diff --git a/Cargo.toml b/Cargo.toml
index 7cd9d62..d60b76c 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -43,6 +43,7 @@ backtrace = { version = "0.3", optional = true }
 subtle = "2.3"
 crossbeam-utils = "0.8"
 ff = "0.9"
+group = "0.9"
 metrics = "0.14.2"
 num_cpus = "1.13"
 rand = "0.8"
diff --git a/benches/hashtocurve.rs b/benches/hashtocurve.rs
index cd3e987..7e6af30 100644
--- a/benches/hashtocurve.rs
+++ b/benches/hashtocurve.rs
@@ -2,7 +2,7 @@
 
 use criterion::{criterion_group, criterion_main, Criterion};
 
-use halo2::arithmetic::Curve;
+use halo2::arithmetic::CurveExt;
 use halo2::pasta::{pallas, vesta};
 
 fn criterion_benchmark(c: &mut Criterion) {
diff --git a/examples/performance_model.rs b/examples/performance_model.rs
index 5f03efb..8cd9c53 100644
--- a/examples/performance_model.rs
+++ b/examples/performance_model.rs
@@ -1,5 +1,6 @@
+use group::Curve;
 use halo2::{
-    arithmetic::{Curve, FieldExt},
+    arithmetic::FieldExt,
     model::ModelRecorder,
     pasta::{EqAffine, Fp},
     plonk::*,
diff --git a/src/arithmetic.rs b/src/arithmetic.rs
index bee2b04..31c972b 100644
--- a/src/arithmetic.rs
+++ b/src/arithmetic.rs
@@ -3,6 +3,7 @@
 
 use crossbeam_utils::thread;
 pub use ff::Field;
+use group::Group as _;
 
 mod curves;
 mod fields;
diff --git a/src/arithmetic/curves.rs b/src/arithmetic/curves.rs
index 21da4f7..97a9527 100644
--- a/src/arithmetic/curves.rs
+++ b/src/arithmetic/curves.rs
@@ -2,8 +2,8 @@
 //! write code that generalizes over a pair of groups.
 
 use core::cmp;
-use core::fmt::Debug;
-use core::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
+use core::ops::{Add, Sub};
+use group::prime::{PrimeCurve, PrimeCurveAffine};
 use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
 
 use super::{FieldExt, Group};
@@ -13,66 +13,26 @@ use std::io::{self, Read, Write};
 /// This trait is a common interface for dealing with elements of an elliptic
 /// curve group in a "projective" form, where that arithmetic is usually more
 /// efficient.
-pub trait Curve:
-    Sized
+pub trait CurveExt:
+    PrimeCurve
+    + group::Group<Scalar = <Self as CurveExt>::ScalarExt>
     + Default
-    + Copy
-    + Clone
-    + Send
-    + Sync
-    + 'static
-    + Debug
-    + Add<Output = Self>
-    + Sub<Output = Self>
-    + Mul<<Self as Curve>::Scalar, Output = Self>
-    + Neg<Output = Self>
-    + for<'a> Add<&'a Self, Output = Self>
-    + for<'a> Sub<&'a Self, Output = Self>
-    + MulAssign<<Self as Curve>::Scalar>
-    + AddAssign
-    + SubAssign
-    + for<'a> AddAssign<&'a Self>
-    + for<'a> SubAssign<&'a Self>
-    + AddAssign<<Self as Curve>::Affine>
-    + SubAssign<<Self as Curve>::Affine>
     + PartialEq
     + cmp::Eq
     + ConditionallySelectable
     + ConstantTimeEq
-    + From<<Self as Curve>::Affine>
-    + Group<Scalar = <Self as Curve>::Scalar>
+    + From<<Self as PrimeCurve>::Affine>
+    + Group<Scalar = <Self as group::Group>::Scalar>
 {
-    /// The representation of a point on this curve in the affine coordinate space.
-    type Affine: CurveAffine<Curve = Self, Scalar = <Self as Curve>::Scalar, Base = <Self as Curve>::Base>
-        + Add<Output = Self>
-        + Sub<Output = Self>
-        + Mul<<Self as Curve>::Scalar, Output = Self>
-        + Neg<Output = <Self as Curve>::Affine>
-        + From<Self>;
     /// The scalar field of this elliptic curve.
-    type Scalar: FieldExt;
+    type ScalarExt: FieldExt;
     /// The base field over which this elliptic curve is constructed.
     type Base: FieldExt;
 
-    /// Obtains the additive identity.
-    fn identity() -> Self;
-
-    /// Obtains the base point of the curve.
-    fn generator() -> Self;
-
-    /// Doubles this element.
-    fn double(&self) -> Self;
-
-    /// Returns whether or not this element is the identity.
-    fn is_identity(&self) -> Choice;
-
     /// Apply the curve endomorphism by multiplying the x-coordinate
     /// by an element of multiplicative order 3.
     fn endo(&self) -> Self;
 
