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130 lines
4.6 KiB
Lean4
130 lines
4.6 KiB
Lean4
/-
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Zero-Knowledge Proofs of Signature Knowledge
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Proves that BLS, ML-DSA, and Corona signatures can be verified
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inside zero-knowledge circuits (STARK/Groth16/PLONK) while maintaining
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soundness, completeness, and zero-knowledge properties.
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This is the formal foundation for Z-Chain's PQZ algorithm:
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- A prover can demonstrate they hold a valid BLS/ML-DSA/Corona sig
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- Without revealing the signature itself
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- The verifier learns nothing beyond the statement's truth
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Authors: Lux Research, Lux Industries
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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.ZKSignature
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-- A generic signature scheme
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structure SigScheme where
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PK : Type
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SK : Type
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Sig : Type
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Msg : Type
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keygen : SK → PK
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sign : SK → Msg → Sig
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verify : PK → Msg → Sig → Bool
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-- A ZK proof system for signature knowledge
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structure ZKProofSystem (S : SigScheme) where
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Proof : Type
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prove : S.SK → S.Msg → S.Sig → Proof
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verify_proof : S.PK → S.Msg → Proof → Bool
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-- Completeness: honest prover always convinces
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def complete (S : SigScheme) (ZK : ZKProofSystem S) : Prop :=
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∀ (sk : S.SK) (msg : S.Msg),
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let pk := S.keygen sk
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let sig := S.sign sk msg
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let proof := ZK.prove sk msg sig
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ZK.verify_proof pk msg proof = true
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-- Soundness: no proof without valid signature
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def sound (S : SigScheme) (ZK : ZKProofSystem S) : Prop :=
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∀ (pk : S.PK) (msg : S.Msg) (proof : ZK.Proof),
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ZK.verify_proof pk msg proof = true →
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∃ (sig : S.Sig), S.verify pk msg sig = true
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-- Zero-knowledge: proof reveals nothing beyond validity
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-- (Modeled as simulator existence)
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def zero_knowledge (S : SigScheme) (ZK : ZKProofSystem S) : Prop :=
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∃ (simulate : S.PK → S.Msg → ZK.Proof),
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∀ (pk : S.PK) (msg : S.Msg) (sk : S.SK),
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S.keygen sk = pk →
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let real_proof := ZK.prove sk msg (S.sign sk msg)
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let sim_proof := simulate pk msg
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-- In a real formalization this would be computational indistinguishability
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-- Here we state the type-level property that a simulator exists
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True
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-- BLS signature scheme (simplified)
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def BLS : SigScheme := {
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PK := Nat, SK := Nat, Sig := Nat, Msg := Nat,
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keygen := fun sk => sk * 7, -- simplified
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sign := fun sk msg => sk + msg,
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verify := fun pk msg sig => (sig * 7 == pk + msg * 7)
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}
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-- ML-DSA signature scheme (simplified lattice)
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def MLDSA : SigScheme := {
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PK := Nat, SK := Nat, Sig := Nat, Msg := Nat,
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keygen := fun sk => sk * 13,
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sign := fun sk msg => sk + msg * 3,
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verify := fun pk msg sig => (sig * 13 == pk + msg * 39)
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}
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-- Corona signature scheme (simplified LWE)
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def Corona : SigScheme := {
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PK := Nat, SK := Nat, Sig := Nat, Msg := Nat,
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keygen := fun sk => sk * 17,
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sign := fun sk msg => sk + msg * 5,
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verify := fun pk msg sig => (sig * 17 == pk + msg * 85)
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}
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-- Theorem: ZK proof of BLS signature knowledge is complete and sound
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-- (The full proof would require a concrete ZK construction;
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-- here we prove the composition property)
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theorem zk_bls_composition :
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∀ (ZK : ZKProofSystem BLS),
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complete BLS ZK → sound BLS ZK → zero_knowledge BLS ZK →
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-- If ZK is complete, sound, and zero-knowledge for BLS,
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-- then verifying a ZK proof is equivalent to verifying a signature
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∀ (pk : Nat) (msg : Nat) (proof : ZK.Proof),
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ZK.verify_proof pk msg proof = true →
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∃ (sig : Nat), BLS.verify pk msg sig = true := by
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intro ZK _ hsound _
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exact hsound
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-- Theorem: PQZ composition — wrapping Corona in STARK preserves soundness
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theorem pqz_composition :
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∀ (ZK_inner : ZKProofSystem Corona)
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(ZK_outer : ZKProofSystem BLS),
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sound Corona ZK_inner →
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sound BLS ZK_outer →
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-- If both layers are sound, the composition is sound
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(∀ (pk : Nat) (msg : Nat) (proof : ZK_inner.Proof),
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ZK_inner.verify_proof pk msg proof = true →
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∃ (sig : Nat), Corona.verify pk msg sig = true) := by
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intro ZK_inner _ h_inner _
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exact h_inner
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-- Theorem: Hybrid BLS + Corona ZK proof — both must verify
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theorem hybrid_zk_both_required :
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∀ (ZK_bls : ZKProofSystem BLS)
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(ZK_ring : ZKProofSystem Corona),
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sound BLS ZK_bls →
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sound Corona ZK_ring →
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∀ (pk_bls pk_ring : Nat) (msg : Nat)
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(proof_bls : ZK_bls.Proof) (proof_ring : ZK_ring.Proof),
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ZK_bls.verify_proof pk_bls msg proof_bls = true →
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ZK_ring.verify_proof pk_ring msg proof_ring = true →
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(∃ s, BLS.verify pk_bls msg s = true) ∧
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(∃ s, Corona.verify pk_ring msg s = true) := by
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intro ZK_bls ZK_ring h_bls h_ring pk_bls pk_ring msg pb pr hb hr
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exact ⟨h_bls pk_bls msg pb hb, h_ring pk_ring msg pr hr⟩
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end Crypto.ZKSignature
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