Function binding conflicting evaluations branch maps to arsdh
KZG.CommitmentScheme.function_binding_conflicting_evaluations_branch_maps_to_arsdh
Plain-language statement
The conflicting-evaluations branch maps a function-binding violation to ARSDH.
Exact Lean statement
lemma function_binding_conflicting_evaluations_branch_maps_to_arsdh {n L : ℕ}
(hn : 1 ≤ n) (hp : p ≥ n + 2) (hpair : pairing g₁ g₂ ≠ 0)
{τ : ZMod p} {srs : Vector G₁ (n + 1) × Vector G₂ 2} {cm : G₁}
{queryOf responseOf : Fin L → ZMod p} {accepts : Fin L → Bool} {proofs : Fin L → G₁}
(hsrs : srs = Groups.PowerSrs.generate (g₁ := g₁) (g₂ := g₂) n τ)
(hgen : srs.1[0] ≠ 1)
(hverify_all : ∀ i : Fin L, accepts i = true →
KZG.verifyOpening (g₁ := g₁) (g₂ := g₂) (pairing := pairing)
srs.2 cm (proofs i) (queryOf i) (responseOf i))
(hFBcond : functionBindingCondExt n L (τ, srs, cm, queryOf, responseOf, accepts, proofs))
{i₁ i₂ : Fin L} (hfc : findConflict queryOf responseOf = some (i₁, i₂)) :
Groups.arsdhCondition n
(τ, (conflictingEvaluationsArsdhOutput (p := p) (G₂ := G₂) hn
({ srs := srs, cm := cm, queryOf := queryOf, responseOf := responseOf,
accepts := accepts, proofs := proofs } :
FunctionBindingExtTranscript (p := p) n L G₁ G₂) i₁ i₂).toTuple)Formal artifact
Lean source
lemma function_binding_conflicting_evaluations_branch_maps_to_arsdh {n L : ℕ} (hn : 1 ≤ n) (hp : p ≥ n + 2) (hpair : pairing g₁ g₂ ≠ 0) {τ : ZMod p} {srs : Vector G₁ (n + 1) × Vector G₂ 2} {cm : G₁} {queryOf responseOf : Fin L → ZMod p} {accepts : Fin L → Bool} {proofs : Fin L → G₁} (hsrs : srs = Groups.PowerSrs.generate (g₁ := g₁) (g₂ := g₂) n τ) (hgen : srs.1[0] ≠ 1) (hverify_all : ∀ i : Fin L, accepts i = true → KZG.verifyOpening (g₁ := g₁) (g₂ := g₂) (pairing := pairing) srs.2 cm (proofs i) (queryOf i) (responseOf i)) (hFBcond : functionBindingCondExt n L (τ, srs, cm, queryOf, responseOf, accepts, proofs)) {i₁ i₂ : Fin L} (hfc : findConflict queryOf responseOf = some (i₁, i₂)) : Groups.arsdhCondition n (τ, (conflictingEvaluationsArsdhOutput (p := p) (G₂ := G₂) hn ({ srs := srs, cm := cm, queryOf := queryOf, responseOf := responseOf, accepts := accepts, proofs := proofs } : FunctionBindingExtTranscript (p := p) n L G₁ G₂) i₁ i₂).toTuple) := by simp only [conflictingEvaluationsArsdhOutput, FunctionBindingArsdhOutput.toTuple, Groups.arsdhCondition, ne_eq, one_div, Finset.union_singleton] have hαβ := find_conflict_successful queryOf responseOf hfc obtain ⟨hα, hβ⟩ := hαβ have h_acc_all : ∀ i ∈ (Finset.univ : Finset (Fin L)), accepts i = true := hFBcond.1 have hai₁ : accepts i₁ = true := h_acc_all i₁ (Finset.mem_univ _) have hai₂ : accepts i₂ = true := h_acc_all i₂ (Finset.mem_univ _) have hverify₁ : KZG.verifyOpening (g₁ := g₁) (g₂ := g₂) (pairing := pairing) srs.2 cm (proofs i₁) (queryOf i₁) (responseOf i₁) := hverify_all i₁ hai₁ have hverify₂ : KZG.verifyOpening (g₁ := g₁) (g₂ := g₂) (pairing := pairing) srs.2 cm (proofs i₂) (queryOf i₂) (responseOf i₂) := hverify_all i₂ hai₂ constructor · simpa [Finset.union_singleton] using choose_s_conflict_size_adjoined hp hn (queryOf i₁) srs hgen · constructor · have hα_ne_τ : queryOf i₁ ≠ τ := conflict_query_ne_tau (g₁ := g₁) (g₂ := g₂) (pairing := pairing) PrimeOrderWith.hCard hn (queryOf i₁) (queryOf i₂) (responseOf i₁) (responseOf i₂) τ cm (proofs i₁) (proofs i₂) hα hβ srs hsrs hgen hpair hverify₁ hverify₂ change (∏ s ∈ insert (queryOf i₁) (chooseSConflict (queryOf i₁) srs hn), (X - C s : CPolynomial (ZMod p))).eval τ ≠ 0 exact choose_s_conflict_insert_eval_ne_zero (g₁ := g₁) (g₂ := g₂) hn (queryOf i₁) τ srs hsrs hα_ne_τ · constructor · exact h1_ne_one (g₁ := g₁) (g₂ := g₂) hp PrimeOrderWith.hCard hn (queryOf i₁) τ srs hsrs hgen · have key := h1_zs_eq_h2 (g₁ := g₁) (g₂ := g₂) (pairing := pairing) hp PrimeOrderWith.hCard hn (queryOf i₁) (queryOf i₂) (responseOf i₁) (responseOf i₂) τ cm (proofs i₁) (proofs i₂) hα hβ srs hsrs hgen hpair hverify₁ hverify₂ simpa [Finset.union_singleton, one_div] using key- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/KZG/FunctionBinding/EvaluationBindingConflict.lean:486-536
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