diff --git a/zksnark/bn128_curve.py b/zksnark/bn128_curve.py
index 9b8940f..c79638b 100644
--- a/zksnark/bn128_curve.py
+++ b/zksnark/bn128_curve.py
@@ -14,6 +14,8 @@ G2 = (FQ2([4, 0]), FQ2([16893045765507297706785249332518927989146279141265438554
 
 # Check that a point is on the curve defined by y**2 == x**3 + b
 def is_on_curve(pt, b):
+    if pt is None:
+        return True
     x, y = pt
     return y**2 - x**3 == b
 
@@ -81,3 +83,10 @@ def twist(pt):
 
 # Check that the twist creates a point that is on the curve
 assert is_on_curve(twist(G2), b12)
+
+# Check that the G12 curve works fine
+
+G12 = twist(G2)
+assert add(add(double(G12), G12), G12) == double(double(G12))
+assert double(G12) != G12
+assert add(multiply(G12, 9), multiply(G12, 5)) == add(multiply(G12, 12), multiply(G12, 2))
diff --git a/zksnark/bn128_pairing.py b/zksnark/bn128_pairing.py
index 41febd3..0f309ce 100644
--- a/zksnark/bn128_pairing.py
+++ b/zksnark/bn128_pairing.py
@@ -4,11 +4,12 @@ from bn128_curve import double, add, multiply, is_on_curve, twist, b, b2, b12, c
 from bn128_field_elements import field_modulus, FQ, FQ2, FQ12
 
 ate_loop_count = 29793968203157093288
-log_ate_loop_count = 64
+log_ate_loop_count = 63
 
 # Create a function representing the line between P1 and P2,
 # and evaluate it at T
 def linefunc(P1, P2, T):
+    assert P1 and P2 and T # No points-at-infinity allowed, sorry
     x1, y1 = P1
     x2, y2 = P2
     xt, yt = T
@@ -28,27 +29,33 @@ def cast_point_to_fq12(pt):
     return (FQ12([x.n] + [0] * 11), FQ12([y.n] + [0] * 11))
 
 # Check consistency of the "line function"
-one, two, three, negone = G1, double(G1), multiply(G1, 3), multiply(G1, curve_order - 1)
+one, two, three = G1, double(G1), multiply(G1, 3)
+negone, negtwo, negthree = multiply(G1, curve_order - 1), multiply(G1, curve_order - 2), multiply(G1, curve_order - 3)
 
 assert linefunc(one, two, one) == FQ(0)
 assert linefunc(one, two, two) == FQ(0)
 assert linefunc(one, two, three) != FQ(0)
+assert linefunc(one, two, negthree) == FQ(0)
 assert linefunc(one, negone, one) == FQ(0)
 assert linefunc(one, negone, negone) == FQ(0)
 assert linefunc(one, negone, two) != FQ(0)
 assert linefunc(one, one, one) == FQ(0)
 assert linefunc(one, one, two) != FQ(0)
+assert linefunc(one, one, negtwo) == FQ(0)
 
 # Main miller loop
 def miller_loop(Q, P):
+    if Q is None or P is None:
+        return FQ12.one()
     R = Q
     f = FQ12.one()
     for i in range(log_ate_loop_count, -1, -1):
-        f = f * f / linefunc(R, R, P)
+        f = f * f * linefunc(R, R, P)
         R = double(R)
         if ate_loop_count & (2**i):
             f = f * linefunc(R, Q, P)
             R = add(R, Q)
+    assert R == multiply(Q, ate_loop_count)
     Q1 = (Q[0] ** field_modulus, Q[1] ** field_modulus)
     nQ2 = (Q[0] ** (field_modulus ** 2), -Q[1] ** (field_modulus ** 2))
     f = f * linefunc(R, Q1, P)