diff --git a/src/ecdsa/ellipticcurve.py b/src/ecdsa/ellipticcurve.py
index a982c1e4..f601f263 100644
--- a/src/ecdsa/ellipticcurve.py
+++ b/src/ecdsa/ellipticcurve.py
@@ -1185,14 +1185,15 @@ def __add__(self, other):
         if self == INFINITY:
             return other
         assert self.__curve == other.__curve
-        if self.__x == other.__x:
-            if (self.__y + other.__y) % self.__curve.p() == 0:
+
+        p = self.__curve.p()
+
+        if (self.__x - other.__x) % p == 0:
+            if (self.__y + other.__y) % p == 0:
                 return INFINITY
             else:
                 return self.double()
 
-        p = self.__curve.p()
-
         l = (
             (other.__y - self.__y)
             * numbertheory.inverse_mod(other.__x - self.__x, p)
@@ -1206,13 +1207,6 @@ def __add__(self, other):
     def __mul__(self, other):
         """Multiply a point by an integer."""
 
-        def leftmost_bit(x):
-            assert x > 0
-            result = 1
-            while result <= x:
-                result = 2 * result
-            return result // 2
-
         e = other
         if e == 0 or (self.__order and e % self.__order == 0):
             return INFINITY
@@ -1221,26 +1215,14 @@ def leftmost_bit(x):
         if e < 0:
             return (-self) * (-e)
 
-        # From X9.62 D.3.2:
-
-        e3 = 3 * e
-        negative_self = Point(
-            self.__curve,
-            self.__x,
-            (-self.__y) % self.__curve.p(),
-            self.__order,
-        )
-        i = leftmost_bit(e3) // 2
-        result = self
-        # print("Multiplying %s by %d (e3 = %d):" % (self, other, e3))
-        while i > 1:
-            result = result.double()
-            if (e3 & i) != 0 and (e & i) == 0:
-                result = result + self
-            if (e3 & i) == 0 and (e & i) != 0:
-                result = result + negative_self
-            # print(". . . i = %d, result = %s" % ( i, result ))
-            i = i // 2
+        i = e
+        temp = self
+        result = INFINITY
+        while i:
+            if i % 2 == 1:
+                result = result + temp
+            temp = temp.double()
+            i = i >> 1
 
         return result
 
@@ -1289,6 +1271,9 @@ def curve(self):
     def order(self):
         return self.__order
 
+    def to_affine(self):
+        return self
+
 
 class PointEdwards(AbstractPoint):
     """Point on Twisted Edwards curve.
diff --git a/src/ecdsa/test_ellipticcurve.py b/src/ecdsa/test_ellipticcurve.py
index 864cf108..bcc0caa4 100644
--- a/src/ecdsa/test_ellipticcurve.py
+++ b/src/ecdsa/test_ellipticcurve.py
@@ -71,6 +71,20 @@ def test_add_and_mult_equivalence(p, m, check):
     assert p * m == check
 
 
+# from https://github.com/tlsfuzzer/python-ecdsa/issues/373
+curve = CurveFp(p=31, a=2, b=3)
+point = Point(curve, 6, 18)
+
+
+@pytest.mark.parametrize(
+    "p, m, check",
+    [(point, n, exp) for n, exp in enumerate(add_n_times(point, 16))],
+    ids=["p31 test with mult {0}".format(i) for i in range(17)],
+)
+def test_add_and_mult_equivalence_2(p, m, check):
+    assert p * m == check
+
+
 class TestCurve(unittest.TestCase):
     @classmethod
     def setUpClass(cls):
@@ -146,6 +160,16 @@ def setUpClass(cls):
         cls.c192 = CurveFp(p, -3, b)
         cls.p192 = Point(cls.c192, Gx, Gy, r)
 
+    def test_points_with_different_curves(self):
+        with self.assertRaises(AssertionError):
+            self.g_23 + self.p192
+
+    def test_add_point_to_negative(self):
+        self.assertIs(self.g_23 + (-self.g_23), INFINITY)
+
+    def test_add_point_to_explicit_negative(self):
+        self.assertIs(self.g_23 + Point(self.c_23, 13, -7), INFINITY)
+
     def test_p192(self):
         # Checking against some sample computations presented
         # in X9.62: