diff --git a/bls-sigs.sage b/bls-sigs.sage
index 7229fe8..cee977e 100644
--- a/bls-sigs.sage
+++ b/bls-sigs.sage
@@ -10,15 +10,15 @@ def hash(m):
 def hash_to_point(m):
     # WARNING this hash-to-point approach should not be used!
     h = hash(m)
-    return G2 * h
+    return e.G2 * h
 
 
-pairing = Pairing()
+e = Pairing()
 
 class Signer:
     def __init__(self):
-        self.sk = F1.random_element()
-        self.pk = self.sk * G1
+        self.sk = e.F1.random_element()
+        self.pk = self.sk * e.G1
 
     def sign(self, m):
         H = hash_to_point(m)
@@ -26,7 +26,7 @@ class Signer:
 
 def verify(pk, s, m):
     H = hash_to_point(m)
-    return pairing.pair(G1, s) == pairing.pair(pk, H)
+    return e.pair(e.G1, s) == e.pair(pk, H)
 
 def aggr(points):
     R = 0
diff --git a/bls12-381.sage b/bls12-381.sage
index 9bbcac4..736e8e8 100644
--- a/bls12-381.sage
+++ b/bls12-381.sage
@@ -3,61 +3,65 @@
 #  
 # ## Example of usage:
 #   load("bls12-381.sage")
-#   pairing = Pairing()
-#   assert pairing.pair(G1 * 3, G2 * 2) == pairing.pair(G1, G2)^6
+#   e = Pairing()
+#   assert e.pair(e.G1 * 3, e.G2 * 2) == e.pair(e.G1, e.G2)^6
 
 
+class Pairing():
+    def __init__(self):
+        # BLS12-381 Parameters
+        # https://github.com/zkcrypto/bls12_381
+        self.p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab
+        self.r = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
+        self.h1 = 0x396c8c005555e1568c00aaab0000aaab
+        self.h2 = 0x5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5
 
-# BLS12-381 Parameters
-# https://github.com/zkcrypto/bls12_381
-p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab
-r = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
-h1 = 0x396c8c005555e1568c00aaab0000aaab
-h2 = 0x5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5
-
-# Define base fields
-F1 = GF(p)
-F2.<u> = GF(p^2, x, x^2 + 1)
-F12.<w> = GF(p^12, x, x^12 - 2*x^6 + 2)
+        # Define base fields
+        self.F1 = GF(self.p)
+        F2.<u> = GF(self.p^2, x, x^2 + 1)
+        self.u = u
+        self.F2 = F2
+        F12.<w> = GF(self.p^12, x, x^12 - 2*x^6 + 2)
+        self.w = w
+        self.F12 = F12
 
-# Define the Elliptic Curves
-E1 = EllipticCurve(F1, [0, 4])
-E2 = EllipticCurve(F2, [0, 4*(1 + u)])
-E12 = EllipticCurve(F12, [0, 4])
+        # Define the Elliptic Curves
+        self.E1 = EllipticCurve(self.F1, [0, 4])
+        self.E2 = EllipticCurve(F2, [0, 4*(1 + self.u)])
+        self.E12 = EllipticCurve(F12, [0, 4])
 
-# Generator of order r in E1 / F1
-G1x = 0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb
-G1y = 0x8b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1
-G1 = E1(G1x, G1y)
+        # Generator of order r in E1 / F1
+        G1x = 0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb
+        G1y = 0x8b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1
+        self.G1 = self.E1(G1x, G1y)
 
-# Generator of order r in E2 / F2
-G2x0 = 0x24aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8
-G2x1 = 0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e
-G2y0 = 0xce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801
-G2y1 = 0x606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be
-G2 = E2(G2x0 + u*G2x1, G2y0 + u*G2y1)
+        # Generator of order r in E2 / F2
+        G2x0 = 0x24aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8
+        G2x1 = 0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e
+        G2y0 = 0xce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801
+        G2y1 = 0x606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be
+        self.G2 = self.E2(G2x0 + self.u*G2x1, G2y0 + self.u*G2y1)
 
 
-class Pairing():
     def lift_E1_to_E12(self, P):
         """
         Lift point on E/F_q to E/F_{q^12} using the natural lift
         """
-        assert P.curve() == E1, "Attempting to lift a point from the wrong curve."
-        return E12(P)
+        assert P.curve() == self.E1, "Attempting to lift a point from the wrong curve."
+        return self.E12(P)
 
     def lift_E2_to_E12(self, P):
         """
         Lift point on E/F_{q^2} to E/F_{q_12} through the sextic twist
         """
-        assert P.curve() == E2, "Attempting to lift a point from the wrong curve."
-        xs, ys = [c.polynomial().coefficients() for c in (h2*P).xy()]
-        nx = F12(xs[0] - xs[1] + w ^ 6*xs[1])
-        ny = F12(ys[0] - ys[1] + w ^ 6*ys[1])
-        return E12(nx / (w ^ 2), ny / (w ^ 3))
+        assert P.curve() == self.E2, "Attempting to lift a point from the wrong curve."
+        xs, ys = [c.polynomial().coefficients() for c in (self.h2*P).xy()]
+        nx = self.F12(xs[0] - xs[1] + self.w ^ 6*xs[1])
+        ny = self.F12(ys[0] - ys[1] + self.w ^ 6*ys[1])
+        return self.E12(nx / (self.w ^ 2), ny / (self.w ^ 3))
 
     def pair(self, A, B):
         A = self.lift_E1_to_E12(A)
         B = self.lift_E2_to_E12(B)
-        return A.ate_pairing(B, r, 12, E12.trace_of_frobenius())
+        return A.ate_pairing(B, self.r, 12, self.E12.trace_of_frobenius())