diff --git a/src/messages.rs b/src/messages.rs
index fe97e93..090a3b3 100644
--- a/src/messages.rs
+++ b/src/messages.rs
@@ -13,7 +13,6 @@ use crate::{
         PakeError, ProtocolError,
     },
     group::Group,
-    hash::Hash,
     key_exchange::traits::{KeyExchange, ToBytes},
     keypair::{Key, KeyPair, SizedBytesExt},
     serialization::{serialize, tokenize},
@@ -21,32 +20,31 @@ use crate::{
 use generic_array::{typenum::Unsigned, GenericArray};
 use generic_bytes::SizedBytes;
 use std::convert::TryFrom;
-use std::marker::PhantomData;
 
 // Messages
 // =========
 
 /// The message sent by the client to the server, to initiate registration
-pub struct RegistrationRequest<Grp> {
+pub struct RegistrationRequest<CS: CipherSuite> {
     /// blinded password information
-    pub(crate) alpha: Grp,
+    pub(crate) alpha: CS::Group,
 }
 
-impl<Grp: Group> TryFrom<&[u8]> for RegistrationRequest<Grp> {
+impl<CS: CipherSuite> TryFrom<&[u8]> for RegistrationRequest<CS> {
     type Error = ProtocolError;
     fn try_from(first_message_bytes: &[u8]) -> Result<Self, Self::Error> {
-        let elem_len = Grp::ElemLen::to_usize();
+        let elem_len = <CS::Group as Group>::ElemLen::to_usize();
         let checked_slice = check_slice_size(first_message_bytes, elem_len, "first_message_bytes")?;
 
         // Check that the message is actually containing an element of the
         // correct subgroup
         let arr = GenericArray::from_slice(&checked_slice[checked_slice.len() - elem_len..]);
-        let alpha = Grp::from_element_slice(arr)?;
+        let alpha = CS::Group::from_element_slice(arr)?;
         Ok(Self { alpha })
     }
 }
 
-impl<Grp: Group> RegistrationRequest<Grp> {
+impl<CS: CipherSuite> RegistrationRequest<CS> {
     /// Byte representation for the registration request
     pub fn to_bytes(&self) -> Vec<u8> {
         self.alpha.to_arr().to_vec()
@@ -59,39 +57,36 @@ impl<Grp: Group> RegistrationRequest<Grp> {
 
     /// Deserialization from bytes
     pub fn deserialize(input: &[u8]) -> Result<Self, ProtocolError> {
-        let checked_slice =
-            check_slice_size(&input, Grp::ElemLen::to_usize(), "first_message_bytes")?;
+        let elem_len = <CS::Group as Group>::ElemLen::to_usize();
+        let checked_slice = check_slice_size(&input, elem_len, "first_message_bytes")?;
         // Check that the message is actually containing an element of the
         // correct subgroup
         let arr = GenericArray::from_slice(checked_slice);
-        let alpha = Grp::from_element_slice(arr)?;
+        let alpha = CS::Group::from_element_slice(arr)?;
         Ok(Self { alpha })
     }
 }
 
 /// The answer sent by the server to the user, upon reception of the
 /// registration attempt
-pub struct RegistrationResponse<Grp> {
+pub struct RegistrationResponse<CS: CipherSuite> {
     /// The server's oprf output
-    pub(crate) beta: Grp,
+    pub(crate) beta: CS::Group,
     /// Server's static public key
     pub(crate) server_s_pk: Vec<u8>,
 }
 
-impl<Grp> TryFrom<&[u8]> for RegistrationResponse<Grp>
-where
-    Grp: Group,
-{
+impl<CS: CipherSuite> TryFrom<&[u8]> for RegistrationResponse<CS> {
     type Error = ProtocolError;
 
     fn try_from(bytes: &[u8]) -> Result<Self, Self::Error> {
-        let elem_len = Grp::ElemLen::to_usize();
+        let elem_len = <CS::Group as Group>::ElemLen::to_usize();
         let checked_slice = check_slice_size_atleast(bytes, elem_len, "second_message_bytes")?;
 
         // Check that the message is actually containing an element of the
         // correct subgroup
         let arr = GenericArray::from_slice(&checked_slice[..elem_len]);
-        let beta = Grp::from_element_slice(arr)?;
+        let beta = CS::Group::from_element_slice(arr)?;
 
         // FIXME check public key bytes
         let server_s_pk = checked_slice[elem_len..].to_vec();
@@ -100,10 +95,7 @@ where
     }
 }
 
-impl<Grp> RegistrationResponse<Grp>
-where
-    Grp: Group,
-{
+impl<CS: CipherSuite> RegistrationResponse<CS> {
     /// Byte representation for the registration response message. This does not
     /// include the envelope credentials format
     pub fn to_bytes(&self) -> Vec<u8> {
@@ -120,15 +112,15 @@ where
 
