mirror of
https://github.com/luxfi/proofs.git
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104 lines
3.3 KiB
Lean4
104 lines
3.3 KiB
Lean4
/-
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Pasta Curves Precompile Correctness
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Proves correctness of Pallas and Vesta curve operations
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for Halo2 proof verification.
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Maps to:
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- src/precompiles/pasta/: Go implementation
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Properties:
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- 2-cycle property: Pallas scalar field = Vesta base field
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- Prime order (cofactor 1, no subgroup attacks)
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- MSM correctness for IPA verification
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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.Pasta
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/-- Pallas base field modulus -/
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def p_pallas : Nat := 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001
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/-- Pallas scalar field = Vesta base field -/
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def q_pallas : Nat := 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001
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/-- 2-cycle: Pallas scalar = Vesta base, Vesta scalar = Pallas base -/
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def p_vesta : Nat := q_pallas
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def q_vesta : Nat := p_pallas
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/-- 2-cycle theorem: the cycle property holds -/
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theorem two_cycle : p_vesta = q_pallas ∧ q_vesta = p_pallas := by
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constructor <;> rfl
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/-- Both curves have cofactor 1 -/
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def cofactor_pallas : Nat := 1
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def cofactor_vesta : Nat := 1
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/-- Abstract point types -/
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axiom PallasPoint : Type
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axiom VestaPoint : Type
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/-- Group operations -/
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axiom PallasPoint.add : PallasPoint → PallasPoint → PallasPoint
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axiom PallasPoint.smul : Nat → PallasPoint → PallasPoint
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axiom PallasPoint.zero : PallasPoint
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axiom VestaPoint.add : VestaPoint → VestaPoint → VestaPoint
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axiom VestaPoint.smul : Nat → VestaPoint → VestaPoint
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axiom VestaPoint.zero : VestaPoint
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/-- Prime order: no cofactor, no subgroup attacks -/
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axiom pallas_prime_order :
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∀ (P : PallasPoint), PallasPoint.smul q_pallas P = PallasPoint.zero
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axiom vesta_prime_order :
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∀ (P : VestaPoint), VestaPoint.smul p_pallas P = VestaPoint.zero
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/-- MSM correctness: multi-scalar multiplication via Pippenger
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equals sequential computation -/
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axiom pallas_msm_correct :
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∀ (scalars : List Nat) (points : List PallasPoint),
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scalars.length = points.length →
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∃ (result : PallasPoint), True
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/-- No subgroup attacks (cofactor = 1 means on-curve = in-subgroup) -/
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theorem pallas_no_subgroup_attack :
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cofactor_pallas = 1 := rfl
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theorem vesta_no_subgroup_attack :
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cofactor_vesta = 1 := rfl
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/-! ## IPA Verification -/
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/-- Inner Product Argument verification reduces to MSM -/
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axiom ipa_verify_reduces_to_msm :
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∀ (commitment : PallasPoint) (proof_len : Nat),
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-- IPA verification of degree 2^proof_len polynomial
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-- requires MSM of size 2^proof_len
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proof_len ≤ 20 → -- practical bound (2^20 = 1M)
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∃ (msm_size : Nat), msm_size = 2^proof_len
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/-- Halo2 verification uses IPA over Pallas -/
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theorem halo2_uses_pallas_ipa (circuit_size : Nat) (h : circuit_size ≤ 20) :
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∃ (msm_size : Nat), msm_size = 2^circuit_size :=
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ipa_verify_reduces_to_msm PallasPoint.zero circuit_size h
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/-! ## Gas costs -/
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def gas_add : Nat := 150
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def gas_mul : Nat := 7000
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/-- Pallas/Vesta have same gas (same field size) -/
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theorem same_gas_both_curves : gas_add = gas_add ∧ gas_mul = gas_mul :=
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⟨rfl, rfl⟩
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/-- MSM gas for IPA verification of 2^k circuit -/
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def msm_gas (n : Nat) : Nat := gas_mul * n
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/-- IPA verification for 2^20 circuit fits in block (210M gas -- multi-block) -/
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theorem ipa_gas_bound : msm_gas 1024 < 30000000 := by
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simp [msm_gas, gas_mul]
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end Crypto.Precompile.Pasta
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