diff --git a/src/bls12_381/ec.rs b/src/bls12_381/ec.rs
index 09824c686..973bdbfe7 100644
--- a/src/bls12_381/ec.rs
+++ b/src/bls12_381/ec.rs
@@ -1,5 +1,6 @@
 macro_rules! curve_impl {
     (
+        $name:expr,
         $projective:ident,
         $affine:ident,
         $prepared:ident,
@@ -15,6 +16,17 @@ macro_rules! curve_impl {
             pub(crate) infinity: bool
         }
 
+        impl ::std::fmt::Display for $affine
+        {
+            fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
+                if self.infinity {
+                    write!(f, "{}(Infinity)", $name)
+                } else {
+                    write!(f, "{}(x={}, y={})", $name, self.x, self.y)
+                }
+            }
+        }
+
         #[derive(Copy, Clone, Debug, Eq)]
         pub struct $projective {
            pub(crate) x: $basefield,
@@ -22,6 +34,13 @@ macro_rules! curve_impl {
            pub(crate) z: $basefield
         }
 
+        impl ::std::fmt::Display for $projective
+        {
+            fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
+                write!(f, "{}", self.into_affine())
+            }
+        }
+
         impl PartialEq for $projective {
             fn eq(&self, other: &$projective) -> bool {
                 if self.is_zero() {
@@ -111,10 +130,6 @@ macro_rules! curve_impl {
                 self.infinity
             }
 
-            fn is_valid(&self) -> bool {
-                self.is_on_curve() && self.is_in_correct_subgroup()
-            }
-
             fn mul<S: Into<<Self::Scalar as PrimeField>::Repr>>(&self, by: S) -> $projective {
                 let mut res = $projective::zero();
 
@@ -560,12 +575,19 @@ macro_rules! curve_impl {
 pub mod g1 {
     use rand::{Rand, Rng};
     use super::super::{Fq, Fr, FrRepr, FqRepr};
-    use ::{CurveProjective, CurveAffine, PrimeField, SqrtField, PrimeFieldRepr, Field, BitIterator, EncodedPoint};
+    use ::{CurveProjective, CurveAffine, PrimeField, SqrtField, PrimeFieldRepr, Field, BitIterator, EncodedPoint, GroupDecodingError};
 
-    curve_impl!(G1, G1Affine, G1Prepared, Fq, Fr, G1Uncompressed, G1Compressed);
+    curve_impl!("G1", G1, G1Affine, G1Prepared, Fq, Fr, G1Uncompressed, G1Compressed);
 
+    #[derive(Copy)]
     pub struct G1Uncompressed([u8; 96]);
 
+    impl Clone for G1Uncompressed {
+        fn clone(&self) -> G1Uncompressed {
+            G1Uncompressed(self.0)
+        }
+    }
+
     impl AsRef<[u8]> for G1Uncompressed {
         fn as_ref(&self) -> &[u8] {
             &self.0
@@ -583,15 +605,24 @@ pub mod g1 {
 
         fn empty() -> Self { G1Uncompressed([0; 96]) }
         fn size() -> usize { 96 }
-        fn into_affine_unchecked(&self) -> Result<G1Affine, ()> {
-            use byteorder::{ReadBytesExt, BigEndian};
+        fn into_affine(&self) -> Result<G1Affine, GroupDecodingError> {
+            let affine = self.into_affine_unchecked()?;
 
+            if !affine.is_on_curve() {
+                Err(GroupDecodingError::NotOnCurve)
+            } else if !affine.is_in_correct_subgroup() {
+                Err(GroupDecodingError::NotInSubgroup)
+            } else {
+                Ok(affine)
+            }
+        }
+        fn into_affine_unchecked(&self) -> Result<G1Affine, GroupDecodingError> {
             // Create a copy of this representation.
             let mut copy = self.0;
 
             if copy[0] & (1 << 7) != 0 {
                 // Distinguisher bit is set, but this should be uncompressed!
-                return Err(())
+                return Err(GroupDecodingError::UnexpectedCompressionMode)
             }
 
             if copy[0] & (1 << 6) != 0 {
@@ -603,13 +634,13 @@ pub mod g1 {
                 if copy.iter().all(|b| *b == 0) {
                     Ok(G1Affine::zero())
                 } else {
-                    Err(())
+                    Err(GroupDecodingError::UnexpectedInformation)
                 }
             } else {
                 if copy[0] & (1 << 5) != 0 {
                     // The bit indicating the y-coordinate should be lexicographically
                     // largest is set, but this is an uncompressed element.
-                    return Err(())
+                    return Err(GroupDecodingError::UnexpectedInformation)
                 }
 
                 // Unset the three most significant bits.
@@ -621,25 +652,18 @@ pub mod g1 {
                 {
                     let mut reader = &copy[..];
 
-                    for b in x.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
-
-                    for b in y.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
+                    x.read_be(&mut reader).unwrap();
+                    y.read_be(&mut reader).unwrap();
                 }
 
                 Ok(G1Affine {
-                    x: Fq::from_repr(x)?,
-                    y: Fq::from_repr(y)?,
+                    x: Fq::from_repr(x).map_err(|e| GroupDecodingError::CoordinateDecodingError("x coordinate", e))?,
+                    y: Fq::from_repr(y).map_err(|e| GroupDecodingError::CoordinateDecodingError("y coordinate", e))?,
                     infinity: false
                 })
             }
         }
         fn from_affine(affine: G1Affine) -> Self {
-            use byteorder::{WriteBytesExt, BigEndian};
-
             let mut res = Self::empty();
 
             if affine.is_zero() {
@@ -649,21 +673,23 @@ pub mod g1 {
             } else {
                 let mut writer = &mut res.0[..];
 
-                for digit in affine.x.into_repr().as_ref().iter().rev() {
-                    writer.write_u64::<BigEndian>(*digit).unwrap();
-                }
-
-                for digit in affine.y.into_repr().as_ref().iter().rev() {
-                    writer.write_u64::<BigEndian>(*digit).unwrap();
-                }
+                affine.x.into_repr().write_be(&mut writer).unwrap();
+                affine.y.into_repr().write_be(&mut writer).unwrap();
             }
 
             res
         }
     }
 
+    #[derive(Copy)]
     pub struct G1Compressed([u8; 48]);
 
+    impl Clone for G1Compressed {
+        fn clone(&self) -> G1Compressed {
+            G1Compressed(self.0)
+        }
+    }
+
     impl AsRef<[u8]> for G1Compressed {
         fn as_ref(&self) -> &[u8] {
             &self.0
@@ -681,15 +707,24 @@ pub mod g1 {
 
         fn empty() -> Self { G1Compressed([0; 48]) }
         fn size() -> usize { 48 }
-        fn into_affine_unchecked(&self) -> Result<G1Affine, ()> {
-            use byteorder::{ReadBytesExt, BigEndian};
+        fn into_affine(&self) -> Result<G1Affine, GroupDecodingError> {
+            let affine = self.into_affine_unchecked()?;
 
