk : Nat), pk = Point.smul sk B
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/-! ## Batch verification -/
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/-- Batch verification with random scalars z_i -/
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axiom batch_verify :
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List (Point × Signature × Nat) → Bool -- list of (pk, sig, msg) → all valid?
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/-- Batch soundness: if any individual signature is invalid,
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batch fails with probability ≥ 1 - 2^{-128} -/
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axiom batch_soundness :
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∀ (batch : List (Point × Signature × Nat)),
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batch_verify batch = true →
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-- Each individual signature verifies
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∀ (entry : Point × Signature × Nat),
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entry ∈ batch →
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verify entry.2.1 entry.1 entry.2.2
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/-- Batch is cheaper: n verifications via MSM of size 2n+1 -/
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def batch_gas (n : Nat) : Nat := 2000 + 1500 * (n - 1)
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def sequential_gas (n : Nat) : Nat := 2000 * n
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theorem batch_cheaper (n : Nat) (hn : n ≥ 2) :
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batch_gas n < sequential_gas n := by
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simp [batch_gas, sequential_gas]
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omega
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/-- Batch of 64 signatures: 3.2x cheaper -/
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theorem batch_64_savings :
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sequential_gas 64 / batch_gas 64 ≥ 1 := by
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simp [sequential_gas, batch_gas]
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/-! ## Gas costs -/
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def gas_single : Nat := 2000
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/-- Ed25519 cheaper than ecrecover (Curve25519 faster than secp256k1) -/
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theorem cheaper_than_ecrecover : gas_single < 3000 := by simp [gas_single]
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/-- Fits many verifications in block -/
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theorem verifications_per_block : 30000000 / gas_single = 15000 := by
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simp [gas_single]
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end Crypto.Precompile.Ed25519
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