diff --git a/src/arith.rs b/src/arith.rs
index 2d816b7..7c09437 100644
--- a/src/arith.rs
+++ b/src/arith.rs
@@ -20,6 +20,12 @@ impl From<[u64; 4]> for U256 {
     }
 }
 
+impl From<u64> for U256 {
+    fn from(d: u64) -> Self {
+        U256::from([d, 0, 0, 0])
+    }
+}
+
 /// 512-bit, stack allocated biginteger for use in extension
 /// field serialization and scalar interpretation.
 #[derive(Copy, Clone, Debug, PartialEq, Eq)]
@@ -74,6 +80,22 @@ impl U512 {
         U512(res)
     }
 
+     pub fn from_slice(s: &[u8]) -> Result<U512, Error> {
+        if s.len() != 64 {
+            return Err(Error::InvalidLength {
+                expected: 32,
+                actual: s.len(),
+            });
+        }
+
+        let mut n = [0; 4];
+        for (l, i) in (0..4).rev().zip((0..4).map(|i| i * 16)) {
+            n[l] = BigEndian::read_u128(&s[i..]);
+        }
+
+        Ok(U512(n))
+    }
+
     /// Get a random U512
     pub fn random<R: Rng>(rng: &mut R) -> U512 {
         U512(rng.gen())
@@ -423,7 +445,6 @@ fn div2(a: &mut [u128; 2]) {
 
 /// Multiply by two
 #[inline]
-#[inline]
 fn mul2(a: &mut [u128; 2]) {
     let tmp = a[0] >> 127;
     a[0] <<= 1;
diff --git a/src/fields/fq2.rs b/src/fields/fq2.rs
index 8f64016..df96d40 100644
--- a/src/fields/fq2.rs
+++ b/src/fields/fq2.rs
@@ -92,6 +92,14 @@ impl Fq2 {
             }
         }
     }
+
+    pub fn real(&self) -> &Fq {
+        &self.c0
+    }
+
+    pub fn imaginary(&self) -> &Fq {
+        &self.c1
+    }
 }
 
 impl FieldElement for Fq2 {
diff --git a/src/groups/mod.rs b/src/groups/mod.rs
index fc247a8..9d911c4 100644
--- a/src/groups/mod.rs
+++ b/src/groups/mod.rs
@@ -689,13 +689,13 @@ fn test_miller_loop() {
 
     assert_eq!(gt,
         Fq12::new(Fq6::new(
-                Fq2::new(Fq::from_str("14551901853310307118181117653102171756020286507151693083446930124375536995872").unwrap(), Fq::from_str("9312135802322424742640599513015426415694425842442244572104764725304978020017").unwrap()), 
-                Fq2::new(Fq::from_str("2008578374540014049115224515107136454624926345291695498760935593377832328658").unwrap(), Fq::from_str("19401931167387470703307774451905975977586101231060812348184567722817888018105").unwrap()), 
+                Fq2::new(Fq::from_str("14551901853310307118181117653102171756020286507151693083446930124375536995872").unwrap(), Fq::from_str("9312135802322424742640599513015426415694425842442244572104764725304978020017").unwrap()),
+                Fq2::new(Fq::from_str("2008578374540014049115224515107136454624926345291695498760935593377832328658").unwrap(), Fq::from_str("19401931167387470703307774451905975977586101231060812348184567722817888018105").unwrap()),
                 Fq2::new(Fq::from_str("15835061253582829097893482726334173316772697321004871665993836763948321578465").unwrap(), Fq::from_str("2434436628082562384254182545550914004674636606111293955202388712261962820365").unwrap())
             ),
             Fq6::new(
-                Fq2::new(Fq::from_str("2874440054453559166574356420729655370224872280550180463983603224123901706537").unwrap(), Fq::from_str("21199736323249863378180814900160978651989782296293186487853700340281870105680").unwrap()), 
-                Fq2::new(Fq::from_str("19165582755854282767090326095669835261356341739532443976394958023142879015770").unwrap(), Fq::from_str("1381947898997178910398427566832118260186305708991760706544743699683050330259").unwrap()), 
+                Fq2::new(Fq::from_str("2874440054453559166574356420729655370224872280550180463983603224123901706537").unwrap(), Fq::from_str("21199736323249863378180814900160978651989782296293186487853700340281870105680").unwrap()),
+                Fq2::new(Fq::from_str("19165582755854282767090326095669835261356341739532443976394958023142879015770").unwrap(), Fq::from_str("1381947898997178910398427566832118260186305708991760706544743699683050330259").unwrap()),
                 Fq2::new(Fq::from_str("282285618133171001983721596014922591835675934808772882476123488581876545578").unwrap(), Fq::from_str("9533292755262567365755835323107174518472361243562718718917822947506880920117").unwrap())
             )
         )
@@ -1017,15 +1017,15 @@ fn predefined_pair() {
     let g2 = AffineG2::new(
         Fq2::new(
             Fq::from_str("10857046999023057135944570762232829481370756359578518086990519993285655852781")
-                .expect("a-coeff of g2 x generator is of the right order"), 
+                .expect("a-coeff of g2 x generator is of the right order"),
             Fq::from_str("11559732032986387107991004021392285783925812861821192530917403151452391805634")
-                .expect("b-coeff of g2 x generator is of the right order"), 
+                .expect("b-coeff of g2 x generator is of the right order"),
         ),
         Fq2::new(
             Fq::from_str("8495653923123431417604973247489272438418190587263600148770280649306958101930")
-                .expect("a-coeff of g2 y generator is of the right order"), 
+                .expect("a-coeff of g2 y generator is of the right order"),
             Fq::from_str("4082367875863433681332203403145435568316851327593401208105741076214120093531")
-                .expect("b-coeff of g2 y generator is of the right order"), 
+                .expect("b-coeff of g2 y generator is of the right order"),
         ),
     ).expect("Point(11559732032986387107991004021392285783925812861821192530917403151452391805634 * i + 10857046999023057135944570762232829481370756359578518086990519993285655852781, 4082367875863433681332203403145435568316851327593401208105741076214120093531 * i + 8495653923123431417604973247489272438418190587263600148770280649306958101930) is a valid generator for G2")
         .to_jacobian();
diff --git a/src/lib.rs b/src/lib.rs
index 8ce0d4f..e64e391 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -10,7 +10,7 @@ mod fields;
 mod groups;
 