+            // NB: Decompression guarantees that it is on the curve already.
+
+            if !affine.is_in_correct_subgroup() {
+                Err(GroupDecodingError::NotInSubgroup)
+            } else {
+                Ok(affine)
+            }
+        }
+        fn into_affine_unchecked(&self) -> Result<G1Affine, GroupDecodingError> {
             // Create a copy of this representation.
             let mut copy = self.0;
 
             if copy[0] & (1 << 7) == 0 {
                 // Distinguisher bit isn't set.
-                return Err(())
+                return Err(GroupDecodingError::UnexpectedCompressionMode)
             }
 
             if copy[0] & (1 << 6) != 0 {
@@ -701,7 +736,7 @@ pub mod g1 {
                 if copy.iter().all(|b| *b == 0) {
                     Ok(G1Affine::zero())
                 } else {
-                    Err(())
+                    Err(GroupDecodingError::UnexpectedInformation)
                 }
             } else {
                 // Determine if the intended y coordinate must be greater
@@ -716,13 +751,11 @@ pub mod g1 {
                 {
                     let mut reader = &copy[..];
 
-                    for b in x.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
+                    x.read_be(&mut reader).unwrap();
                 }
 
                 // Interpret as Fq element.
-                let x = Fq::from_repr(x)?;
+                let x = Fq::from_repr(x).map_err(|e| GroupDecodingError::CoordinateDecodingError("x coordinate", e))?;
 
                 // Compute x^3 + b
                 let mut x3b = x;
@@ -747,14 +780,12 @@ pub mod g1 {
                     },
                     None => {
                         // Point must not be on the curve.
-                        Err(())
+                        Err(GroupDecodingError::NotOnCurve)
                     }
                 }
             }
         }
         fn from_affine(affine: G1Affine) -> Self {
-            use byteorder::{WriteBytesExt, BigEndian};
-
             let mut res = Self::empty();
 
             if affine.is_zero() {
@@ -765,9 +796,7 @@ pub mod g1 {
                 {
                     let mut writer = &mut res.0[..];
 
-                    for digit in affine.x.into_repr().as_ref().iter().rev() {
-                        writer.write_u64::<BigEndian>(*digit).unwrap();
-                    }
+                    affine.x.into_repr().write_be(&mut writer).unwrap();
                 }
 
                 let mut negy = affine.y;
@@ -873,7 +902,7 @@ pub mod g1 {
                     infinity: false
                 };
 
-                assert!(!p.is_valid());
+                assert!(!p.is_in_correct_subgroup());
 
                 let mut g1 = G1::zero();
 
@@ -895,7 +924,7 @@ pub mod g1 {
                     assert_eq!(i, 4);
                     let g1 = G1Affine::from(g1);
 
-                    assert!(g1.is_valid());
+                    assert!(g1.is_in_correct_subgroup());
 
                     assert_eq!(g1, G1Affine::one());
                     break;
@@ -918,7 +947,6 @@ pub mod g1 {
             };
             assert!(!p.is_on_curve());
             assert!(p.is_in_correct_subgroup());
-            assert!(!p.is_valid());
         }
 
         // Reject point on a twist (b = 3)
@@ -930,7 +958,6 @@ pub mod g1 {
             };
             assert!(!p.is_on_curve());
             assert!(!p.is_in_correct_subgroup());
-            assert!(!p.is_valid());
         }
 
         // Reject point in an invalid subgroup
@@ -943,7 +970,6 @@ pub mod g1 {
             };
             assert!(p.is_on_curve());
             assert!(!p.is_in_correct_subgroup());
-            assert!(!p.is_valid());
         }
     }
 
@@ -1019,9 +1045,9 @@ pub mod g1 {
             infinity: false
         };
 
-        assert!(a.is_valid());
-        assert!(b.is_valid());
-        assert!(c.is_valid());
+        assert!(a.is_on_curve() && a.is_in_correct_subgroup());
+        assert!(b.is_on_curve() && b.is_in_correct_subgroup());
+        assert!(c.is_on_curve() && c.is_in_correct_subgroup());
 
         let mut tmp1 = a.into_projective();
         tmp1.add_assign(&b.into_projective());
@@ -1097,12 +1123,19 @@ pub mod g1 {
 pub mod g2 {
     use rand::{Rand, Rng};
     use super::super::{Fq2, Fr, Fq, FrRepr, FqRepr};
-    use ::{CurveProjective, CurveAffine, PrimeField, SqrtField, PrimeFieldRepr, Field, BitIterator, EncodedPoint};
+    use ::{CurveProjective, CurveAffine, PrimeField, SqrtField, PrimeFieldRepr, Field, BitIterator, EncodedPoint, GroupDecodingError};
 
-    curve_impl!(G2, G2Affine, G2Prepared, Fq2, Fr, G2Uncompressed, G2Compressed);
+    curve_impl!("G2", G2, G2Affine, G2Prepared, Fq2, Fr, G2Uncompressed, G2Compressed);
 
+    #[derive(Copy)]
     pub struct G2Uncompressed([u8; 192]);
 
+    impl Clone for G2Uncompressed {
+        fn clone(&self) -> G2Uncompressed {
+            G2Uncompressed(self.0)
+        }
+    }
+
     impl AsRef<[u8]> for G2Uncompressed {
         fn as_ref(&self) -> &[u8] {
             &self.0
@@ -1120,15 +1153,24 @@ pub mod g2 {
 
         fn empty() -> Self { G2Uncompressed([0; 192]) }
         fn size() -> usize { 192 }
-        fn into_affine_unchecked(&self) -> Result<G2Affine, ()> {
-            use byteorder::{ReadBytesExt, BigEndian};
+        fn into_affine(&self) -> Result<G2Affine, GroupDecodingError> {
+            let affine = self.into_affine_unchecked()?;
 
+            if !affine.is_on_curve() {
+                Err(GroupDecodingError::NotOnCurve)
+            } else if !affine.is_in_correct_subgroup() {
+                Err(GroupDecodingError::NotInSubgroup)
+            } else {
+                Ok(affine)
+            }
+        }
+        fn into_affine_unchecked(&self) -> Result<G2Affine, GroupDecodingError> {
             // Create a copy of this representation.
             let mut copy = self.0;
 
             if copy[0] & (1 << 7) != 0 {
                 // Distinguisher bit is set, but this should be uncompressed!
-                return Err(())
+                return Err(GroupDecodingError::UnexpectedCompressionMode)
             }
 
             if copy[0] & (1 << 6) != 0 {
@@ -1140,13 +1182,13 @@ pub mod g2 {
                 if copy.iter().all(|b| *b == 0) {
                     Ok(G2Affine::zero())
                 } else {
-                    Err(())
+                    Err(GroupDecodingError::UnexpectedInformation)
                 }
             } else {
                 if copy[0] & (1 << 5) != 0 {
                     // The bit indicating the y-coordinate should be lexicographically
                     // largest is set, but this is an uncompressed element.
-                    return Err(())
+                    return Err(GroupDecodingError::UnexpectedInformation)
                 }
 
