diff --git a/finite-field.cabal b/finite-field.cabal
--- a/finite-field.cabal
+++ b/finite-field.cabal
@@ -1,5 +1,5 @@
 Name:		finite-field
-Version:	0.6.0
+Version:	0.7.0
 License:	BSD3
 License-File:	COPYING
 Author:		Masahiro Sakai (masahiro.sakai@gmail.com)
@@ -11,8 +11,18 @@
   This is an implementation of finite fields.
   Currently only prime fields are supported.
   .
+  Changes in 0.7.0
+  .
+  * use extended GCD to compute reciprocals
+  .
+  * conform with the addition of SomeNat type to type-level-numbers-0.1.1.0.
+  .
   Changes in 0.6.0
   .
+  * add Hashable instance
+  .
+  Changes in 0.6.0
+  .
   * add allValues to FiniteField class
   .
   Changes in 0.5.0
@@ -34,7 +44,7 @@
 Library
   Hs-source-dirs: src
   Build-Depends:
-     base >=4 && <5, template-haskell, deepseq, type-level-numbers, algebra >=3.1
+     base >=4 && <5, template-haskell, deepseq, hashable, type-level-numbers >=0.1.1.0 && <0.2.0.0, algebra >=3.1
   Default-Language: Haskell2010
   Other-Extensions:
      DeriveDataTypeable
@@ -43,11 +53,11 @@
      Rank2Types
      GADTs
      TemplateHaskell
+     BangPatterns
   Exposed-Modules:
      Data.FiniteField
      Data.FiniteField.Base
      Data.FiniteField.PrimeField
-     Data.FiniteField.SomeNat
 
 Test-suite TestPrimeField
   Type:              exitcode-stdio-1.0
@@ -64,6 +74,6 @@
       QuickCheck >=2 && <3,
       finite-field,
       primes,
-      type-level-numbers
+      type-level-numbers >=0.1.1.0 && <0.2.0.0
   Default-Language: Haskell2010
   Other-Extensions: TemplateHaskell
diff --git a/src/Data/FiniteField/PrimeField.hs b/src/Data/FiniteField/PrimeField.hs
--- a/src/Data/FiniteField/PrimeField.hs
+++ b/src/Data/FiniteField/PrimeField.hs
@@ -1,4 +1,4 @@
-{-# LANGUAGE ScopedTypeVariables, MultiParamTypeClasses, DeriveDataTypeable, TemplateHaskell #-}
+{-# LANGUAGE ScopedTypeVariables, MultiParamTypeClasses, DeriveDataTypeable, TemplateHaskell, BangPatterns #-}
 {-# OPTIONS_GHC -Wall #-}
 -----------------------------------------------------------------------------
 -- |
@@ -8,7 +8,7 @@
 --
 -- Maintainer  :  masahiro.sakai@gmail.com
 -- Stability   :  provisional
--- Portability :  non-portable (ScopedTypeVariables, MultiParamTypeClasses, DeriveDataTypeable, TemplateHaskell)
+-- Portability :  non-portable (ScopedTypeVariables, MultiParamTypeClasses, DeriveDataTypeable, TemplateHaskell, BangPatterns)
 --
 -- Finite field of prime order p, Fp = Z/pZ.
 --
@@ -27,6 +27,7 @@
 
 import Prelude hiding (toInteger)
 import Control.DeepSeq
+import Data.Hashable
 import Data.Ratio (denominator, numerator)
 import Data.Typeable
 import qualified Language.Haskell.TH as TH
@@ -66,7 +67,13 @@
 
 instance TL.Nat p => Fractional (PrimeField p) where
   fromRational r = fromInteger (numerator r) / fromInteger (denominator r)
-  recip a = a ^ (TL.toInt (undefined :: p) - 2 :: Integer)
+--  recip a = a ^ (TL.toInt (undefined :: p) - 2 :: Integer)
+  recip (PrimeField a) =
+    case exgcd a p of
+      (_, r, _) -> fromInteger r
+    where
+      p :: Integer
+      p = TL.toInt (undefined :: p)
 
 instance TL.Nat p => Bounded (PrimeField p) where
   minBound = PrimeField 0
@@ -88,6 +95,25 @@
   char _    = TL.toInt (undefined :: p)
   pthRoot a = a
   allValues = [minBound .. maxBound]
+
+instance TL.Nat p => Hashable (PrimeField p) where
+  hashWithSalt s (PrimeField a) =
+    s `hashWithSalt` (TL.toInt (undefined :: p) :: Int) `hashWithSalt` a
+
+-- | Extended GCD algorithm
+exgcd :: (Eq a, Integral a) => a -> a -> (a, a, a)
+exgcd f1 f2 = f $ go f1 f2 1 0 0 1
+  where
+    go !r0 !r1 !s0 !s1 !t0 !t1
+      | r1 == 0   = (r0, s0, t0)
+      | otherwise = go r1 r2 s1 s2 t1 t2
+      where
+        (q, r2) = r0 `divMod` r1
+        s2 = s0 - q*s1
+        t2 = t0 - q*t1
+    f (g,u,v)
+      | g < 0 = (-g, -u, -v)
+      | otherwise = (g,u,v)
 
