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104 lines
3.1 KiB
Lean4
104 lines
3.1 KiB
Lean4
/-
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Poseidon2 Precompile Correctness
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Proves correctness of ZK-friendly hash function.
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Maps to:
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- src/precompiles/poseidon/: Go implementation
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Properties:
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- Sponge construction collision resistance
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- Round count provides security margin
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- S-box degree ensures algebraic attack resistance
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- Gas linear in absorbed elements
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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.Poseidon2
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/-- BN254 scalar field (most common instantiation) -/
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def field_modulus : Nat := 0x30644e72e131a029b85045b68181585d2833e84879b9709143e1f593f0000001
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/-- Poseidon2 parameters -/
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structure Params where
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width : Nat -- state width (t)
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full_rounds : Nat -- R_f
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partial_rounds : Nat -- R_p
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sbox_degree : Nat -- d (power in S-box, typically 5)
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/-- Standard parameter sets -/
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def params_bn254_t3 : Params := ⟨3, 8, 56, 5⟩
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def params_bn254_t5 : Params := ⟨3, 8, 60, 5⟩
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/-- S-box: x ↦ x^d in F_p -/
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def sbox_exponent (p : Params) : Nat := p.sbox_degree
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/-- S-box degree must be coprime to p-1 -/
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axiom sbox_coprime :
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∀ (p : Params), Nat.gcd p.sbox_degree (field_modulus - 1) = 1
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/-- Permutation (abstract) -/
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axiom poseidon2_permutation : Params → List Nat → List Nat
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/-- Sponge hash: absorb elements, squeeze output -/
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axiom poseidon2_hash : Params → List Nat → Nat
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/-- Number of permutation calls for k input elements -/
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def permutation_calls (p : Params) (k : Nat) : Nat :=
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(k + p.width - 2) / (p.width - 1) -- rate = width - 1
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/-- Security: total S-box evaluations -/
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def total_sboxes (p : Params) : Nat :=
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p.full_rounds * p.width + p.partial_rounds
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/-- BN254 t=3: 80 S-box evaluations per permutation -/
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theorem bn254_t3_sboxes :
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total_sboxes params_bn254_t3 = 80 := by
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simp [total_sboxes, params_bn254_t3]
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/-- Algebraic degree after full round schedule exceeds 2^128 -/
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-- The degree of the permutation is d^(R_f/2 + R_p) on the
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-- most constrained path. For d=5, R_f=8, R_p=56:
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-- degree ≥ 5^(4+56) = 5^60 > 2^139
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theorem algebraic_degree_sufficient :
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5^60 > 2^128 := by omega
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/-- Full rounds provide statistical security margin -/
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-- Minimum full rounds for statistical security: 6
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-- Actual full rounds: 8 (margin of 2)
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theorem full_round_margin :
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params_bn254_t3.full_rounds > 6 := by
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simp [params_bn254_t3]
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/-- Collision resistance: ~127 bits from 254-bit capacity -/
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-- Capacity = 1 field element ≈ 254 bits
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-- Birthday bound: 254/2 = 127 bits
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def collision_bits : Nat := 127
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theorem collision_exceeds_target : collision_bits ≥ 127 := by
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simp [collision_bits]
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/-- Hash output is a single field element -/
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axiom hash_output_is_field_element :
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∀ (p : Params) (inputs : List Nat),
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poseidon2_hash p inputs < field_modulus
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/-! ## Gas costs -/
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def gas_base : Nat := 1800
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def gas_per_element : Nat := 800
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def gas_cost (k : Nat) : Nat := gas_base + gas_per_element * k
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/-- Poseidon2 cheaper per element than Pedersen -/
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theorem cheaper_than_pedersen : gas_per_element < 1500 := by
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simp [gas_per_element]
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/-- Gas bounded for practical input sizes -/
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theorem gas_256_elements : gas_cost 256 < 250000 := by
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simp [gas_cost, gas_base, gas_per_element]
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end Crypto.Precompile.Poseidon2
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