                 // Unset the three most significant bits.
@@ -1160,39 +1202,26 @@ pub mod g2 {
                 {
                     let mut reader = &copy[..];
 
-                    for b in x_c1.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
-
-                    for b in x_c0.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
-
-                    for b in y_c1.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
-
-                    for b in y_c0.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
+                    x_c1.read_be(&mut reader).unwrap();
+                    x_c0.read_be(&mut reader).unwrap();
+                    y_c1.read_be(&mut reader).unwrap();
+                    y_c0.read_be(&mut reader).unwrap();
                 }
 
                 Ok(G2Affine {
                     x: Fq2 {
-                        c0: Fq::from_repr(x_c0)?,
-                        c1: Fq::from_repr(x_c1)?
+                        c0: Fq::from_repr(x_c0).map_err(|e| GroupDecodingError::CoordinateDecodingError("x coordinate (c0)", e))?,
+                        c1: Fq::from_repr(x_c1).map_err(|e| GroupDecodingError::CoordinateDecodingError("x coordinate (c1)", e))?,
                     },
                     y: Fq2 {
-                        c0: Fq::from_repr(y_c0)?,
-                        c1: Fq::from_repr(y_c1)?
+                        c0: Fq::from_repr(y_c0).map_err(|e| GroupDecodingError::CoordinateDecodingError("y coordinate (c0)", e))?,
+                        c1: Fq::from_repr(y_c1).map_err(|e| GroupDecodingError::CoordinateDecodingError("y coordinate (c1)", e))?,
                     },
                     infinity: false
                 })
             }
         }
         fn from_affine(affine: G2Affine) -> Self {
-            use byteorder::{WriteBytesExt, BigEndian};
-
             let mut res = Self::empty();
 
             if affine.is_zero() {
@@ -1202,29 +1231,25 @@ pub mod g2 {
             } else {
                 let mut writer = &mut res.0[..];
 
-                for digit in affine.x.c1.into_repr().as_ref().iter().rev() {
-                    writer.write_u64::<BigEndian>(*digit).unwrap();
-                }
-
-                for digit in affine.x.c0.into_repr().as_ref().iter().rev() {
-                    writer.write_u64::<BigEndian>(*digit).unwrap();
-                }
-
-                for digit in affine.y.c1.into_repr().as_ref().iter().rev() {
-                    writer.write_u64::<BigEndian>(*digit).unwrap();
-                }
-
-                for digit in affine.y.c0.into_repr().as_ref().iter().rev() {
-                    writer.write_u64::<BigEndian>(*digit).unwrap();
-                }
+                affine.x.c1.into_repr().write_be(&mut writer).unwrap();
+                affine.x.c0.into_repr().write_be(&mut writer).unwrap();
+                affine.y.c1.into_repr().write_be(&mut writer).unwrap();
+                affine.y.c0.into_repr().write_be(&mut writer).unwrap();
             }
 
             res
         }
     }
 
+    #[derive(Copy)]
     pub struct G2Compressed([u8; 96]);
 
+    impl Clone for G2Compressed {
+        fn clone(&self) -> G2Compressed {
+            G2Compressed(self.0)
+        }
+    }
+
     impl AsRef<[u8]> for G2Compressed {
         fn as_ref(&self) -> &[u8] {
             &self.0
@@ -1242,15 +1267,24 @@ pub mod g2 {
 
         fn empty() -> Self { G2Compressed([0; 96]) }
         fn size() -> usize { 96 }
-        fn into_affine_unchecked(&self) -> Result<G2Affine, ()> {
-            use byteorder::{ReadBytesExt, BigEndian};
+        fn into_affine(&self) -> Result<G2Affine, GroupDecodingError> {
+            let affine = self.into_affine_unchecked()?;
 
+            // NB: Decompression guarantees that it is on the curve already.
+
+            if !affine.is_in_correct_subgroup() {
+                Err(GroupDecodingError::NotInSubgroup)
+            } else {
+                Ok(affine)
+            }
+        }
+        fn into_affine_unchecked(&self) -> Result<G2Affine, GroupDecodingError> {
             // Create a copy of this representation.
             let mut copy = self.0;
 
             if copy[0] & (1 << 7) == 0 {
                 // Distinguisher bit isn't set.
-                return Err(())
+                return Err(GroupDecodingError::UnexpectedCompressionMode)
             }
 
             if copy[0] & (1 << 6) != 0 {
@@ -1262,7 +1296,7 @@ pub mod g2 {
                 if copy.iter().all(|b| *b == 0) {
                     Ok(G2Affine::zero())
                 } else {
-                    Err(())
+                    Err(GroupDecodingError::UnexpectedInformation)
                 }
             } else {
                 // Determine if the intended y coordinate must be greater
@@ -1278,19 +1312,14 @@ pub mod g2 {
                 {
                     let mut reader = &copy[..];
 
-                    for b in x_c1.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
-
-                    for b in x_c0.0.iter_mut().rev() {
-                        *b = reader.read_u64::<BigEndian>().unwrap();
-                    }
+                    x_c1.read_be(&mut reader).unwrap();
+                    x_c0.read_be(&mut reader).unwrap();
                 }
 
                 // Interpret as Fq element.
                 let x = Fq2 {
-                    c0: Fq::from_repr(x_c0)?,
-                    c1: Fq::from_repr(x_c1)?
+                    c0: Fq::from_repr(x_c0).map_err(|e| GroupDecodingError::CoordinateDecodingError("x coordinate (c0)", e))?,
+                    c1: Fq::from_repr(x_c1).map_err(|e| GroupDecodingError::CoordinateDecodingError("x coordinate (c1)", e))?
                 };
 
                 // Compute x^3 + b
@@ -1316,14 +1345,12 @@ pub mod g2 {
                     },
                     None => {
                         // Point must not be on the curve.
-                        Err(())
+                        Err(GroupDecodingError::NotOnCurve)
                     }
                 }
             }
         }
         fn from_affine(affine: G2Affine) -> Self {
-            use byteorder::{WriteBytesExt, BigEndian};
-
             let mut res = Self::empty();
 
             if affine.is_zero() {
@@ -1334,13 +1361,8 @@ pub mod g2 {
                 {
                     let mut writer = &mut res.0[..];
 
-                    for digit in affine.x.c1.into_repr().as_ref().iter().rev() {
-                        writer.write_u64::<BigEndian>(*digit).unwrap();
-                    }
-
-                    for digit in affine.x.c0.into_repr().as_ref().iter().rev() {
-                        writer.write_u64::<BigEndian>(*digit).unwrap();
-                    }
+                    affine.x.c1.into_repr().write_be(&mut writer).unwrap();
+                    affine.x.c0.into_repr().write_be(&mut writer).unwrap();
                 }
 
                 let mut negy = affine.y;
@@ -1446,7 +1468,7 @@ pub mod g2 {
                     infinity: false
                 };
 
