diff --git a/Benchs/Bench.hs b/Benchs/Bench.hs
--- a/Benchs/Bench.hs
+++ b/Benchs/Bench.hs
@@ -36,46 +36,50 @@
         ecdsaSignatureP = fst $ ECDSA.sign rng ecdsaPrivatekeyP SHA1.hash bs
         ecdsaSignatureB = fst $ ECDSA.sign rng ecdsaPrivatekeyB SHA1.hash bs
     defaultMain
-        [ bgroup "RSA PKCS15"
-            [ bench "encryption" $ nf (right . fst . PKCS15.encrypt rng rsaPublickey) bs
-            , bgroup "decryption"
-                [ bench "slow" $ nf (right . PKCS15.decrypt Nothing privateKeySlow) encryptedMsgPKCS
-                , bench "fast" $ nf (right . PKCS15.decrypt Nothing rsaPrivatekey) encryptedMsgPKCS
-                , bench "slow+blinding" $ nf (right . PKCS15.decrypt (Just blinder) privateKeySlow) encryptedMsgPKCS
-                , bench "fast+blinding" $ nf (right . PKCS15.decrypt (Just blinder) rsaPrivatekey) encryptedMsgPKCS
+        [ bgroup "RSA"
+            [ bgroup "PKCS15"
+                [ bench "encryption" $ nf (right . fst . PKCS15.encrypt rng rsaPublickey) bs
+                , bgroup "decryption"
+                    [ bench "slow" $ nf (right . PKCS15.decrypt Nothing privateKeySlow) encryptedMsgPKCS
+                    , bench "fast" $ nf (right . PKCS15.decrypt Nothing rsaPrivatekey) encryptedMsgPKCS
+                    , bench "slow+blinding" $ nf (right . PKCS15.decrypt (Just blinder) privateKeySlow) encryptedMsgPKCS
+                    , bench "fast+blinding" $ nf (right . PKCS15.decrypt (Just blinder) rsaPrivatekey) encryptedMsgPKCS
+                    ]
+                , bgroup "signing"
+                    [ bench "slow" $ nf (right . PKCS15.sign Nothing hashDescrSHA1 privateKeySlow) bs
+                    , bench "fast" $ nf (right . PKCS15.sign Nothing hashDescrSHA1 rsaPrivatekey) bs
+                    , bench "slow+blinding" $ nf (right . PKCS15.sign (Just blinder) hashDescrSHA1 privateKeySlow) bs
+                    , bench "fast+blinding" $ nf (right . PKCS15.sign (Just blinder) hashDescrSHA1 rsaPrivatekey) bs
+                    ]
+                , bench "verify" $ nf (PKCS15.verify hashDescrSHA1 rsaPublickey bs) signedMsgPKCS
                 ]
-            , bgroup "signing"
-                [ bench "slow" $ nf (right . PKCS15.sign Nothing hashDescrSHA1 privateKeySlow) bs
-                , bench "fast" $ nf (right . PKCS15.sign Nothing hashDescrSHA1 rsaPrivatekey) bs
-                , bench "slow+blinding" $ nf (right . PKCS15.sign (Just blinder) hashDescrSHA1 privateKeySlow) bs
-                , bench "fast+blinding" $ nf (right . PKCS15.sign (Just blinder) hashDescrSHA1 rsaPrivatekey) bs
+            , bgroup "OAEP"
+                [ bench "encryption" $ nf (right . fst . OAEP.encrypt rng oaepParams rsaPublickey) bs
+                , bgroup "decryption"
+                    [ bench "slow" $ nf (right . OAEP.decrypt Nothing oaepParams privateKeySlow) encryptedMsgOAEP
+                    , bench "fast" $ nf (right . OAEP.decrypt Nothing oaepParams rsaPrivatekey) encryptedMsgOAEP
+                    , bench "slow+blinding" $ nf (right . OAEP.decrypt (Just blinder) oaepParams privateKeySlow) encryptedMsgOAEP
+                    , bench "fast+blinding" $ nf (right . OAEP.decrypt (Just blinder) oaepParams rsaPrivatekey) encryptedMsgOAEP
+                    ]
                 ]
-            , bench "verify" $ nf (PKCS15.verify hashDescrSHA1 rsaPublickey bs) signedMsgPKCS
-            ]
-        , bgroup "RSA OAEP"
-            [ bench "encryption" $ nf (right . fst . OAEP.encrypt rng oaepParams rsaPublickey) bs
-            , bgroup "decryption"
-                [ bench "slow" $ nf (right . OAEP.decrypt Nothing oaepParams privateKeySlow) encryptedMsgOAEP
-                , bench "fast" $ nf (right . OAEP.decrypt Nothing oaepParams rsaPrivatekey) encryptedMsgOAEP
-                , bench "slow+blinding" $ nf (right . OAEP.decrypt (Just blinder) oaepParams privateKeySlow) encryptedMsgOAEP
-                , bench "fast+blinding" $ nf (right . OAEP.decrypt (Just blinder) oaepParams rsaPrivatekey) encryptedMsgOAEP
+            , bgroup "PSS"
+                [ bgroup "signing"
+                    [ bench "slow" $ nf (right . fst . PSS.sign rng Nothing pssParams privateKeySlow) bs
+                    , bench "fast" $ nf (right . fst . PSS.sign rng Nothing pssParams rsaPrivatekey) bs
+                    , bench "slow+blinding" $ nf (right . fst . PSS.sign rng (Just blinder) pssParams privateKeySlow) bs
+                    , bench "fast+blinding" $ nf (right . fst . PSS.sign rng (Just blinder) pssParams rsaPrivatekey) bs
+                    ]
+                , bench "verify" $ nf (PSS.verify pssParams rsaPublickey bs) signedMsgPSS
                 ]
             ]
-        , bgroup "RSA PSS"
-            [ bgroup "signing"
-                [ bench "slow" $ nf (right . fst . PSS.sign rng Nothing pssParams privateKeySlow) bs
-                , bench "fast" $ nf (right . fst . PSS.sign rng Nothing pssParams rsaPrivatekey) bs
-                , bench "slow+blinding" $ nf (right . fst . PSS.sign rng (Just blinder) pssParams privateKeySlow) bs
-                , bench "fast+blinding" $ nf (right . fst . PSS.sign rng (Just blinder) pssParams rsaPrivatekey) bs
+        , bgroup "ECDSA"
+            [ bgroup "secp160r1"
+                [ bench "sign" $ nf (fst . ECDSA.sign rng ecdsaPrivatekeyP SHA1.hash) bs
+                , bench "verify"  $ nf (ECDSA.verify SHA1.hash ecdsaPublickeyP ecdsaSignatureP) bs
                 ]
-            , bench "verify" $ nf (PSS.verify pssParams rsaPublickey bs) signedMsgPSS
-            ]
-        , bgroup "ECDSA secp160r1"
-            [ bench "sign" $ nf (fst . ECDSA.sign rng ecdsaPrivatekeyP SHA1.hash) bs
-            , bench "verify"  $ nf (ECDSA.verify SHA1.hash ecdsaPublickeyP ecdsaSignatureP) bs
-            ]
-        , bgroup "ECDSA sect163k1"
-            [ bench "sign" $ nf (fst . ECDSA.sign rng ecdsaPrivatekeyB SHA1.hash) bs
-            , bench "verify"  $ nf (ECDSA.verify SHA1.hash ecdsaPublickeyB ecdsaSignatureB) bs
+            , bgroup "sect163k1"
+                [ bench "sign" $ nf (fst . ECDSA.sign rng ecdsaPrivatekeyB SHA1.hash) bs
+                , bench "verify"  $ nf (ECDSA.verify SHA1.hash ecdsaPublickeyB ecdsaSignatureB) bs
+                ]
             ]
         ]
diff --git a/Crypto/PubKey/ECC/Prim.hs b/Crypto/PubKey/ECC/Prim.hs
--- a/Crypto/PubKey/ECC/Prim.hs
+++ b/Crypto/PubKey/ECC/Prim.hs
@@ -5,6 +5,7 @@
     ( pointAdd
     , pointDouble
     , pointMul
+    , isPointAtInfinity
     ) where
 