-    /// Converts this element into its affine form.
-    fn to_affine(&self) -> Self::Affine;
-
     /// Return the Jacobian coordinates of this point.
     fn jacobian_coordinates(&self) -> (Self::Base, Self::Base, Self::Base);
 
@@ -84,10 +44,10 @@ pub trait Curve:
     /// # Example
     ///
     /// ```
-    /// use halo2::arithmetic::Curve;
-    /// fn pedersen_commitment<C: Curve>(
-    ///     x: <C as Curve>::Scalar,
-    ///     r: <C as Curve>::Scalar,
+    /// use halo2::arithmetic::CurveExt;
+    /// fn pedersen_commitment<C: CurveExt>(
+    ///     x: C::ScalarExt,
+    ///     r: C::ScalarExt,
     /// ) -> C::Affine {
     ///     let hasher = C::hash_to_curve("z.cash:example_pedersen_commitment");
     ///     let g = hasher(b"g");
@@ -101,10 +61,6 @@ pub trait Curve:
     /// always be true unless an "unchecked" API was used.
     fn is_on_curve(&self) -> Choice;
 
-    /// Converts many elements into their affine form. Panics if the
-    /// sizes of the slices are different.
-    fn batch_normalize(v: &[Self], target: &mut [Self::Affine]);
-
     /// Returns the curve constant a.
     fn a() -> Self::Base;
 
@@ -119,36 +75,16 @@ pub trait Curve:
 /// This trait is the affine counterpart to `Curve` and is used for
 /// serialization, storage in memory, and inspection of $x$ and $y$ coordinates.
 pub trait CurveAffine:
-    Sized
+    PrimeCurveAffine<Scalar = <Self as CurveAffine>::ScalarExt>
     + Default
-    + Copy
-    + Clone
-    + Send
-    + Sync
-    + 'static
-    + Debug
-    + Add<Output = <Self as CurveAffine>::Curve>
-    + Sub<Output = <Self as CurveAffine>::Curve>
-    + Mul<<Self as CurveAffine>::Scalar, Output = <Self as CurveAffine>::Curve>
-    + Neg<Output = Self>
-    + PartialEq
-    + cmp::Eq
+    + Add<Output = <Self as PrimeCurveAffine>::Curve>
+    + Sub<Output = <Self as PrimeCurveAffine>::Curve>
     + ConditionallySelectable
     + ConstantTimeEq
-    + From<<Self as CurveAffine>::Curve>
+    + From<<Self as PrimeCurveAffine>::Curve>
 {
-    /// The representation of a point on this curve in the projective coordinate space.
-    type Curve: Curve<
-            Affine = Self,
-            Scalar = <Self as CurveAffine>::Scalar,
-            Base = <Self as CurveAffine>::Base,
-        > + Mul<<Self as CurveAffine>::Scalar, Output = <Self as CurveAffine>::Curve>
-        + MulAssign<<Self as CurveAffine>::Scalar>
-        + AddAssign<Self>
-        + SubAssign<Self>
-        + From<Self>;
     /// The scalar field of this elliptic curve.
-    type Scalar: FieldExt;
+    type ScalarExt: FieldExt;
     /// The base field over which this elliptic curve is constructed.
     type Base: FieldExt;
 
@@ -159,18 +95,6 @@ pub trait CurveAffine:
     /// CURVE_ID used for hash-to-curve.
     const CURVE_ID: &'static str;
 
-    /// Obtains the additive identity.
-    fn identity() -> Self;
-
-    /// Obtains the base point of the curve.
-    fn generator() -> Self;
-
-    /// Returns whether or not this element is the identity.
-    fn is_identity(&self) -> Choice;
-
-    /// Converts this element into its projective form.
-    fn to_curve(&self) -> Self::Curve;
-
     /// Gets the $(x, y)$ coordinates of this point.
     fn get_xy(&self) -> CtOption<(Self::Base, Self::Base)>;
 
@@ -182,27 +106,19 @@ pub trait CurveAffine:
     /// always be true unless an "unchecked" API was used.
     fn is_on_curve(&self) -> Choice;
 
-    /// Attempts to obtain a group element from its compressed 32-byte little
-    /// endian representation.
-    fn from_bytes(bytes: &[u8; 32]) -> CtOption<Self>;
-
     /// Reads a compressed element from the buffer and attempts to parse it
     /// using `from_bytes`.
     fn read<R: Read>(reader: &mut R) -> io::Result<Self> {
-        let mut compressed = [0u8; 32];
-        reader.read_exact(&mut compressed[..])?;
+        let mut compressed = Self::Repr::default();
+        reader.read_exact(compressed.as_mut())?;
         Option::from(Self::from_bytes(&compressed))
             .ok_or_else(|| io::Error::new(io::ErrorKind::Other, "invalid point encoding in proof"))
     }
 