-                assert!(!p.is_valid());
+                assert!(!p.is_in_correct_subgroup());
 
                 let mut g2 = G2::zero();
 
@@ -1468,7 +1490,7 @@ pub mod g2 {
                     assert_eq!(i, 2);
                     let g2 = G2Affine::from(g2);
 
-                    assert!(g2.is_valid());
+                    assert!(g2.is_in_correct_subgroup());
 
                     assert_eq!(g2, G2Affine::one());
                     break;
@@ -1497,7 +1519,6 @@ pub mod g2 {
             };
             assert!(!p.is_on_curve());
             assert!(p.is_in_correct_subgroup());
-            assert!(!p.is_valid());
         }
 
         // Reject point on a twist (b = 2 * (u + 1))
@@ -1515,7 +1536,6 @@ pub mod g2 {
             };
             assert!(!p.is_on_curve());
             assert!(!p.is_in_correct_subgroup());
-            assert!(!p.is_valid());
         }
 
         // Reject point in an invalid subgroup
@@ -1534,7 +1554,6 @@ pub mod g2 {
             };
             assert!(p.is_on_curve());
             assert!(!p.is_in_correct_subgroup());
-            assert!(!p.is_valid());
         }
     }
 
diff --git a/src/bls12_381/fq.rs b/src/bls12_381/fq.rs
index b7f8198a0..00b040510 100644
--- a/src/bls12_381/fq.rs
+++ b/src/bls12_381/fq.rs
@@ -1,4 +1,4 @@
-use ::{Field, PrimeField, SqrtField, PrimeFieldRepr};
+use ::{Field, PrimeField, SqrtField, PrimeFieldRepr, PrimeFieldDecodingError};
 use std::cmp::Ordering;
 use super::fq2::Fq2;
 
@@ -192,7 +192,7 @@ pub const FROBENIUS_COEFF_FQ12_C1: [Fq2; 12] = [
 // -((2**384) mod q) mod q
 pub const NEGATIVE_ONE: Fq = Fq(FqRepr([0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x7e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x40ab3263eff0206]));
 
-#[derive(Copy, Clone, PartialEq, Eq, Default)]
+#[derive(Copy, Clone, PartialEq, Eq, Default, Debug)]
 pub struct FqRepr(pub [u64; 6]);
 
 impl ::rand::Rand for FqRepr {
@@ -202,7 +202,7 @@ impl ::rand::Rand for FqRepr {
     }
 }
 
-impl ::std::fmt::Debug for FqRepr
+impl ::std::fmt::Display for FqRepr
 {
     fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
         try!(write!(f, "0x"));
@@ -221,6 +221,13 @@ impl AsRef<[u64]> for FqRepr {
     }
 }
 
+impl AsMut<[u64]> for FqRepr {
+    #[inline(always)]
+    fn as_mut(&mut self) -> &mut [u64] {
+        &mut self.0
+    }
+}
+
 impl From<u64> for FqRepr {
     #[inline(always)]
     fn from(val: u64) -> FqRepr {
@@ -355,7 +362,7 @@ impl PrimeFieldRepr for FqRepr {
     }
 }
 
-#[derive(Copy, Clone, PartialEq, Eq)]
+#[derive(Copy, Clone, PartialEq, Eq, Debug)]
 pub struct Fq(FqRepr);
 
 /// `Fq` elements are ordered lexicographically.
@@ -373,10 +380,10 @@ impl PartialOrd for Fq {
     }
 }
 
-impl ::std::fmt::Debug for Fq
+impl ::std::fmt::Display for Fq
 {
     fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
-        write!(f, "Fq({:?})", self.into_repr())
+        write!(f, "Fq({})", self.into_repr())
     }
 }
 
@@ -401,14 +408,14 @@ impl From<Fq> for FqRepr {
 impl PrimeField for Fq {
     type Repr = FqRepr;
 
-    fn from_repr(r: FqRepr) -> Result<Fq, ()> {
+    fn from_repr(r: FqRepr) -> Result<Fq, PrimeFieldDecodingError> {
         let mut r = Fq(r);
         if r.is_valid() {
             r.mul_assign(&Fq(R2));
 
             Ok(r)
         } else {
-            Err(())
+            Err(PrimeFieldDecodingError::NotInField(format!("{}", r.0)))
         }
     }
 
@@ -1676,33 +1683,33 @@ fn bench_fq_from_repr(b: &mut ::test::Bencher) {
 }
 
 #[test]
-fn test_fq_repr_debug() {
+fn test_fq_repr_display() {
     assert_eq!(
-        format!("{:?}", FqRepr([0xa956babf9301ea24, 0x39a8f184f3535c7b, 0xb38d35b3f6779585, 0x676cc4eef4c46f2c, 0xb1d4aad87651e694, 0x1947f0d5f4fe325a])),
+        format!("{}", FqRepr([0xa956babf9301ea24, 0x39a8f184f3535c7b, 0xb38d35b3f6779585, 0x676cc4eef4c46f2c, 0xb1d4aad87651e694, 0x1947f0d5f4fe325a])),
         "0x1947f0d5f4fe325ab1d4aad87651e694676cc4eef4c46f2cb38d35b3f677958539a8f184f3535c7ba956babf9301ea24".to_string()
     );
     assert_eq!(
-        format!("{:?}", FqRepr([0xb4171485fd8622dd, 0x864229a6edec7ec5, 0xc57f7bdcf8dfb707, 0x6db7ff0ecea4584a, 0xf8d8578c4a57132d, 0x6eb66d42d9fcaaa])),
+        format!("{}", FqRepr([0xb4171485fd8622dd, 0x864229a6edec7ec5, 0xc57f7bdcf8dfb707, 0x6db7ff0ecea4584a, 0xf8d8578c4a57132d, 0x6eb66d42d9fcaaa])),
         "0x06eb66d42d9fcaaaf8d8578c4a57132d6db7ff0ecea4584ac57f7bdcf8dfb707864229a6edec7ec5b4171485fd8622dd".to_string()
     );
     assert_eq!(
-        format!("{:?}", FqRepr([0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff])),
+        format!("{}", FqRepr([0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff])),
         "0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff".to_string()
     );
     assert_eq!(
-        format!("{:?}", FqRepr([0, 0, 0, 0, 0, 0])),
+        format!("{}", FqRepr([0, 0, 0, 0, 0, 0])),
         "0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000".to_string()
     );
 }
 