 import Data.Maybe
@@ -12,6 +13,8 @@
 import Crypto.Number.F2m
 import Crypto.Types.PubKey.ECC
 
+--TODO: Extract helper function for `fromMaybe PointO...`
+
 -- | Elliptic Curve point addition.
 --
 -- /WARNING:/ Vulnerable to timing attacks.
@@ -22,10 +25,11 @@
 pointAdd c@(CurveFP (CurvePrime pr _)) p@(Point xp yp) q@(Point xq yq)
     | p == Point xq (-yq) = PointO
     | p == q = pointDouble c p
-    | otherwise = let s  = divmod (yp - yq) (xp - xq) pr
-                      xr = (s ^ (2::Int) - xp - xq) `mod` pr
-                      yr = (s * (xp - xr) - yp) `mod` pr
-                  in Point xr yr
+    | otherwise = fromMaybe PointO $ do
+                      s <- divmod (yp - yq) (xp - xq) pr
+                      let xr = (s ^ (2::Int) - xp - xq) `mod` pr
+                          yr = (s * (xp - xr) - yp) `mod` pr
+                      return $ Point xr yr
 pointAdd c@(CurveF2m (CurveBinary fx cc)) p@(Point xp yp) q@(Point xq yq)
     | p == Point xq (xq `addF2m` yq) = PointO
     | p == q = pointDouble c p
@@ -41,11 +45,11 @@
 -- /WARNING:/ Vulnerable to timing attacks.
 pointDouble :: Curve -> Point -> Point
 pointDouble _ PointO = PointO
-pointDouble (CurveFP (CurvePrime pr cc)) (Point xp yp) =
-    let l = divmod (3 * xp ^ (2::Int) + a) (2 * yp) pr
-        xr = (l ^ (2::Int) - 2 * xp) `mod` pr
+pointDouble (CurveFP (CurvePrime pr cc)) (Point xp yp) = fromMaybe PointO $ do
+    l <- divmod (3 * xp ^ (2::Int) + a) (2 * yp) pr
+    let xr = (l ^ (2::Int) - 2 * xp) `mod` pr
         yr = (l * (xp - xr) - yp) `mod` pr
-    in Point xr yr
+    return $ Point xr yr
   where a = ecc_a cc
 pointDouble (CurveF2m (CurveBinary fx cc)) (Point xp yp) = fromMaybe PointO $ do
     s <- return . addF2m xp =<< divF2m fx yp xp
@@ -58,11 +62,21 @@
 --
 -- /WARNING:/ Vulnerable to timing attacks.
 pointMul :: Curve -> Integer -> Point -> Point
-pointMul c n p
+pointMul _ _ PointO = PointO
+pointMul c n p@(Point xp yp)
+    | n <  0 = pointMul c (-n) (Point xp (-yp))
+    | n == 0 = PointO
     | n == 1 = p
     | odd n = pointAdd c p (pointMul c (n - 1) p)
     | otherwise = pointMul c (n `div` 2) (pointDouble c p)
 