-    /// Obtains the compressed, 32-byte little endian representation of this
-    /// element.
-    fn to_bytes(&self) -> [u8; 32];
-
     /// Writes an element in compressed form to the buffer.
     fn write<W: Write>(&self, writer: &mut W) -> io::Result<()> {
         let compressed = self.to_bytes();
-        writer.write_all(&compressed[..])
+        writer.write_all(compressed.as_ref())
     }
 
     /// Attempts to obtain a group element from its uncompressed 64-byte little
diff --git a/src/pasta.rs b/src/pasta.rs
index 3046bf4..4f170c6 100644
--- a/src/pasta.rs
+++ b/src/pasta.rs
@@ -15,7 +15,8 @@ pub use fields::*;
 
 #[test]
 fn test_endo_consistency() {
-    use crate::arithmetic::{Curve, FieldExt};
+    use crate::arithmetic::{CurveExt, FieldExt};
+    use group::Group;
 
     let a = pallas::Point::generator();
     assert_eq!(a * pallas::Scalar::ZETA, a.endo());
diff --git a/src/pasta/curves.rs b/src/pasta/curves.rs
index fe266cc..552944e 100644
--- a/src/pasta/curves.rs
+++ b/src/pasta/curves.rs
@@ -3,12 +3,18 @@
 
 use core::cmp;
 use core::fmt::Debug;
+use core::iter::Sum;
 use core::ops::{Add, Mul, Neg, Sub};
 use ff::Field;
+use group::{
+    prime::{PrimeCurve, PrimeCurveAffine, PrimeGroup},
+    Curve as _, Group as _, GroupEncoding,
+};
+use rand::RngCore;
 use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
 
 use super::{Fp, Fq};
-use crate::arithmetic::{Curve, CurveAffine, FieldExt, Group};
+use crate::arithmetic::{CurveAffine, CurveExt, FieldExt, Group};
 
 macro_rules! new_curve_impl {
     (($($privacy:tt)*), $name:ident, $name_affine:ident, $iso:ident, $base:ident, $scalar:ident, $blake2b_personalization:literal,
@@ -50,10 +56,21 @@ macro_rules! new_curve_impl {
             }
         }
 
-        impl Curve for $name {
-            type Affine = $name_affine;
+        impl group::Group for $name {
             type Scalar = $scalar;
-            type Base = $base;
+
+            fn random(mut rng: impl RngCore) -> Self {
+                loop {
+                    let mut buf = [0; 64];
+                    rng.fill_bytes(&mut buf);
+                    let p: Option<$name_affine> = $name_affine::from_bytes_wide(&buf).into();
+                    if let Some(p) = p {
+                        if !bool::from(p.is_identity()) {
+                            break p.to_curve();
+                        }
+                    }
+                }
+            }
 
             impl_projective_curve_specific!($name, $base, $curve_type);
 
@@ -68,22 +85,11 @@ macro_rules! new_curve_impl {
             fn is_identity(&self) -> Choice {
                 self.z.ct_is_zero()
             }
+        }
 
-            fn to_affine(&self) -> Self::Affine {
-                let zinv = self.z.invert().unwrap_or($base::zero());
-                let zinv2 = zinv.square();
-                let x = self.x * zinv2;
-                let zinv3 = zinv2 * zinv;
-                let y = self.y * zinv3;
-
-                let tmp = $name_affine {
-                    x,
-                    y,
-                    infinity: Choice::from(0u8),
-                };
-
-                $name_affine::conditional_select(&tmp, &$name_affine::identity(), zinv.ct_is_zero())
-            }
+        impl CurveExt for $name {
+            type ScalarExt = $scalar;
+            type Base = $base;
 
             impl_projective_curve_ext!($name, $name_affine, $iso, $base, $curve_type);
 
@@ -115,8 +121,12 @@ macro_rules! new_curve_impl {
                     .ct_eq(&(z6 * $name::curve_constant_b()))
                     | self.z.ct_is_zero()
             }
+        }
 