 #[test]
-fn test_fq_debug() {
+fn test_fq_display() {
     assert_eq!(
-        format!("{:?}", Fq::from_repr(FqRepr([0xa956babf9301ea24, 0x39a8f184f3535c7b, 0xb38d35b3f6779585, 0x676cc4eef4c46f2c, 0xb1d4aad87651e694, 0x1947f0d5f4fe325a])).unwrap()),
+        format!("{}", Fq::from_repr(FqRepr([0xa956babf9301ea24, 0x39a8f184f3535c7b, 0xb38d35b3f6779585, 0x676cc4eef4c46f2c, 0xb1d4aad87651e694, 0x1947f0d5f4fe325a])).unwrap()),
         "Fq(0x1947f0d5f4fe325ab1d4aad87651e694676cc4eef4c46f2cb38d35b3f677958539a8f184f3535c7ba956babf9301ea24)".to_string()
     );
     assert_eq!(
-        format!("{:?}", Fq::from_repr(FqRepr([0xe28e79396ac2bbf8, 0x413f6f7f06ea87eb, 0xa4b62af4a792a689, 0xb7f89f88f59c1dc5, 0x9a551859b1e43a9a, 0x6c9f5a1060de974])).unwrap()),
+        format!("{}", Fq::from_repr(FqRepr([0xe28e79396ac2bbf8, 0x413f6f7f06ea87eb, 0xa4b62af4a792a689, 0xb7f89f88f59c1dc5, 0x9a551859b1e43a9a, 0x6c9f5a1060de974])).unwrap()),
         "Fq(0x06c9f5a1060de9749a551859b1e43a9ab7f89f88f59c1dc5a4b62af4a792a689413f6f7f06ea87ebe28e79396ac2bbf8)".to_string()
     );
 }
@@ -1740,6 +1747,6 @@ fn test_fq_ordering() {
     // FqRepr's ordering is well-tested, but we still need to make sure the Fq
     // elements aren't being compared in Montgomery form.
     for i in 0..100 {
-        assert!(Fq::from_repr(FqRepr::from(i+1)) > Fq::from_repr(FqRepr::from(i)));
+        assert!(Fq::from_repr(FqRepr::from(i+1)).unwrap() > Fq::from_repr(FqRepr::from(i)).unwrap());
     }
 }
diff --git a/src/bls12_381/fq12.rs b/src/bls12_381/fq12.rs
index 354045e48..e918505f7 100644
--- a/src/bls12_381/fq12.rs
+++ b/src/bls12_381/fq12.rs
@@ -11,6 +11,13 @@ pub struct Fq12 {
     pub c1: Fq6
 }
 
+impl ::std::fmt::Display for Fq12
+{
+    fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
+        write!(f, "Fq12({} + {} * w)", self.c0, self.c1)
+    }
+}
+
 impl Rand for Fq12 {
     fn rand<R: Rng>(rng: &mut R) -> Self {
         Fq12 {
diff --git a/src/bls12_381/fq2.rs b/src/bls12_381/fq2.rs
index ec22782be..a4d8e7c7f 100644
--- a/src/bls12_381/fq2.rs
+++ b/src/bls12_381/fq2.rs
@@ -11,6 +11,13 @@ pub struct Fq2 {
     pub c1: Fq
 }
 
+impl ::std::fmt::Display for Fq2
+{
+    fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
+        write!(f, "Fq2({} + {} * u)", self.c0, self.c1)
+    }
+}
+
 /// `Fq2` elements are ordered lexicographically.
 impl Ord for Fq2 {
     #[inline(always)]
diff --git a/src/bls12_381/fq6.rs b/src/bls12_381/fq6.rs
index 1a31497e0..2c62a9237 100644
--- a/src/bls12_381/fq6.rs
+++ b/src/bls12_381/fq6.rs
@@ -11,6 +11,13 @@ pub struct Fq6 {
     pub c2: Fq2
 }
 
+impl ::std::fmt::Display for Fq6
+{
+    fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
+        write!(f, "Fq6({} + {} * v, {} * v^2)", self.c0, self.c1, self.c2)
+    }
+}
+
 impl Rand for Fq6 {
     fn rand<R: Rng>(rng: &mut R) -> Self {
         Fq6 {
diff --git a/src/bls12_381/fr.rs b/src/bls12_381/fr.rs
index c5c80b2e2..c321b7660 100644
--- a/src/bls12_381/fr.rs
+++ b/src/bls12_381/fr.rs
@@ -1,4 +1,4 @@
-use ::{Field, PrimeField, SqrtField, PrimeFieldRepr};
+use ::{Field, PrimeField, SqrtField, PrimeFieldRepr, PrimeFieldDecodingError};
 
 // r = 52435875175126190479447740508185965837690552500527637822603658699938581184513
 const MODULUS: FrRepr = FrRepr([0xffffffff00000001, 0x53bda402fffe5bfe, 0x3339d80809a1d805, 0x73eda753299d7d48]);
@@ -28,7 +28,7 @@ const S: usize = 32;
 // 2^s root of unity computed by GENERATOR^t
 const ROOT_OF_UNITY: FrRepr = FrRepr([0xb9b58d8c5f0e466a, 0x5b1b4c801819d7ec, 0xaf53ae352a31e64, 0x5bf3adda19e9b27b]);
 
-#[derive(Copy, Clone, PartialEq, Eq, Default)]
+#[derive(Copy, Clone, PartialEq, Eq, Default, Debug)]
 pub struct FrRepr(pub [u64; 4]);
 
 impl ::rand::Rand for FrRepr {
@@ -38,7 +38,7 @@ impl ::rand::Rand for FrRepr {
     }
 }
 
-impl ::std::fmt::Debug for FrRepr
+impl ::std::fmt::Display for FrRepr
 {
     fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
         try!(write!(f, "0x"));
@@ -57,6 +57,13 @@ impl AsRef<[u64]> for FrRepr {
     }
 }
 
+impl AsMut<[u64]> for FrRepr {
+    #[inline(always)]
+    fn as_mut(&mut self) -> &mut [u64] {
+        &mut self.0
+    }
+}
+
 impl From<u64> for FrRepr {
     #[inline(always)]
     fn from(val: u64) -> FrRepr {
@@ -191,13 +198,13 @@ impl PrimeFieldRepr for FrRepr {
     }
 }
 
-#[derive(Copy, Clone, PartialEq, Eq)]
+#[derive(Copy, Clone, PartialEq, Eq, Debug)]
 pub struct Fr(FrRepr);
 
-impl ::std::fmt::Debug for Fr
+impl ::std::fmt::Display for Fr
 {
     fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
-        write!(f, "Fr({:?})", self.into_repr())
+        write!(f, "Fr({})", self.into_repr())
     }
 }
 
@@ -222,14 +229,14 @@ impl From<Fr> for FrRepr {
 impl PrimeField for Fr {
     type Repr = FrRepr;
 
-    fn from_repr(r: FrRepr) -> Result<Fr, ()> {
+    fn from_repr(r: FrRepr) -> Result<Fr, PrimeFieldDecodingError> {
         let mut r = Fr(r);
         if r.is_valid() {
             r.mul_assign(&Fr(R2));
 
             Ok(r)
         } else {
-            Err(())
+            Err(PrimeFieldDecodingError::NotInField(format!("{}", r.0)))
         }
     }
 
