diff --git a/src/ciphersuite.rs b/src/ciphersuite.rs
index 9f96968..ce159e2 100644
--- a/src/ciphersuite.rs
+++ b/src/ciphersuite.rs
@@ -7,19 +7,22 @@
 
 use crate::{
     errors::InternalPakeError,
-    group::Group,
     keypair::{Key, KeyPair},
+    map_to_curve::GroupWithMapToCurve,
     slow_hash::SlowHash,
 };
-use generic_array::typenum::{U32, U64};
+
 use rand_core::{CryptoRng, RngCore};
 
 /// Configures the underlying primitives used in OPAQUE
-/// * Group: a finite cyclic group along with a point representation
-/// * KeyFormat: a keypair type composed of public and private components
-/// * SlowHash: a slow hashing function, typically used for password hashing
+/// * `Group`: a finite cyclic group along with a point representation, along
+///   with an extension trait PasswordToCurve that allows some customization on
+///   how to hash a password to a curve point. See `group::Group` and
+///   `map_to_curve::GroupWithMapToCurve`.
+/// * `KeyFormat`: a keypair type composed of public and private components
+/// * `SlowHash`: a slow hashing function, typically used for password hashing
 pub trait CipherSuite {
-    type Group: Group<ScalarLen = U32, UniformBytesLen = U64>;
+    type Group: GroupWithMapToCurve;
     type KeyFormat: KeyPair<Repr = Key> + PartialEq;
     type SlowHash: SlowHash;
 
diff --git a/src/envelope.rs b/src/envelope.rs
index bb6bbb5..7f481e2 100644
--- a/src/envelope.rs
+++ b/src/envelope.rs
@@ -11,14 +11,16 @@ use generic_array::{
 use hkdf::Hkdf;
 use hmac::{Hmac, Mac, NewMac};
 use rand_core::{CryptoRng, RngCore};
-use sha2::{Digest, Sha256};
+use sha2::Sha256;
 
 // Constant string used as salt for HKDF computation
 const STR_ENVU: &[u8] = b"EnvU";
 
 /// The length of the "export key" output by the client registration
 /// and login finish steps
-pub(crate) type ExportKeySize = <Sha256 as Digest>::OutputSize;
+pub(crate) type ExportKeySize = U32;
+
+const NONCE_LEN: usize = 32;
 
 /// This struct is an instantiation of the envelope as described in
 /// https://tools.ietf.org/html/draft-krawczyk-cfrg-opaque-06#section-4
@@ -33,42 +35,32 @@ pub(crate) type ExportKeySize = <Sha256 as Digest>::OutputSize;
 pub(crate) struct Envelope {
     nonce: Vec<u8>,
     ciphertext: Vec<u8>,
-    hmac: Vec<u8>,
+    hmac: GenericArray<u8, U32>,
 }
 
-type NonceLen = U32;
-
 impl Envelope {
     /// The additional number of bytes added to the plaintext
     pub(crate) fn additional_size() -> usize {
-        Self::nonce_size() + <Sha256 as Digest>::OutputSize::to_usize()
+        NONCE_LEN + U32::to_usize()
     }
 
     fn hmac_key_size() -> usize {
-        <Sha256 as Digest>::OutputSize::to_usize()
+        U32::to_usize()
     }
 
     fn hmac_size() -> usize {
-        <Sha256 as Digest>::OutputSize::to_usize()
-    }
-
-    fn nonce_size() -> usize {
-        NonceLen::to_usize()
+        U32::to_usize()
     }
 
     fn export_key_size() -> usize {
         ExportKeySize::to_usize()
     }
 
-    pub(crate) fn new(
-        nonce: Vec<u8>,
-        ciphertext: Vec<u8>,
-        hmac: &GenericArray<u8, <Sha256 as Digest>::OutputSize>,
-    ) -> Self {
+    pub(crate) fn new(nonce: Vec<u8>, ciphertext: Vec<u8>, hmac: GenericArray<u8, U32>) -> Self {
         Self {
             nonce,
             ciphertext,
-            hmac: hmac.to_vec(),
+            hmac,
         }
     }
 
@@ -76,13 +68,13 @@ impl Envelope {
     /// nonce             | ciphertext       | hmac
     /// nonce_size bytes  | variable length  | hmac_size bytes
     pub(crate) fn from_bytes(bytes: &[u8]) -> Result<Self, InternalPakeError> {
-        let ciphertext_start = Self::nonce_size();
+        let ciphertext_start = NONCE_LEN;
         let ciphertext_end = bytes.len() - Self::hmac_size();
 