+-- | Check if a point is the point at infinity.
+isPointAtInfinity :: Point -> Bool
+isPointAtInfinity PointO = True
+isPointAtInfinity _      = False
+
 -- | div and mod
-divmod :: Integer -> Integer -> Integer -> Integer
-divmod y x m = y * fromJust (inverse (x `mod` m) m) `mod` m
+divmod :: Integer -> Integer -> Integer -> Maybe Integer
+divmod y x m = do
+    i <- inverse (x `mod` m) m
+    return $ y * i `mod` m
diff --git a/Tests/EccArithmetic.hs b/Tests/EccArithmetic.hs
new file mode 100644
--- /dev/null
+++ b/Tests/EccArithmetic.hs
@@ -0,0 +1,43 @@
+module EccArithmetic (properties_ecc_arithmetic) where
+
+import Test.Framework.Providers.QuickCheck2 (testProperty)
+import Test.QuickCheck
+import Test.QuickCheck.Test
+
+import Crypto.Types.PubKey.ECC
+import Crypto.PubKey.ECC.Generate
+import Crypto.PubKey.ECC.Prim
+
+data GeneratePoint1 = GeneratePoint1 Curve Point
+    deriving (Show,Eq)
+data GeneratePoint2 = GeneratePoint2 Curve Point Point
+    deriving (Show,Eq)
+
+arbitraryPoint curve = do
+    sec <- choose (1,n)
+    return $ pointMul curve sec g
+  where common = common_curve curve
+        n      = ecc_n common
+        g      = ecc_g common
+
+instance Arbitrary Curve where
+    arbitrary = elements $ map getCurveByName $ enumFrom SEC_p112r1
+
+instance Arbitrary GeneratePoint1 where
+    arbitrary = do
+        curve <- arbitrary
+        p1    <- arbitraryPoint curve
+        return $ GeneratePoint1 curve p1
+
+instance Arbitrary GeneratePoint2 where
+    arbitrary = do
+        curve <- arbitrary
+        p1    <- arbitraryPoint curve
+        p2    <- arbitraryPoint curve
+        return $ GeneratePoint2 curve p1 p2
+
+properties_ecc_arithmetic =
+    [ testProperty "commutative" $ \(GeneratePoint2 curve p1 p2) ->
+        pointAdd curve p1 p2 == pointAdd curve p2 p1
+    , testProperty "add-neutral" $ \(GeneratePoint1 curve p1) -> pointAdd curve p1 PointO == p1
+    ]
diff --git a/Tests/Tests.hs b/Tests/Tests.hs
--- a/Tests/Tests.hs
+++ b/Tests/Tests.hs
@@ -33,6 +33,7 @@
 import RNG
 import KAT
 import PregenKeys
+import EccArithmetic
 
 withAleasInteger :: Rng -> Seed -> (Rng -> (a,Rng)) -> a
 withAleasInteger rng (Seed i) f = fst $ f $ reseed (i2osp (if i < 0 then -i else i)) rng
@@ -136,9 +137,6 @@
                    let q = generateQ curve d
                    return $ ECDSA.KeyPair curve q d
 
-instance Arbitrary Curve where
-    arbitrary = elements $ map getCurveByName $ enumFrom SEC_p112r1
-
 instance Arbitrary DH.PrivateNumber where
     arbitrary = fromIntegral <$> (suchThat (arbitrary :: Gen Integer) (\x -> x >= 1))
 
@@ -173,6 +171,7 @@
 
 asymOtherTests = testGroup "assymetric other tests"
     [ testProperty "DH valid" prop_dh_valid
+    , testGroup "ECC arithmetic" properties_ecc_arithmetic
     ]
 
 main = defaultMain
diff --git a/crypto-pubkey.cabal b/crypto-pubkey.cabal
--- a/crypto-pubkey.cabal
+++ b/crypto-pubkey.cabal
@@ -1,5 +1,5 @@
 Name:                crypto-pubkey
-Version:             0.2.5
+Version:             0.2.6
 Description:
     Public Key cryptography
     .