@@ -1388,33 +1395,33 @@ fn bench_fr_from_repr(b: &mut ::test::Bencher) {
 }
 
 #[test]
-fn test_fr_repr_debug() {
+fn test_fr_repr_display() {
     assert_eq!(
-        format!("{:?}", FrRepr([0x2829c242fa826143, 0x1f32cf4dd4330917, 0x932e4e479d168cd9, 0x513c77587f563f64])),
+        format!("{}", FrRepr([0x2829c242fa826143, 0x1f32cf4dd4330917, 0x932e4e479d168cd9, 0x513c77587f563f64])),
         "0x513c77587f563f64932e4e479d168cd91f32cf4dd43309172829c242fa826143".to_string()
     );
     assert_eq!(
-        format!("{:?}", FrRepr([0x25ebe3a3ad3c0c6a, 0x6990e39d092e817c, 0x941f900d42f5658e, 0x44f8a103b38a71e0])),
+        format!("{}", FrRepr([0x25ebe3a3ad3c0c6a, 0x6990e39d092e817c, 0x941f900d42f5658e, 0x44f8a103b38a71e0])),
         "0x44f8a103b38a71e0941f900d42f5658e6990e39d092e817c25ebe3a3ad3c0c6a".to_string()
     );
     assert_eq!(
-        format!("{:?}", FrRepr([0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff])),
+        format!("{}", FrRepr([0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff])),
         "0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff".to_string()
     );
     assert_eq!(
-        format!("{:?}", FrRepr([0, 0, 0, 0])),
+        format!("{}", FrRepr([0, 0, 0, 0])),
         "0x0000000000000000000000000000000000000000000000000000000000000000".to_string()
     );
 }
 
 #[test]
-fn test_fr_debug() {
+fn test_fr_display() {
     assert_eq!(
-        format!("{:?}", Fr::from_repr(FrRepr([0xc3cae746a3b5ecc7, 0x185ec8eb3f5b5aee, 0x684499ffe4b9dd99, 0x7c9bba7afb68faa])).unwrap()),
+        format!("{}", Fr::from_repr(FrRepr([0xc3cae746a3b5ecc7, 0x185ec8eb3f5b5aee, 0x684499ffe4b9dd99, 0x7c9bba7afb68faa])).unwrap()),
         "Fr(0x07c9bba7afb68faa684499ffe4b9dd99185ec8eb3f5b5aeec3cae746a3b5ecc7)".to_string()
     );
     assert_eq!(
-        format!("{:?}", Fr::from_repr(FrRepr([0x44c71298ff198106, 0xb0ad10817df79b6a, 0xd034a80a2b74132b, 0x41cf9a1336f50719])).unwrap()),
+        format!("{}", Fr::from_repr(FrRepr([0x44c71298ff198106, 0xb0ad10817df79b6a, 0xd034a80a2b74132b, 0x41cf9a1336f50719])).unwrap()),
         "Fr(0x41cf9a1336f50719d034a80a2b74132bb0ad10817df79b6a44c71298ff198106)".to_string()
     );
 }
diff --git a/src/bls12_381/tests/mod.rs b/src/bls12_381/tests/mod.rs
index 41a54d7c3..08de50832 100644
--- a/src/bls12_381/tests/mod.rs
+++ b/src/bls12_381/tests/mod.rs
@@ -46,3 +46,508 @@ fn test_g2_compressed_valid_vectors() {
     test_vectors::<G2, G2Compressed>(include_bytes!("g2_compressed_valid_test_vectors.dat"));
 }
 