         Ok(Self::new(
             bytes[..ciphertext_start].to_vec(),
             bytes[ciphertext_start..ciphertext_end].to_vec(),
-            GenericArray::from_slice(&bytes[ciphertext_end..]),
+            GenericArray::clone_from_slice(&bytes[ciphertext_end..]),
         ))
     }
 
@@ -98,7 +90,7 @@ impl Envelope {
         aad: &[u8],
         rng: &mut R,
     ) -> Result<(Self, GenericArray<u8, ExportKeySize>), InternalPakeError> {
-        let mut nonce = vec![0u8; Self::nonce_size()];
+        let mut nonce = vec![0u8; NONCE_LEN];
         rng.fill_bytes(&mut nonce);
 
         let h = Hkdf::<Sha256>::new(Some(&nonce), &key);
@@ -121,7 +113,7 @@ impl Envelope {
         hmac.update(&aad);
 
         Ok((
-            Self::new(nonce, ciphertext.to_vec(), &hmac.finalize().into_bytes()),
+            Self::new(nonce, ciphertext.to_vec(), hmac.finalize().into_bytes()),
             *GenericArray::from_slice(&export_key),
         ))
     }
diff --git a/src/group.rs b/src/group.rs
index 44547d2..27d0b39 100644
--- a/src/group.rs
+++ b/src/group.rs
@@ -140,7 +140,7 @@ impl Group for EdwardsPoint {
         *GenericArray::from_slice(c.as_bytes())
     }
 
-    type UniformBytesLen = U64;
+    type UniformBytesLen = U32;
     fn hash_to_curve(uniform_bytes: &GenericArray<u8, Self::UniformBytesLen>) -> Self {
         let mut result = [0u8; 32];
         let mut counter = 0;
diff --git a/src/key_exchange.rs b/src/key_exchange.rs
index 39ca876..494f5a9 100644
--- a/src/key_exchange.rs
+++ b/src/key_exchange.rs
@@ -9,7 +9,7 @@ use crate::{
     sized_bytes_using_constant_and_try_from,
 };
 use generic_array::{
-    typenum::{U64, U96},
+    typenum::{U32, U64, U96},
     GenericArray,
 };
 use hkdf::Hkdf;
@@ -199,9 +199,9 @@ struct TripleDHComponents {
 
 // Consists of a shared secret, followed by two mac keys
 type TripleDHDerivationResult = (
-    GenericArray<u8, <Sha256 as Digest>::OutputSize>,
-    GenericArray<u8, <Sha256 as Digest>::OutputSize>,
-    GenericArray<u8, <Sha256 as Digest>::OutputSize>,
+    GenericArray<u8, U32>,
+    GenericArray<u8, U32>,
+    GenericArray<u8, U32>,
 );
 
 // Internal function which takes the public and private components of the client and server keypairs, along
diff --git a/src/lib.rs b/src/lib.rs
index cea3e78..2a824eb 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -380,9 +380,13 @@ pub mod opaque;
 
 pub mod ciphersuite;
 mod envelope;
+
 mod group;
+mod map_to_curve;
+
 mod key_exchange;
 pub mod keypair;
+
 mod oprf;
 pub mod slow_hash;
 
diff --git a/src/map_to_curve.rs b/src/map_to_curve.rs
new file mode 100644
index 0000000..8992d9a
--- /dev/null
+++ b/src/map_to_curve.rs
@@ -0,0 +1,23 @@
+use crate::group::Group;
+use curve25519_dalek::{edwards::EdwardsPoint, ristretto::RistrettoPoint};
+
+use generic_array::GenericArray;
+use hkdf::Hkdf;
+
+pub trait GroupWithMapToCurve: Group {
+    fn map_to_curve(password: &[u8], pepper: Option<&[u8]>) -> Self;
+}
+
+impl GroupWithMapToCurve for RistrettoPoint {
+    fn map_to_curve(password: &[u8], pepper: Option<&[u8]>) -> Self {
+        let (hashed_input, _) = Hkdf::<sha2::Sha512>::extract(pepper, password);
+        <Self as Group>::hash_to_curve(GenericArray::from_slice(&hashed_input))
+    }
+}
+
+impl GroupWithMapToCurve for EdwardsPoint {
+    fn map_to_curve(password: &[u8], pepper: Option<&[u8]>) -> Self {
+        let (hashed_input, _) = Hkdf::<sha2::Sha256>::extract(pepper, password);
+        <Self as Group>::hash_to_curve(GenericArray::from_slice(&hashed_input))
+    }
+}
diff --git a/src/oprf.rs b/src/oprf.rs
index 98c066d..e4f145d 100644
--- a/src/oprf.rs
+++ b/src/oprf.rs
@@ -3,11 +3,10 @@
 // This source code is licensed under the MIT license found in the
 // LICENSE file in the root directory of this source tree.
 