 use fields::FieldElement;
-use groups::GroupElement;
+use groups::{GroupElement, G1Params, G2Params, GroupParams};
 
 use std::ops::{Add, Mul, Neg, Sub};
 use rand::Rng;
@@ -146,6 +146,15 @@ impl Fq {
         a.to_big_endian(slice)
             .map_err(|_| FieldError::InvalidSliceLength)
     }
+    pub fn from_u256(u256: arith::U256) -> Result<Self, FieldError> {
+        Ok(Fq(fields::Fq::new(u256).ok_or(FieldError::NotMember)?))
+    }
+    pub fn into_u256(self) -> arith::U256 {
+        (self.0).into()
+    }
+    pub fn modulus() -> arith::U256 {
+        fields::Fq::modulus()
+    }
 }
 
 impl Add<Fq> for Fq {
@@ -180,6 +189,8 @@ impl Mul for Fq {
     }
 }
 
+#[derive(Copy, Clone, Debug, PartialEq, Eq)]
+#[repr(C)]
 pub struct Fq2(fields::Fq2);
 
 impl Fq2 {
@@ -187,6 +198,10 @@ impl Fq2 {
         Fq2(fields::Fq2::one())
     }
 
+    pub fn i() -> Fq2 {
+        Fq2::new(Fq::zero(), Fq::one())
+    }
+
     pub fn zero() -> Fq2 {
         Fq2(fields::Fq2::zero())
     }
@@ -199,6 +214,51 @@ impl Fq2 {
     pub fn is_zero(&self) -> bool {
         self.0.is_zero()
     }
+
+    pub fn pow(&self, exp: arith::U256) -> Self {
+        Fq2(self.0.pow(exp))
+    }
+
+    pub fn real(&self) -> Fq {
+        Fq(*self.0.real())
+    }
+
+    pub fn imaginary(&self) -> Fq {
+        Fq(*self.0.imaginary())
+    }
+}
+
+
+impl Add<Fq2> for Fq2 {
+    type Output = Self;
+
+    fn add(self, other: Self) -> Self {
+        Fq2(self.0 + other.0)
+    }
+}
+
+impl Sub<Fq2> for Fq2 {
+    type Output = Self;
+
+    fn sub(self, other: Self) -> Self {
+        Fq2(self.0 - other.0)
+    }
+}
+
+impl Neg for Fq2 {
+    type Output = Self;
+
+    fn neg(self) -> Self {
+        Fq2(-self.0)
+    }
+}
+
+impl Mul for Fq2 {
+    type Output = Self;
+
+    fn mul(self, other: Self) -> Self {
+        Fq2(self.0 * other.0)
+    }
 }
 
 pub trait Group
@@ -253,6 +313,10 @@ impl G1 {
     pub fn set_z(&mut self, z: Fq) {
         *self.0.z_mut() = z.0
     }
+
+    pub fn b() -> Fq {
+        Fq(G1Params::coeff_b())
+    }
 }
 
 impl Group for G1 {
@@ -380,6 +444,10 @@ impl G2 {
     pub fn set_z(&mut self, z: Fq2) {
         *self.0.z_mut() = z.0
     }
+
+    pub fn b() -> Fq2 {
+        Fq2(G2Params::coeff_b())
+    }
 }
 
 impl Group for G2 {