-            fn batch_normalize(p: &[Self], q: &mut [Self::Affine]) {
+        impl group::Curve for $name {
+            type AffineRepr = $name_affine;
+
+            fn batch_normalize(p: &[Self], q: &mut [Self::AffineRepr]) {
                 assert_eq!(p.len(), q.len());
 
                 let mut acc = $base::one();
@@ -153,6 +163,45 @@ macro_rules! new_curve_impl {
                     *q = $name_affine::conditional_select(&q, &$name_affine::identity(), skip);
                 }
             }
+
+            fn to_affine(&self) -> Self::AffineRepr {
+                let zinv = self.z.invert().unwrap_or($base::zero());
+                let zinv2 = zinv.square();
+                let x = self.x * zinv2;
+                let zinv3 = zinv2 * zinv;
+                let y = self.y * zinv3;
+
+                let tmp = $name_affine {
+                    x,
+                    y,
+                    infinity: Choice::from(0u8),
+                };
+
+                $name_affine::conditional_select(&tmp, &$name_affine::identity(), zinv.ct_is_zero())
+            }
+        }
+
+        impl PrimeGroup for $name {}
+
+        impl PrimeCurve for $name {
+            type Affine = $name_affine;
+        }
+
+        impl GroupEncoding for $name {
+            type Repr = [u8; 32];
+
+            fn from_bytes(bytes: &Self::Repr) -> CtOption<Self> {
+                $name_affine::from_bytes(bytes).map(Self::from)
+            }
+
+            fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption<Self> {
+                // We can't avoid curve checks when parsing a compressed encoding.
+                $name_affine::from_bytes(bytes).map(Self::from)
+            }
+
+            fn to_bytes(&self) -> Self::Repr {
+                $name_affine::from(self).to_bytes()
+            }
         }
 
         impl<'a> From<&'a $name_affine> for $name {
@@ -233,6 +282,18 @@ macro_rules! new_curve_impl {
             }
         }
 
+        impl<T> Sum<T> for $name
+        where
+            T: core::borrow::Borrow<$name>,
+        {
+            fn sum<I>(iter: I) -> Self
+            where
+                I: Iterator<Item = T>,
+            {
+                iter.fold(Self::identity(), |acc, item| acc + item.borrow())
+            }
+        }
+
         impl<'a, 'b> Add<&'a $name> for &'b $name {
             type Output = $name;
 
@@ -473,13 +534,9 @@ macro_rules! new_curve_impl {
             }
         }
 
-        impl CurveAffine for $name_affine {
+        impl PrimeCurveAffine for $name_affine {
             type Curve = $name;
             type Scalar = $scalar;
-            type Base = $base;
-
-            const BLAKE2B_PERSONALIZATION: &'static [u8; 16] = $blake2b_personalization;
-            const CURVE_ID: &'static str = $curve_id;
 
             impl_affine_curve_specific!($name, $base, $curve_type);
 
@@ -495,12 +552,6 @@ macro_rules! new_curve_impl {
                 self.infinity
             }
 
-            fn is_on_curve(&self) -> Choice {
-                // y^2 - x^3 - ax ?= b
-                (self.y.square() - (self.x.square() + &$name::curve_constant_a()) * self.x).ct_eq(&$name::curve_constant_b())
-                    | self.infinity
-            }
-
             fn to_curve(&self) -> Self::Curve {
                 $name {
                     x: self.x,
@@ -508,17 +559,10 @@ macro_rules! new_curve_impl {
                     z: $base::conditional_select(&$base::one(), &$base::zero(), self.infinity),
                 }
             }
+        }
 
-            fn get_xy(&self) -> CtOption<(Self::Base, Self::Base)> {
-                CtOption::new((self.x, self.y), !self.is_identity())
-            }
-
-            fn from_xy(x: Self::Base, y: Self::Base) -> CtOption<Self> {
-                let p = $name_affine {
-                    x, y, infinity: 0u8.into()
-                };
-                CtOption::new(p, p.is_on_curve())
-            }
+        impl GroupEncoding for $name_affine {
+            type Repr = [u8; 32];
 
             fn from_bytes(bytes: &[u8; 32]) -> CtOption<Self> {
                 let mut tmp = *bytes;
@@ -546,6 +590,11 @@ macro_rules! new_curve_impl {
                 })
             }
 
+            fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption<Self> {
+                // We can't avoid curve checks when parsing a compressed encoding.
+                Self::from_bytes(bytes)
+            }
+
             fn to_bytes(&self) -> [u8; 32] {
                 // TODO: not constant time
                 if bool::from(self.is_identity()) {
@@ -558,6 +607,31 @@ macro_rules! new_curve_impl {
                     xbytes
                 }
             }
+        }
+
+        impl CurveAffine for $name_affine {
+            type ScalarExt = $scalar;
+            type Base = $base;
+
+            const BLAKE2B_PERSONALIZATION: &'static [u8; 16] = $blake2b_personalization;
+            const CURVE_ID: &'static str = $curve_id;
+
+            fn is_on_curve(&self) -> Choice {
+                // y^2 - x^3 - ax ?= b
+                (self.y.square() - (self.x.square() + &$name::curve_constant_a()) * self.x).ct_eq(&$name::curve_constant_b())
+                    | self.infinity
+            }
+
+            fn get_xy(&self) -> CtOption<(Self::Base, Self::Base)> {
+                CtOption::new((self.x, self.y), !self.is_identity())
+            }
+
+            fn from_xy(x: Self::Base, y: Self::Base) -> CtOption<Self> {
+                let p = $name_affine {
+                    x, y, infinity: 0u8.into()
+                };
+                CtOption::new(p, p.is_on_curve())
+            }
 
             fn from_bytes_wide(bytes: &[u8; 64]) -> CtOption<Self> {
                 let mut xbytes = [0u8; 32];
diff --git a/src/pasta/hashtocurve.rs b/src/pasta/hashtocurve.rs
index d3acfe8..3fae7b8 100644
--- a/src/pasta/hashtocurve.rs
+++ b/src/pasta/hashtocurve.rs
@@ -3,7 +3,7 @@
 
 use subtle::ConstantTimeEq;
 