     /// Deserialization from bytes
     pub fn deserialize(input: &[u8]) -> Result<Self, ProtocolError> {
-        let checked_slice =
-            check_slice_size_atleast(&input, Grp::ElemLen::to_usize(), "second_message_bytes")?;
+        let elem_len = <CS::Group as Group>::ElemLen::to_usize();
+        let checked_slice = check_slice_size_atleast(&input, elem_len, "second_message_bytes")?;
 
         // Check that the message is actually containing an element of the
         // correct subgroup
-        let arr = GenericArray::from_slice(&checked_slice[..Grp::ElemLen::to_usize()]);
-        let beta = Grp::from_element_slice(arr)?;
+        let arr = GenericArray::from_slice(&checked_slice[..elem_len]);
+        let beta = CS::Group::from_element_slice(arr)?;
 
-        let (server_s_pk, remainder) = tokenize(&checked_slice[Grp::ElemLen::to_usize()..], 2)?;
+        let (server_s_pk, remainder) = tokenize(&checked_slice[elem_len..], 2)?;
         if !remainder.is_empty() {
             return Err(PakeError::SerializationError.into());
         }
@@ -139,38 +131,36 @@ where
 
 /// The final message from the client, containing sealed cryptographic
 /// identifiers
-pub struct RegistrationUpload<D: Hash, G: Group> {
+pub struct RegistrationUpload<CS: CipherSuite> {
     /// The "envelope" generated by the user, containing sealed
     /// cryptographic identifiers
-    pub(crate) envelope: Envelope<D>,
+    pub(crate) envelope: Envelope<CS::Hash>,
     /// The user's public key
     pub(crate) client_s_pk: Key,
-    pub(crate) _g: PhantomData<G>,
 }
 
-impl<D: Hash, G: Group> TryFrom<&[u8]> for RegistrationUpload<D, G> {
+impl<CS: CipherSuite> TryFrom<&[u8]> for RegistrationUpload<CS> {
     type Error = ProtocolError;
 
     fn try_from(third_message_bytes: &[u8]) -> Result<Self, Self::Error> {
         let key_len = <Key as SizedBytes>::Len::to_usize();
-        let envelope_size = key_len + Envelope::<D>::additional_size();
+        let envelope_size = key_len + Envelope::<CS::Hash>::additional_size();
         let checked_bytes = check_slice_size(
             third_message_bytes,
             envelope_size + key_len,
             "third_message",
         )?;
         let unchecked_client_s_pk = Key::from_bytes(&checked_bytes[envelope_size..])?;
-        let client_s_pk = KeyPair::<G>::check_public_key(unchecked_client_s_pk)?;
+        let client_s_pk = KeyPair::<CS::Group>::check_public_key(unchecked_client_s_pk)?;
 
         Ok(Self {
-            envelope: Envelope::<D>::from_bytes(&checked_bytes[..envelope_size])?,
+            envelope: Envelope::<CS::Hash>::from_bytes(&checked_bytes[..envelope_size])?,
             client_s_pk,
-            _g: PhantomData,
         })
     }
 }
 
-impl<D: Hash, G: Group> RegistrationUpload<D, G> {
+impl<CS: CipherSuite> RegistrationUpload<CS> {
     /// Serialization into bytes
     pub fn serialize(&self) -> Vec<u8> {
         let mut message: Vec<u8> = Vec::new();
@@ -182,7 +172,7 @@ impl<D: Hash, G: Group> RegistrationUpload<D, G> {
     /// Deserialization from bytes
     pub fn deserialize(input: &[u8]) -> Result<Self, ProtocolError> {
         let (client_s_pk, remainder) = tokenize(&input, 2)?;
-        let (envelope, remainder) = Envelope::<D>::deserialize(&remainder)?;
+        let (envelope, remainder) = Envelope::<CS::Hash>::deserialize(&remainder)?;
 
         if !remainder.is_empty() {
             return Err(PakeError::SerializationError.into());
@@ -190,8 +180,7 @@ impl<D: Hash, G: Group> RegistrationUpload<D, G> {
 