-use crate::{errors::InternalPakeError, group::Group};
-use generic_array::{typenum::U64, GenericArray};
+use crate::{errors::InternalPakeError, group::Group, map_to_curve::GroupWithMapToCurve};
+use generic_array::{typenum::U32, GenericArray};
 use hkdf::Hkdf;
 use rand_core::{CryptoRng, RngCore};
-use sha2::{Digest, Sha256};
 
 pub(crate) struct OprfClientBytes<Grp: Group> {
     pub(crate) alpha: Grp,
@@ -18,15 +17,14 @@ pub(crate) struct OprfClientBytes<Grp: Group> {
 /// message is sent from the client (who holds the input) to the server (who holds the OPRF key).
 /// The client can also pass in an optional "pepper" string to be mixed in with the input through
 /// an HKDF computation.
-pub(crate) fn generate_oprf1<R: RngCore + CryptoRng, G: Group<UniformBytesLen = U64>>(
+pub(crate) fn generate_oprf1<R: RngCore + CryptoRng, G: GroupWithMapToCurve>(
     input: &[u8],
     pepper: Option<&[u8]>,
     blinding_factor_rng: &mut R,
 ) -> Result<OprfClientBytes<G>, InternalPakeError> {
-    let (hashed_input, _) = Hkdf::<Sha256>::extract(pepper, &input);
-    let curve_input: Vec<u8> = [hashed_input.as_slice(), &[0u8; 32]].concat();
+    let mapped_point = G::map_to_curve(input, pepper);
     let blinding_factor = G::random_scalar(blinding_factor_rng);
-    let alpha = G::hash_to_curve(GenericArray::from_slice(&curve_input)) * &blinding_factor;
+    let alpha = mapped_point * &blinding_factor;
     Ok(OprfClientBytes {
         alpha,
         blinding_factor,
@@ -48,10 +46,10 @@ pub(crate) fn generate_oprf3<G: Group>(
     input: &[u8],
     point: G,
     blinding_factor: &G::Scalar,
-) -> Result<GenericArray<u8, <Sha256 as Digest>::OutputSize>, InternalPakeError> {
+) -> Result<GenericArray<u8, U32>, InternalPakeError> {
     let unblinded = point * &G::scalar_invert(&blinding_factor);
     let ikm: Vec<u8> = [&unblinded.to_arr()[..], input].concat();
-    let (prk, _) = Hkdf::<Sha256>::extract(None, &ikm);
+    let (prk, _) = Hkdf::<sha2::Sha256>::extract(None, &ikm);
     Ok(prk)
 }
 
@@ -66,14 +64,14 @@ mod tests {
     use generic_array::{arr, GenericArray};
     use hkdf::Hkdf;
     use rand_core::OsRng;
+    use sha2::{Digest, Sha256, Sha512};
 
     fn prf(
         input: &[u8],
         oprf_key: &[u8; 32],
     ) -> GenericArray<u8, <RistrettoPoint as Group>::ElemLen> {
-        let (hashed_input, _) = Hkdf::<Sha256>::extract(None, &input);
-        let curve_input: Vec<u8> = [hashed_input.as_slice(), &[0u8; 32]].concat();
-        let point = RistrettoPoint::hash_to_curve(GenericArray::from_slice(&curve_input));
+        let (hashed_input, _) = Hkdf::<Sha512>::extract(None, &input);
+        let point = RistrettoPoint::hash_to_curve(GenericArray::from_slice(&hashed_input));
         let scalar =
             RistrettoPoint::from_scalar_slice(GenericArray::from_slice(&oprf_key[..])).unwrap();
         let res = point * scalar;
@@ -114,14 +112,12 @@ mod tests {
         } = generate_oprf1::<_, RistrettoPoint>(&input, None, &mut rng).unwrap();
         let res = generate_oprf3::<RistrettoPoint>(&input, alpha, &blinding_factor).unwrap();
 