-use crate::arithmetic::{Curve, FieldExt};
+use crate::arithmetic::{CurveExt, FieldExt};
 
 /// Hashes over a message and writes the output to all of `buf`.
 pub fn hash_to_field<F: FieldExt>(
@@ -72,7 +72,7 @@ pub fn hash_to_field<F: FieldExt>(
 }
 
 /// Implements a degree 3 isogeny map.
-pub fn iso_map<F: FieldExt, C: Curve<Base = F>, I: Curve<Base = F>>(
+pub fn iso_map<F: FieldExt, C: CurveExt<Base = F>, I: CurveExt<Base = F>>(
     p: &I,
     iso: &[C::Base; 13],
 ) -> C {
@@ -99,7 +99,7 @@ pub fn iso_map<F: FieldExt, C: Curve<Base = F>, I: Curve<Base = F>>(
     C::new_jacobian(xo, yo, zo).unwrap()
 }
 
-pub fn map_to_curve_simple_swu<F: FieldExt, C: Curve<Base = F>, I: Curve<Base = F>>(
+pub fn map_to_curve_simple_swu<F: FieldExt, C: CurveExt<Base = F>, I: CurveExt<Base = F>>(
     u: &F,
     theta: F,
     z: F,
diff --git a/src/pasta/pallas.rs b/src/pasta/pallas.rs
index 729b264..c3094ef 100644
--- a/src/pasta/pallas.rs
+++ b/src/pasta/pallas.rs
@@ -16,7 +16,8 @@ pub type Affine = EpAffine;
 
 #[test]
 fn test_iso_map() {
-    use crate::arithmetic::Curve;
+    use crate::arithmetic::CurveExt;
+    use group::Group;
 
     // This is a regression test (it's the same input to iso_map as for hash_to_curve
     // with domain prefix "z.cash:test", Shake128, and input b"hello"). We don't
@@ -64,7 +65,8 @@ fn test_iso_map() {
 
 #[test]
 fn test_iso_map_identity() {
-    use crate::arithmetic::Curve;
+    use crate::arithmetic::CurveExt;
+    use group::Group;
 
     let r = super::IsoEp::new_jacobian(
         Base::from_raw([
@@ -97,7 +99,7 @@ fn test_iso_map_identity() {
 
 #[test]
 fn test_map_to_curve_simple_swu() {
-    use crate::arithmetic::Curve;
+    use crate::arithmetic::CurveExt;
     use crate::pasta::curves::IsoEp;
     use crate::pasta::hashtocurve::map_to_curve_simple_swu;
 
@@ -131,7 +133,8 @@ fn test_map_to_curve_simple_swu() {
 
 #[test]
 fn test_hash_to_curve() {
-    use crate::arithmetic::Curve;
+    use crate::arithmetic::CurveExt;
+    use group::Group;
 
     // This test vector is chosen so that the first map_to_curve_simple_swu takes the gx1 square
     // "branch" and the second takes the gx1 non-square "branch" (opposite to the Vesta test vector).
diff --git a/src/pasta/vesta.rs b/src/pasta/vesta.rs
index d41d60f..c958804 100644
--- a/src/pasta/vesta.rs
+++ b/src/pasta/vesta.rs
@@ -16,7 +16,7 @@ pub type Affine = EqAffine;
 
 #[test]
 fn test_map_to_curve_simple_swu() {
-    use crate::arithmetic::Curve;
+    use crate::arithmetic::CurveExt;
     use crate::pasta::curves::IsoEq;
     use crate::pasta::hashtocurve::map_to_curve_simple_swu;
 
@@ -50,7 +50,7 @@ fn test_map_to_curve_simple_swu() {
 
 #[test]
 fn test_hash_to_curve() {
-    use crate::arithmetic::Curve;
+    use crate::arithmetic::CurveExt;
 
