Project documentation
Completeness theorem for Merkle trees: for any full binary tree with 2 ^ n leaves, and for any index i, the honestly-generated opening proof verifies against the honestly-built root with probability one.
Exact Lean statement
theorem completeness [DecidableEq α] [Inhabited α] [SampleableType α] {n : ℕ}
[(spec α).DecidableEq]
(leaves : List.Vector α (2 ^ n)) (i : Fin (2 ^ n))
(preexisting_cache : (spec α).QueryCache) :
Pr[fun v => v.1 = true | (simulateQ (spec α).randomOracle (do
let cache ← buildMerkleTree α n leaves
let proof := generateProof α i cache
let verified ← (verifyProof (m := OracleComp (spec α)) α i leaves[i]
(getRoot α cache) proof)
return verified)).run preexisting_cache] = 1Formal artifact
Lean source
theorem completeness [DecidableEq α] [Inhabited α] [SampleableType α] {n : ℕ} [(spec α).DecidableEq] (leaves : List.Vector α (2 ^ n)) (i : Fin (2 ^ n)) (preexisting_cache : (spec α).QueryCache) : Pr[fun v => v.1 = true | (simulateQ (spec α).randomOracle (do let cache ← buildMerkleTree α n leaves let proof := generateProof α i cache let verified ← (verifyProof (m := OracleComp (spec α)) α i leaves[i] (getRoot α cache) proof) return verified)).run preexisting_cache] = 1 := by refine (probEvent_eq_one_simulateQ_randomOracle_run_iff (spec := spec α) (p := fun b : Bool => b = true) _ _).mpr ?_ intro f _hf simp only [evalWithAnswerFn, verifyProof, simulateQ_bind, simulateQ_pure, simulateQ_buildMerkleTree_eq, simulateQ_getPutativeRoot_eq] change ((_ : α) == _) = true exact beq_iff_eq.mpr (functional_completeness α leaves i f)- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/CryptoFoundations/MerkleTree/Vector/Completeness.lean:101-117
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