-        let (hashed_input, _) = Hkdf::<Sha256>::extract(None, &input);
-        let mut curve_input: Vec<u8> = Vec::new();
-        curve_input.extend_from_slice(&hashed_input);
-        curve_input.extend_from_slice(&[0u8; 32]);
+        let (hashed_input, _) = Hkdf::<Sha512>::extract(None, &input);
+
         // This is because RistrettoPoint is on an obsolete sha2 version
         let mut bits = [0u8; 64];
         let mut hasher = sha2::Sha512::new();
-        hasher.update(&curve_input[..]);
+        Digest::update(&mut hasher, &hashed_input[..]);
         bits.copy_from_slice(&hasher.finalize());
 
         let point = RistrettoPoint::from_uniform_bytes(&bits);
diff --git a/src/slow_hash.rs b/src/slow_hash.rs
index 7a502a5..537d521 100644
--- a/src/slow_hash.rs
+++ b/src/slow_hash.rs
@@ -7,33 +7,26 @@
 
 use crate::errors::InternalPakeError;
 
-use generic_array::GenericArray;
-use sha2::{Digest, Sha256};
+use generic_array::{typenum::U32, GenericArray};
 
 /// Used for the slow hashing function in OPAQUE
 pub trait SlowHash {
     /// Computes the slow hashing function
-    fn hash(
-        input: GenericArray<u8, <Sha256 as Digest>::OutputSize>,
-    ) -> Result<Vec<u8>, InternalPakeError>;
+    fn hash(input: GenericArray<u8, U32>) -> Result<Vec<u8>, InternalPakeError>;
 }
 
 /// A no-op hash which simply returns its input
 pub struct NoOpHash;
 
 impl SlowHash for NoOpHash {
-    fn hash(
-        input: GenericArray<u8, <Sha256 as Digest>::OutputSize>,
-    ) -> Result<Vec<u8>, InternalPakeError> {
+    fn hash(input: GenericArray<u8, U32>) -> Result<Vec<u8>, InternalPakeError> {
         Ok(input.to_vec())
     }
 }
 
 #[cfg(feature = "slow-hash")]
 impl SlowHash for scrypt::ScryptParams {
-    fn hash(
-        input: GenericArray<u8, <Sha256 as Digest>::OutputSize>,
-    ) -> Result<Vec<u8>, InternalPakeError> {
+    fn hash(input: GenericArray<u8, U32>) -> Result<Vec<u8>, InternalPakeError> {
         let params = scrypt::ScryptParams::new(15, 8, 1).unwrap();
         let mut output = [0u8; 32];
         scrypt::scrypt(&input, &[], &params, &mut output)
diff --git a/src/tests/opaque_ke_test.rs b/src/tests/opaque_ke_test.rs
index 60c81e9..fc13a3f 100644
--- a/src/tests/opaque_ke_test.rs
+++ b/src/tests/opaque_ke_test.rs
@@ -247,13 +247,14 @@ fn generate_parameters<CS: CipherSuite>() -> TestVectorParameters {
     )
     .unwrap();
     let r1_bytes = r1.to_bytes().to_vec();
-    let blinding_factor_bytes = *CS::Group::scalar_as_bytes(&client_registration.blinding_factor);
+    let blinding_factor_bytes =
+        CS::Group::scalar_as_bytes(&client_registration.blinding_factor).clone();
     let client_registration_state = client_registration.to_bytes().to_vec();
 
     let mut oprf_key_rng = CycleRng::new(oprf_key_raw.to_vec());
     let (r2, server_registration) = ServerRegistration::<CS>::start(r1, &mut oprf_key_rng).unwrap();
     let r2_bytes = r2.to_bytes().to_vec();
-    let oprf_key_bytes = *CS::Group::scalar_as_bytes(&server_registration.oprf_key);
+    let oprf_key_bytes = CS::Group::scalar_as_bytes(&server_registration.oprf_key).clone();
     let server_registration_state = server_registration.to_bytes().to_vec();
 
     let mut client_s_sk_and_nonce: Vec<u8> = Vec::new();