     // This test vector is chosen so that the first map_to_curve_simple_swu takes the gx1 non-square
     // "branch" and the second takes the gx1 square "branch" (opposite to the Pallas test vector).
diff --git a/src/plonk.rs b/src/plonk.rs
index bb9fd31..95719c4 100644
--- a/src/plonk.rs
+++ b/src/plonk.rs
@@ -44,7 +44,7 @@ impl<C: CurveAffine> VerifyingKey<C> {
     /// Writes a verifying key to a buffer.
     pub fn write<W: io::Write>(&self, writer: &mut W) -> io::Result<()> {
         for commitment in &self.fixed_commitments {
-            writer.write_all(&commitment.to_bytes())?;
+            writer.write_all(commitment.to_bytes().as_ref())?;
         }
         for permutation in &self.permutations {
             permutation.write(writer)?;
@@ -190,7 +190,7 @@ type ChallengeX<F> = ChallengeScalar<F, X>;
 
 #[test]
 fn test_proving() {
-    use crate::arithmetic::{Curve, FieldExt};
+    use crate::arithmetic::FieldExt;
     use crate::dev::MockProver;
     use crate::pasta::{EqAffine, Fp};
     use crate::poly::{
@@ -199,6 +199,7 @@ fn test_proving() {
     };
     use crate::transcript::{Blake2bRead, Blake2bWrite};
     use circuit::{Advice, Column, Fixed};
+    use group::Curve;
     use std::marker::PhantomData;
     const K: u32 = 5;
 
diff --git a/src/plonk/keygen.rs b/src/plonk/keygen.rs
index a206913..18cbafb 100644
--- a/src/plonk/keygen.rs
+++ b/src/plonk/keygen.rs
@@ -1,10 +1,11 @@
 use ff::Field;
+use group::Curve;
 
 use super::{
     circuit::{Advice, Assignment, Circuit, Column, ConstraintSystem, Fixed},
     permutation, Error, LagrangeCoeff, Polynomial, ProvingKey, VerifyingKey,
 };
-use crate::arithmetic::{Curve, CurveAffine};
+use crate::arithmetic::CurveAffine;
 use crate::poly::{
     commitment::{Blind, Params},
     EvaluationDomain, Rotation,
diff --git a/src/plonk/lookup/prover.rs b/src/plonk/lookup/prover.rs
index 1eee099..76693dc 100644
--- a/src/plonk/lookup/prover.rs
+++ b/src/plonk/lookup/prover.rs
@@ -4,7 +4,7 @@ use super::super::{
 };
 use super::Argument;
 use crate::{
-    arithmetic::{eval_polynomial, parallelize, BatchInvert, Curve, CurveAffine, FieldExt},
+    arithmetic::{eval_polynomial, parallelize, BatchInvert, CurveAffine, FieldExt},
     poly::{
         commitment::{Blind, Params},
         multiopen::ProverQuery,
@@ -13,6 +13,7 @@ use crate::{
     transcript::TranscriptWrite,
 };
 use ff::Field;
+use group::Curve;
 use std::{
     collections::BTreeMap,
     iter,
@@ -86,7 +87,7 @@ impl<F: FieldExt> Argument<F> {
         transcript: &mut T,
     ) -> Result<Permuted<C>, Error>
     where
-        C: CurveAffine<Scalar = F>,
+        C: CurveAffine<ScalarExt = F>,
         C::Curve: Mul<F, Output = C::Curve> + MulAssign<F>,
     {
         // Closure to get values of expressions and compress them
diff --git a/src/plonk/permutation/keygen.rs b/src/plonk/permutation/keygen.rs
index 072f6c5..dfc1f55 100644
--- a/src/plonk/permutation/keygen.rs
+++ b/src/plonk/permutation/keygen.rs
@@ -1,8 +1,9 @@
 use ff::Field;
+use group::Curve;
 
 use super::{Argument, ProvingKey, VerifyingKey};
 use crate::{
-    arithmetic::{Curve, CurveAffine, FieldExt},
+    arithmetic::{CurveAffine, FieldExt},
     plonk::{circuit::ConstraintSystem, Error},
     poly::{
         commitment::{Blind, Params},
diff --git a/src/plonk/permutation/prover.rs b/src/plonk/permutation/prover.rs
index b0a3f30..eca39a2 100644
--- a/src/plonk/permutation/prover.rs
+++ b/src/plonk/permutation/prover.rs
@@ -1,10 +1,11 @@
 use ff::Field;
+use group::Curve;
 use std::iter;
 
 use super::super::circuit::Any;
 use super::{Argument, ProvingKey};
 use crate::{
-    arithmetic::{eval_polynomial, parallelize, BatchInvert, Curve, CurveAffine, FieldExt},
+    arithmetic::{eval_polynomial, parallelize, BatchInvert, CurveAffine, FieldExt},
     plonk::{self, ChallengeBeta, ChallengeGamma, ChallengeX, Error},
     poly::{
         commitment::{Blind, Params},
diff --git a/src/plonk/prover.rs b/src/plonk/prover.rs
index f9f1e3a..e919fd9 100644
--- a/src/plonk/prover.rs
+++ b/src/plonk/prover.rs
@@ -1,4 +1,5 @@
 use ff::Field;
+use group::Curve;
 use std::iter;
 