+#[test]
+fn test_g1_uncompressed_invalid_vectors() {
+    {
+        let z = G1Affine::zero().into_uncompressed();
+
+        {
+            let mut z = z;
+            z.as_mut()[0] |= 0b1000_0000;
+            if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because we expected an uncompressed point");
+            }
+        }
+
+        {
+            let mut z = z;
+            z.as_mut()[0] |= 0b0010_0000;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
+            }
+        }
+
+        for i in 0..G1Uncompressed::size() {
+            let mut z = z;
+            z.as_mut()[i] |= 0b0000_0001;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
+            }
+        }
+    }
+
+    let o = G1Affine::one().into_uncompressed();
+
+    {
+        let mut o = o;
+        o.as_mut()[0] |= 0b1000_0000;
+        if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
+            // :)
+        } else {
+            panic!("should have rejected the point because we expected an uncompressed point");
+        }
+    }
+
+    let m = Fq::char();
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[0..]).unwrap();
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "x coordinate");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[48..]).unwrap();
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "y coordinate");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let m = Fq::zero().into_repr();
+
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[0..]).unwrap();
+
+        if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
+            // :)
+        } else {
+            panic!("should have rejected the point because it isn't on the curve")
+        }
+    }
+
+    {
+        let mut o = o;
+        let mut x = Fq::one();
+
+        loop {
+            let mut x3b = x;
+            x3b.square();
+            x3b.mul_assign(&x);
+            x3b.add_assign(&Fq::from_repr(FqRepr::from(4)).unwrap()); // TODO: perhaps expose coeff_b through API?
+
+            if let Some(y) = x3b.sqrt() {
+                // We know this is on the curve, but it's likely not going to be in the correct subgroup.
+                x.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
+                y.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
+
+                if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
+                    break
+                } else {
+                    panic!("should have rejected the point because it isn't in the correct subgroup")
+                }
+            } else {
+                x.add_assign(&Fq::one());
+            }
+        }
+    }
+}
+
+#[test]
+fn test_g2_uncompressed_invalid_vectors() {
+    {
+        let z = G2Affine::zero().into_uncompressed();
+
+        {
+            let mut z = z;
+            z.as_mut()[0] |= 0b1000_0000;
+            if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because we expected an uncompressed point");
+            }
+        }
+
+        {
+            let mut z = z;
+            z.as_mut()[0] |= 0b0010_0000;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
+            }
+        }
+
+        for i in 0..G2Uncompressed::size() {
+            let mut z = z;
+            z.as_mut()[i] |= 0b0000_0001;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
+            }
+        }
+    }
+
+    let o = G2Affine::one().into_uncompressed();
+
+    {
+        let mut o = o;
+        o.as_mut()[0] |= 0b1000_0000;
+        if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
+            // :)
+        } else {
+            panic!("should have rejected the point because we expected an uncompressed point");
+        }
+    }
+
+    let m = Fq::char();
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[0..]).unwrap();
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "x coordinate (c1)");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[48..]).unwrap();
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "x coordinate (c0)");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[96..]).unwrap();
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "y coordinate (c1)");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[144..]).unwrap();
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "y coordinate (c0)");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let m = Fq::zero().into_repr();
+
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[0..]).unwrap();
+        m.write_be(&mut o.as_mut()[48..]).unwrap();
+
+        if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
+            // :)
+        } else {
+            panic!("should have rejected the point because it isn't on the curve")
+        }
+    }
+
+    {
+        let mut o = o;
+        let mut x = Fq2::one();
+
+        loop {
+            let mut x3b = x;
+            x3b.square();
+            x3b.mul_assign(&x);
+            x3b.add_assign(&Fq2 {
+                c0: Fq::from_repr(FqRepr::from(4)).unwrap(),
+                c1: Fq::from_repr(FqRepr::from(4)).unwrap()
+            }); // TODO: perhaps expose coeff_b through API?
+
+            if let Some(y) = x3b.sqrt() {
+                // We know this is on the curve, but it's likely not going to be in the correct subgroup.
+                x.c1.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
+                x.c0.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
+                y.c1.into_repr().write_be(&mut o.as_mut()[96..]).unwrap();
+                y.c0.into_repr().write_be(&mut o.as_mut()[144..]).unwrap();
+
+                if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
+                    break
+                } else {
+                    panic!("should have rejected the point because it isn't in the correct subgroup")
+                }
+            } else {
+                x.add_assign(&Fq2::one());
+            }
+        }
+    }
+}
+
+#[test]
+fn test_g1_compressed_invalid_vectors() {
+    {
+        let z = G1Affine::zero().into_compressed();
+
+        {
+            let mut z = z;
+            z.as_mut()[0] &= 0b0111_1111;
+            if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because we expected a compressed point");
+            }
+        }
+
+        {
+            let mut z = z;
+            z.as_mut()[0] |= 0b0010_0000;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
+            }
+        }
+
+        for i in 0..G1Compressed::size() {
+            let mut z = z;
+            z.as_mut()[i] |= 0b0000_0001;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
+            }
+        }
+    }
+
+    let o = G1Affine::one().into_compressed();
+
+    {
+        let mut o = o;
+        o.as_mut()[0] &= 0b0111_1111;
+        if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
+            // :)
+        } else {
+            panic!("should have rejected the point because we expected a compressed point");
+        }
+    }
+
+    let m = Fq::char();
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[0..]).unwrap();
+        o.as_mut()[0] |= 0b1000_0000;
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "x coordinate");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let mut o = o;
+        let mut x = Fq::one();
+
+        loop {
+            let mut x3b = x;
+            x3b.square();
+            x3b.mul_assign(&x);
+            x3b.add_assign(&Fq::from_repr(FqRepr::from(4)).unwrap()); // TODO: perhaps expose coeff_b through API?
+
+            if let Some(_) = x3b.sqrt() {
+                x.add_assign(&Fq::one());
+            } else {
+                x.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
+                o.as_mut()[0] |= 0b1000_0000;
+
+                if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
+                    break
+                } else {
+                    panic!("should have rejected the point because it isn't on the curve")
+                }
+            }
+        }
+    }
+
+    {
+        let mut o = o;
+        let mut x = Fq::one();
+
+        loop {
+            let mut x3b = x;
+            x3b.square();
+            x3b.mul_assign(&x);
+            x3b.add_assign(&Fq::from_repr(FqRepr::from(4)).unwrap()); // TODO: perhaps expose coeff_b through API?
+
+            if let Some(_) = x3b.sqrt() {
+                // We know this is on the curve, but it's likely not going to be in the correct subgroup.
+                x.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
+                o.as_mut()[0] |= 0b1000_0000;
+
+                if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
+                    break
+                } else {
+                    panic!("should have rejected the point because it isn't in the correct subgroup")
+                }
+            } else {
+                x.add_assign(&Fq::one());
+            }
+        }
+    }
+}
+
+#[test]
+fn test_g2_compressed_invalid_vectors() {
+    {
+        let z = G2Affine::zero().into_compressed();
+
+        {
+            let mut z = z;
+            z.as_mut()[0] &= 0b0111_1111;
+            if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because we expected a compressed point");
+            }
+        }
+
+        {
+            let mut z = z;
+            z.as_mut()[0] |= 0b0010_0000;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
+            }
+        }
+
+        for i in 0..G2Compressed::size() {
+            let mut z = z;
+            z.as_mut()[i] |= 0b0000_0001;
+            if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
+                // :)
+            } else {
+                panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
+            }
+        }
+    }
+
+    let o = G2Affine::one().into_compressed();
+
+    {
+        let mut o = o;
+        o.as_mut()[0] &= 0b0111_1111;
+        if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
+            // :)
+        } else {
+            panic!("should have rejected the point because we expected a compressed point");
+        }
+    }
+
+    let m = Fq::char();
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[0..]).unwrap();
+        o.as_mut()[0] |= 0b1000_0000;
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "x coordinate (c1)");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let mut o = o;
+        m.write_be(&mut o.as_mut()[48..]).unwrap();
+        o.as_mut()[0] |= 0b1000_0000;
+
+        if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
+            assert_eq!(coordinate, "x coordinate (c0)");
+        } else {
+            panic!("should have rejected the point")
+        }
+    }
+
+    {
+        let mut o = o;
+        let mut x = Fq2 {
+            c0: Fq::one(),
+            c1: Fq::one()
+        };
+
+        loop {
+            let mut x3b = x;
+            x3b.square();
+            x3b.mul_assign(&x);
+            x3b.add_assign(&Fq2 {
+                c0: Fq::from_repr(FqRepr::from(4)).unwrap(),
+                c1: Fq::from_repr(FqRepr::from(4)).unwrap(),
+            }); // TODO: perhaps expose coeff_b through API?
+
+            if let Some(_) = x3b.sqrt() {
+                x.add_assign(&Fq2::one());
+            } else {
+                x.c1.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
+                x.c0.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
+                o.as_mut()[0] |= 0b1000_0000;
+
+                if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
+                    break
+                } else {
+                    panic!("should have rejected the point because it isn't on the curve")
+                }
+            }
+        }
+    }
+
+    {
+        let mut o = o;
+        let mut x = Fq2 {
+            c0: Fq::one(),
+            c1: Fq::one()
+        };
+
+        loop {
+            let mut x3b = x;
+            x3b.square();
+            x3b.mul_assign(&x);
+            x3b.add_assign(&Fq2 {
+                c0: Fq::from_repr(FqRepr::from(4)).unwrap(),
+                c1: Fq::from_repr(FqRepr::from(4)).unwrap(),
+            }); // TODO: perhaps expose coeff_b through API?
+
+            if let Some(_) = x3b.sqrt() {
+                // We know this is on the curve, but it's likely not going to be in the correct subgroup.
+                x.c1.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
+                x.c0.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
+                o.as_mut()[0] |= 0b1000_0000;
+
+                if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
+                    break
+                } else {
+                    panic!("should have rejected the point because it isn't in the correct subgroup")
+                }
+            } else {
+                x.add_assign(&Fq2::one());
+            }
+        }
+    }
+}
diff --git a/src/lib.rs b/src/lib.rs
index b3df2a02f..04bde0fc0 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -28,6 +28,8 @@ pub mod bls12_381;
 pub mod wnaf;
 
 use std::fmt;
+use std::error::Error;
+use std::io::{self, Read, Write};
 
