feat: add proof of elgamal binding

* refactor: rename parameters

* feat: add proof of elgamal binding

Author: ashandoak

VerifiedCommitments/ElgamalCommitments.lean

@@ -382,4 +382,44 @@ theorem computational_hiding_from_ddh (ε : ENNReal)
   exact DDH_hard (D_from_adversary hA)
 
 
+/- ========================================
+   BINDING PROPERTY
+   ======================================== -/
+
+lemma ordg_eq_q : orderOf params.g = params.q := by
+  have h_card_zpow : Fintype.card (Subgroup.zpowers params.g) = orderOf params.g := Fintype.card_zpowers
+  have h_card_eq : Fintype.card (Subgroup.zpowers params.g) = Fintype.card G := by
+    have : Function.Bijective (Subtype.val : Subgroup.zpowers params.g → G) := by
+      constructor
+      · exact Subtype.val_injective
+      · intro x
+        use ⟨x, params.g_gen_G x⟩
+    exact Fintype.card_of_bijective this
+  rw [← h_card_zpow, h_card_eq, params.card_eq]
+
+
+theorem perfect_binding : Commitment.perfect_binding (@scheme G params) := by
+  unfold Commitment.perfect_binding
+  intro h c m m' o o'
+  unfold scheme verify
+  simp only [ite_eq_left_iff, zero_ne_one, imp_false, Decidable.not_not]
+  intro hc
+  simp [hc]
+  congr! with hc₁ hc₂
+
+
+  have h_congr : o.val ≡ o'.val [MOD params.q] := by
+      simpa [ordg_eq_q] using (pow_eq_pow_iff_modEq.mp hc₁)
+
+  have o_eq : o = o' := by
+    have eq_cast : ((o.val : ℕ) : ZMod params.q) = ((o'.val : ℕ) : ZMod params.q) :=
+      ZMod.natCast_eq_natCast_iff o.val o'.val params.q |>.mpr h_congr
+    simp at eq_cast
+    exact eq_cast
+
+  subst o_eq
+  simp at hc₂
+  exact hc₂
+
+
 end Elgamal