 use super::{
@@ -6,7 +7,7 @@ use super::{
     lookup, permutation, vanishing, ChallengeBeta, ChallengeGamma, ChallengeTheta, ChallengeX,
     ChallengeY, Error, ProvingKey,
 };
-use crate::arithmetic::{eval_polynomial, Curve, CurveAffine, FieldExt};
+use crate::arithmetic::{eval_polynomial, CurveAffine, FieldExt};
 use crate::poly::{
     commitment::{Blind, Params},
     multiopen::{self, ProverQuery},
diff --git a/src/plonk/vanishing/prover.rs b/src/plonk/vanishing/prover.rs
index bf8094e..441615e 100644
--- a/src/plonk/vanishing/prover.rs
+++ b/src/plonk/vanishing/prover.rs
@@ -1,6 +1,8 @@
+use group::Curve;
+
 use super::Argument;
 use crate::{
-    arithmetic::{eval_polynomial, Curve, CurveAffine, FieldExt},
+    arithmetic::{eval_polynomial, CurveAffine, FieldExt},
     plonk::{ChallengeX, ChallengeY, Error},
     poly::{
         commitment::{Blind, Params},
diff --git a/src/poly/commitment.rs b/src/poly/commitment.rs
index 5bbc350..7ed1ac3 100644
--- a/src/poly/commitment.rs
+++ b/src/poly/commitment.rs
@@ -6,10 +6,10 @@
 use blake2b_simd::{Params as Blake2bParams, State as Blake2bState};
 
 use super::{Coeff, LagrangeCoeff, Polynomial};
-use crate::arithmetic::{best_fft, best_multiexp, parallelize, Curve, CurveAffine, FieldExt};
+use crate::arithmetic::{best_fft, best_multiexp, parallelize, CurveAffine, FieldExt, Group};
 
 use ff::{Field, PrimeField};
-use std::convert::TryInto;
+use group::{prime::PrimeCurveAffine, Curve};
 use std::ops::{Add, AddAssign, Mul, MulAssign};
 
 mod msm;
@@ -36,7 +36,10 @@ pub struct Params<C: CurveAffine> {
 impl<C: CurveAffine> Params<C> {
     /// Initializes parameters for the curve, given a random oracle to draw
     /// points from.
-    pub fn new(k: u32) -> Self {
+    pub fn new(k: u32) -> Self
+    where
+        <C as PrimeCurveAffine>::Curve: Group,
+    {
         // This is usually a limitation on the curve, but we also want 32-bit
         // architectures to be supported.
         assert!(k < 32);
@@ -50,7 +53,9 @@ impl<C: CurveAffine> Params<C> {
             loop {
                 let mut hasher = hasher.clone();
                 hasher.update(&(trial.to_le_bytes())[..]);
-                let p = C::from_bytes(&hasher.finalize().as_bytes().try_into().unwrap());
+                let mut repr = C::Repr::default();
+                repr.as_mut().copy_from_slice(hasher.finalize().as_bytes());
+                let p = C::from_bytes(&repr);
                 if bool::from(p.is_some()) {
                     break p.unwrap();
                 }
@@ -83,7 +88,7 @@ impl<C: CurveAffine> Params<C> {
 
         // Let's evaluate all of the Lagrange basis polynomials
         // using an inverse FFT.
-        let mut alpha_inv = C::Scalar::ROOT_OF_UNITY_INV;
+        let mut alpha_inv = <<C as PrimeCurveAffine>::Curve as Group>::Scalar::ROOT_OF_UNITY_INV;
         for _ in k..C::Scalar::S {
             alpha_inv = alpha_inv.square();
         }
@@ -191,13 +196,13 @@ impl<C: CurveAffine> Params<C> {
     pub fn write<W: io::Write>(&self, writer: &mut W) -> io::Result<()> {
         writer.write_all(&self.k.to_le_bytes())?;
         for g_element in &self.g {
-            writer.write_all(&g_element.to_bytes())?;
+            writer.write_all(g_element.to_bytes().as_ref())?;
         }
         for g_lagrange_element in &self.g_lagrange {
-            writer.write_all(&g_lagrange_element.to_bytes())?;
+            writer.write_all(g_lagrange_element.to_bytes().as_ref())?;
         }
-        writer.write_all(&self.h.to_bytes())?;
-        writer.write_all(&self.u.to_bytes())?;
+        writer.write_all(self.h.to_bytes().as_ref())?;
+        writer.write_all(self.u.to_bytes().as_ref())?;
 