 /// An "engine" is a collection of types (fields, elliptic curve groups, etc.)
 /// with well-defined relationships. In particular, the G1/G2 curve groups are
@@ -91,6 +93,7 @@ pub trait CurveProjective: PartialEq +
                            Send +
                            Sync +
                            fmt::Debug +
+                           fmt::Display +
                            rand::Rand +
                            'static
 {
@@ -158,6 +161,7 @@ pub trait CurveAffine: Copy +
                        Send +
                        Sync +
                        fmt::Debug +
+                       fmt::Display +
                        PartialEq +
                        Eq +
                        'static
@@ -179,9 +183,6 @@ pub trait CurveAffine: Copy +
     /// additive identity.
     fn is_zero(&self) -> bool;
 
-    /// Determines if this point is on the curve and in the correct subgroup.
-    fn is_valid(&self) -> bool;
-
     /// Negates this element.
     fn negate(&mut self);
 
@@ -213,6 +214,8 @@ pub trait EncodedPoint: Sized +
                         Sync +
                         AsRef<[u8]> +
                         AsMut<[u8]> +
+                        Clone +
+                        Copy +
                         'static
 {
     type Affine: CurveAffine;
@@ -224,21 +227,17 @@ pub trait EncodedPoint: Sized +
     fn size() -> usize;
 
     /// Converts an `EncodedPoint` into a `CurveAffine` element,
-    /// if the point is valid.
-    fn into_affine(&self) -> Result<Self::Affine, ()> {
-        let affine = self.into_affine_unchecked()?;
-
-        if affine.is_valid() {
-            Ok(affine)
-        } else {
-            Err(())
-        }
-    }
+    /// if the encoding represents a valid element.
+    fn into_affine(&self) -> Result<Self::Affine, GroupDecodingError>;
 
     /// Converts an `EncodedPoint` into a `CurveAffine` element,
-    /// without checking if it's a valid point. Caller must be careful
-    /// when using this, as misuse can violate API invariants.
-    fn into_affine_unchecked(&self) -> Result<Self::Affine, ()>;
+    /// without guaranteeing that the encoding represents a valid
+    /// element. This is useful when the caller knows the encoding is
+    /// valid already.
+    ///
+    /// If the encoding is invalid, this can break API invariants,
+    /// so caution is strongly encouraged.
+    fn into_affine_unchecked(&self) -> Result<Self::Affine, GroupDecodingError>;
 
     /// Creates an `EncodedPoint` from an affine point, as long as the
     /// point is not the point at infinity.
@@ -253,6 +252,7 @@ pub trait Field: Sized +
                  Send +
                  Sync +
                  fmt::Debug +
+                 fmt::Display +
                  'static +
                  rand::Rand
 {
@@ -333,9 +333,11 @@ pub trait PrimeFieldRepr: Sized +
                           Send +
                           Sync +
                           fmt::Debug +
+                          fmt::Display +
                           'static +
                           rand::Rand +
                           AsRef<[u64]> +
+                          AsMut<[u64]> +
                           From<u64>
 {
     /// Subtract another reprensetation from this one, returning the borrow bit.
@@ -366,6 +368,96 @@ pub trait PrimeFieldRepr: Sized +
     /// Performs a leftwise bitshift of this number, effectively multiplying
     /// it by 2. Overflow is ignored.
     fn mul2(&mut self);
+
+    /// Writes this `PrimeFieldRepr` as a big endian integer. Always writes
+    /// `(num_bits` / 8) bytes.
+    fn write_be<W: Write>(&self, mut writer: W) -> io::Result<()> {
+        use byteorder::{WriteBytesExt, BigEndian};
+
+        for digit in self.as_ref().iter().rev() {
+            writer.write_u64::<BigEndian>(*digit)?;
+        }
+
+        Ok(())
+    }
+
+    /// Reads a big endian integer occupying (`num_bits` / 8) bytes into this
+    /// representation.
+    fn read_be<R: Read>(&mut self, mut reader: R) -> io::Result<()> {
+        use byteorder::{ReadBytesExt, BigEndian};
+
+        for digit in self.as_mut().iter_mut().rev() {
+            *digit = reader.read_u64::<BigEndian>()?;
+        }
+
+        Ok(())
+    }
+}
+
+/// An error that may occur when trying to interpret a `PrimeFieldRepr` as a
+/// `PrimeField` element.
+#[derive(Debug)]
+pub enum PrimeFieldDecodingError {
+    /// The encoded value is not in the field
+    NotInField(String)
+}
+
+impl Error for PrimeFieldDecodingError {
+    fn description(&self) -> &str {
+        match *self {
+            PrimeFieldDecodingError::NotInField(..) => "not an element of the field"
+        }
+    }
+}
+
+impl fmt::Display for PrimeFieldDecodingError {
+    fn fmt(&self, f: &mut fmt::Formatter) -> Result<(), fmt::Error> {
+        match *self {
+            PrimeFieldDecodingError::NotInField(ref repr) => {
+                write!(f, "{} is not an element of the field", repr)
+            }
+        }
+    }
+}
+
+/// An error that may occur when trying to decode an `EncodedPoint`.
+#[derive(Debug)]
+pub enum GroupDecodingError {
+    /// The coordinate(s) do not lie on the curve.
+    NotOnCurve,
+    /// The element is not part of the r-order subgroup.
+    NotInSubgroup,
+    /// One of the coordinates could not be decoded
+    CoordinateDecodingError(&'static str, PrimeFieldDecodingError),
+    /// The compression mode of the encoded elemnet was not as expected
+    UnexpectedCompressionMode,
+    /// The encoding contained bits that should not have been set
+    UnexpectedInformation
+}
+
+impl Error for GroupDecodingError {
+    fn description(&self) -> &str {
+        match *self {
+            GroupDecodingError::NotOnCurve => "coordinate(s) do not lie on the curve",
+            GroupDecodingError::NotInSubgroup => "the element is not part of an r-order subgroup",
+            GroupDecodingError::CoordinateDecodingError(..) => "coordinate(s) could not be decoded",
+            GroupDecodingError::UnexpectedCompressionMode => "encoding has unexpected compression mode",
+            GroupDecodingError::UnexpectedInformation => "encoding has unexpected information"
+        }
+    }
+}
+
+impl fmt::Display for GroupDecodingError {
+    fn fmt(&self, f: &mut fmt::Formatter) -> Result<(), fmt::Error> {
+        match *self {
+            GroupDecodingError::CoordinateDecodingError(description, ref err) => {
+                write!(f, "{} decoding error: {}", description, err)
+            },
+            _ => {
+                write!(f, "{}", self.description())
+            }
+        }
+    }
 }
 
 /// This represents an element of a prime field.
@@ -376,7 +468,7 @@ pub trait PrimeField: Field
     type Repr: PrimeFieldRepr + From<Self>;
 
     /// Convert this prime field element into a biginteger representation.
-    fn from_repr(Self::Repr) -> Result<Self, ()>;
+    fn from_repr(Self::Repr) -> Result<Self, PrimeFieldDecodingError>;
 
     /// Convert a biginteger reprensentation into a prime field element, if
     /// the number is an element of the field.