Perf: KZG verification in a single point

internal/generator/kzg/template/kzg.go.tmpl

@@ -177,28 +177,20 @@ func Verify(commitment *Digest, proof *OpeningProof, point fr.Element, vk Verify
 	var negH {{ .CurvePackage }}.G1Affine
 	negH.Neg(&proof.H)
 
-	// [α-a]G₂
-	var alphaMinusaG2Jac, genG2Jac, alphaG2Jac {{ .CurvePackage }}.G2Jac
+	// [f(α) - f(a) + a*H(α)]G₁
+	var totalG1 {{ .CurvePackage }}.G1Jac
 	var pointBigInt big.Int
 	point.BigInt(&pointBigInt)
-	genG2Jac.FromAffine(&vk.G2[0])
-	alphaG2Jac.FromAffine(&vk.G2[1])
-	alphaMinusaG2Jac.ScalarMultiplication(&genG2Jac, &pointBigInt).
-		Neg(&alphaMinusaG2Jac).
-		AddAssign(&alphaG2Jac)
+	totalG1.ScalarMultiplicationAffine(&proof.H, &pointBigInt)
+	totalG1.AddAssign(&fminusfaG1Jac)
+	var totalG1Aff {{ .CurvePackage }}.G1Affine
+	totalG1Aff.FromJacobian(&totalG1)
 
-	// [α-a]G₂
-	var xminusaG2Aff {{ .CurvePackage }}.G2Affine
-	xminusaG2Aff.FromJacobian(&alphaMinusaG2Jac)
 
-	// [f(α) - f(a)]G₁
-	var fminusfaG1Aff {{ .CurvePackage }}.G1Affine
-	fminusfaG1Aff.FromJacobian(&fminusfaG1Jac)
-
-	// e([f(α) - f(a)]G₁, G₂).e([-H(α)]G₁, [α-a]G₂) ==? 1
+	// e([f(α)-f(a)+aH(α)]G₁], G₂).e([-H(α)]G₁, [α]G₂) == 1
 	check, err := {{ .CurvePackage }}.PairingCheck(
-		[]{{ .CurvePackage }}.G1Affine{fminusfaG1Aff, negH},
-		[]{{ .CurvePackage }}.G2Affine{vk.G2[0], xminusaG2Aff},
+		[]{{ .CurvePackage }}.G1Affine{totalG1Aff, negH},
+		[]{{ .CurvePackage }}.G2Affine{vk.G2[0], vk.G2[1]},
 	)
 	if err != nil {
 		return err