         Ok(())
     }
diff --git a/src/poly/commitment/msm.rs b/src/poly/commitment/msm.rs
index 090fbb9..91184a1 100644
--- a/src/poly/commitment/msm.rs
+++ b/src/poly/commitment/msm.rs
@@ -1,6 +1,7 @@
 use super::Params;
-use crate::arithmetic::{best_multiexp, parallelize, Curve, CurveAffine};
+use crate::arithmetic::{best_multiexp, parallelize, CurveAffine};
 use ff::Field;
+use group::Group;
 
 /// A multiscalar multiplication in the polynomial commitment scheme
 #[derive(Debug, Clone)]
diff --git a/src/poly/commitment/prover.rs b/src/poly/commitment/prover.rs
index 71995b5..db3e97c 100644
--- a/src/poly/commitment/prover.rs
+++ b/src/poly/commitment/prover.rs
@@ -3,10 +3,11 @@ use ff::Field;
 use super::super::{Coeff, Polynomial};
 use super::{Blind, Params};
 use crate::arithmetic::{
-    best_multiexp, compute_inner_product, eval_polynomial, parallelize, Curve, CurveAffine,
-    FieldExt,
+    best_multiexp, compute_inner_product, eval_polynomial, parallelize, CurveAffine, FieldExt,
 };
 use crate::transcript::{Challenge, ChallengeScalar, TranscriptWrite};
+
+use group::Curve;
 use std::io;
 
 /// Create a polynomial commitment opening proof for the polynomial defined
diff --git a/src/poly/commitment/verifier.rs b/src/poly/commitment/verifier.rs
index da057fd..2665223 100644
--- a/src/poly/commitment/verifier.rs
+++ b/src/poly/commitment/verifier.rs
@@ -1,10 +1,11 @@
 use ff::Field;
+use group::Curve;
 
 use super::super::Error;
 use super::{Params, MSM};
 use crate::transcript::{Challenge, ChallengeScalar, TranscriptRead};
 
-use crate::arithmetic::{best_multiexp, BatchInvert, Curve, CurveAffine};
+use crate::arithmetic::{best_multiexp, BatchInvert, CurveAffine};
 
 /// A guard returned by the verifier
 #[derive(Debug, Clone)]
diff --git a/src/poly/multiopen.rs b/src/poly/multiopen.rs
index c5937e4..2ed4ee7 100644
--- a/src/poly/multiopen.rs
+++ b/src/poly/multiopen.rs
@@ -206,8 +206,10 @@ where
 
 #[test]
 fn test_roundtrip() {
+    use group::Curve;
+
     use super::commitment::{Blind, Params};
-    use crate::arithmetic::{eval_polynomial, Curve, FieldExt};
+    use crate::arithmetic::{eval_polynomial, FieldExt};
     use crate::pasta::{EqAffine, Fp};
 
     const K: u32 = 4;
diff --git a/src/poly/multiopen/prover.rs b/src/poly/multiopen/prover.rs
index 7cf4703..9dbb647 100644
--- a/src/poly/multiopen/prover.rs
+++ b/src/poly/multiopen/prover.rs
@@ -7,10 +7,11 @@ use super::{
     Query,
 };
 
-use crate::arithmetic::{eval_polynomial, kate_division, Curve, CurveAffine, FieldExt};
+use crate::arithmetic::{eval_polynomial, kate_division, CurveAffine, FieldExt};
 use crate::transcript::TranscriptWrite;
 
 use ff::Field;
+use group::Curve;
 use std::io;
 use std::marker::PhantomData;
 
diff --git a/src/transcript.rs b/src/transcript.rs
index 86052b6..c3da9a9 100644
--- a/src/transcript.rs
+++ b/src/transcript.rs
@@ -69,8 +69,8 @@ impl<R: Read, C: CurveAffine> Blake2bRead<R, C> {
 
 impl<R: Read, C: CurveAffine> TranscriptRead<C> for Blake2bRead<R, C> {
     fn read_point(&mut self) -> io::Result<C> {
-        let mut compressed = [0u8; 32];
-        self.reader.read_exact(&mut compressed[..])?;
+        let mut compressed = C::Repr::default();
+        self.reader.read_exact(compressed.as_mut())?;
         let point: C = Option::from(C::from_bytes(&compressed)).ok_or_else(|| {
             io::Error::new(io::ErrorKind::Other, "invalid point encoding in proof")
         })?;
@@ -154,7 +154,7 @@ impl<W: Write, C: CurveAffine> TranscriptWrite<C> for Blake2bWrite<W, C> {
     fn write_point(&mut self, point: C) -> io::Result<()> {
         self.common_point(point)?;
         let compressed = point.to_bytes();
-        self.writer.write_all(&compressed[..])
+        self.writer.write_all(compressed.as_ref())
     }
     fn write_scalar(&mut self, scalar: C::Scalar) -> io::Result<()> {
         self.common_scalar(scalar)?;