 -- ---------------------------------------------------------------------------
 
diff --git a/src/Data/FiniteField/SomeNat.hs b/src/Data/FiniteField/SomeNat.hs
deleted file mode 100644
--- a/src/Data/FiniteField/SomeNat.hs
+++ /dev/null
@@ -1,58 +0,0 @@
-{-# LANGUAGE ScopedTypeVariables, Rank2Types, GADTs, DeriveDataTypeable #-}
-{-# OPTIONS_GHC -Wall #-}
------------------------------------------------------------------------------
--- |
--- Module      :  Data.FiniteField.SomeNat
--- Copyright   :  (c) Masahiro Sakai 2013
--- License     :  BSD-style
---
--- Maintainer  :  masahiro.sakai@gmail.com
--- Stability   :  provisional
--- Portability :  non-portable (ScopedTypeVariables, Rank2Types, GADTs, DeriveDataTypeable)
---
--- Utility for type-level manipulation of natural numbers
---
------------------------------------------------------------------------------
-module Data.FiniteField.SomeNat
-  ( SomeNat (..)
-  , fromInteger
-  ) where
-
-import Prelude hiding (fromInteger)
-import Control.DeepSeq
-import Data.Bits
-import Data.Typeable
-import TypeLevel.Number.Nat
-
-data SomeNat where
-  SomeNat :: Nat n => n -> SomeNat
-  deriving Typeable
-
-instance Show SomeNat where
-  showsPrec d (SomeNat n) = showParen (d > 10) $
-    showString "fromInteger " . shows (toInt n :: Integer)
-
-instance NFData SomeNat
-
-fromInteger :: Integer -> SomeNat
-fromInteger a | a < 0  = error "Data.FiniteField.SomeNat.fromInteger: negative number"
-fromInteger 0 = SomeNat (undefined :: Z)
-fromInteger a = f a (\n -> SomeNat n) (\n -> SomeNat n)
-  where
-    f :: Integer
-      -> (forall n m. (Nat n, n ~ O m) => n -> SomeNat)
-      -> (forall n m. (Nat n, n ~ I m) => n -> SomeNat)
-      -> SomeNat
-    f 1 _  k1 = k1 (undefined :: I Z)
-    f x k0 k1 = f (x `shiftR` 1) k0' k1'
-      where
-        k0' :: forall n m. (Nat n, n ~ O m) => n -> SomeNat
-        k0' _ =
-          if testBit x 0
-          then k1 (undefined :: I n)
-          else k0 (undefined :: O n)
-        k1' :: forall n m. (Nat n, n ~ I m) => n -> SomeNat
-        k1' _ =
-          if testBit x 0
-          then k1 (undefined :: I n)
-          else k0 (undefined :: O n)
diff --git a/test/TestPrimeField.hs b/test/TestPrimeField.hs
--- a/test/TestPrimeField.hs
+++ b/test/TestPrimeField.hs
@@ -11,8 +11,6 @@
 import Data.Numbers.Primes (primes)
 
 import Data.FiniteField
-import Data.FiniteField.SomeNat (SomeNat (..))
-import qualified Data.FiniteField.SomeNat as SomeNat
 import TypeLevel.Number.Nat
 
 -- ----------------------------------------------------------------------
@@ -121,11 +119,6 @@
 
 -- ----------------------------------------------------------------------
 
-prop_intToSomeNat = do
-  forAll arbitrary $ \n ->
-    case SomeNat.fromInteger (abs n) of
-      SomeNat m -> abs n == toInt m
-
 case_primeFieldT = a @?= 1
   where
     a :: $(primeField 15485867)
@@ -136,7 +129,7 @@
 smallPrimes :: Gen SomeNat
 smallPrimes = do
   i <- choose (0, 2^(16::Int))
-  return $ SomeNat.fromInteger $ primes !! i
+  return $ withNat SomeNat (primes !! i)
 
 instance Nat p => Arbitrary (PrimeField p) where
   arbitrary = liftM fromInteger arbitrary