         Ok(Self {
             envelope,
-            client_s_pk: KeyPair::<G>::check_public_key(Key::from_bytes(&client_s_pk)?)?,
-            _g: PhantomData,
+            client_s_pk: KeyPair::<CS::Group>::check_public_key(Key::from_bytes(&client_s_pk)?)?,
         })
     }
 }
diff --git a/src/opaque.rs b/src/opaque.rs
index 5b053e7..9f41bd2 100644
--- a/src/opaque.rs
+++ b/src/opaque.rs
@@ -96,7 +96,7 @@ impl Default for ClientRegistrationFinishParameters {
 /// Contains the fields that are returned by a client registration start
 pub struct ClientRegistrationStartResult<CS: CipherSuite> {
     /// The registration request message to be sent to the server
-    pub message: RegistrationRequest<CS::Group>,
+    pub message: RegistrationRequest<CS>,
     /// The client state that must be persisted in order to complete registration
     pub state: ClientRegistration<CS>,
 }
@@ -132,19 +132,18 @@ impl<CS: CipherSuite> ClientRegistration<CS> {
         let (token, alpha) = oprf::blind::<R, CS::Group, CS::Hash>(&password, blinding_factor_rng)?;
 
         Ok(ClientRegistrationStartResult {
-            message: RegistrationRequest::<CS::Group> { alpha },
+            message: RegistrationRequest::<CS> { alpha },
             state: Self { token },
         })
     }
 }
 
 /// Contains the fields that are returned by a client registration finish
-pub struct ClientRegistrationFinishResult<D: Hash, G: Group> {
+pub struct ClientRegistrationFinishResult<CS: CipherSuite> {
     /// The registration upload message to be sent to the server
-    pub message: RegistrationUpload<D, G>,
+    pub message: RegistrationUpload<CS>,
     /// The export key output by client registration
-    pub export_key: GenericArray<u8, <D as Digest>::OutputSize>,
-    _g: PhantomData<G>,
+    pub export_key: GenericArray<u8, <CS::Hash as Digest>::OutputSize>,
 }
 
 impl<CS: CipherSuite> ClientRegistration<CS> {
@@ -182,9 +181,9 @@ impl<CS: CipherSuite> ClientRegistration<CS> {
     pub fn finish<R: CryptoRng + RngCore>(
         self,
         rng: &mut R,
-        r2: RegistrationResponse<CS::Group>,
+        r2: RegistrationResponse<CS>,
         params: ClientRegistrationFinishParameters,
-    ) -> Result<ClientRegistrationFinishResult<CS::Hash, CS::Group>, ProtocolError> {
+    ) -> Result<ClientRegistrationFinishResult<CS>, ProtocolError> {
         let optional_ids = match params {
             ClientRegistrationFinishParameters::WithIdentifiers(id_u, id_s) => Some((id_u, id_s)),
             ClientRegistrationFinishParameters::Default => None,
@@ -206,10 +205,8 @@ impl<CS: CipherSuite> ClientRegistration<CS> {
             message: RegistrationUpload {
                 envelope,
                 client_s_pk: client_static_keypair.public().clone(),
-                _g: PhantomData,
             },
             export_key,
-            _g: PhantomData,
         })
     }
 }
@@ -245,7 +242,7 @@ impl<CS: CipherSuite> Drop for ClientLogin<CS> {
 /// Contains the fields that are returned by a server registration start
 pub struct ServerRegistrationStartResult<CS: CipherSuite> {
     /// The registration resposne message to send to the client
-    pub message: RegistrationResponse<CS::Group>,
+    pub message: RegistrationResponse<CS>,
     /// The state that the server must keep in order to complete registration
     pub state: ServerRegistration<CS>,
 }
@@ -336,7 +333,7 @@ impl<CS: CipherSuite> ServerRegistration<CS> {
     /// ```
     pub fn start<R: RngCore + CryptoRng>(
         rng: &mut R,
-        message: RegistrationRequest<CS::Group>,
+        message: RegistrationRequest<CS>,
         server_s_pk: &Key,
     ) -> Result<ServerRegistrationStartResult<CS>, ProtocolError> {
         // RFC: generate oprf_key (salt) and v_u = g^oprf_key
@@ -388,10 +385,7 @@ impl<CS: CipherSuite> ServerRegistration<CS> {
     /// let client_record = server_registration_start_result.state.finish(client_registration_finish_result.message)?;
     /// # Ok::<(), ProtocolError>(())
     /// ```
-    pub fn finish(
-        self,
-        message: RegistrationUpload<CS::Hash, CS::Group>,
-    ) -> Result<Self, ProtocolError> {
+    pub fn finish(self, message: RegistrationUpload<CS>) -> Result<Self, ProtocolError> {
         Ok(Self {
             envelope: Some(message.envelope),
             client_s_pk: Some(message.client_s_pk),
diff --git a/src/serialization/tests.rs b/src/serialization/tests.rs
index ba2972e..22bb3c9 100644
--- a/src/serialization/tests.rs
+++ b/src/serialization/tests.rs
@@ -103,7 +103,7 @@ fn register_first_message_roundtrip() {
     let mut input = Vec::new();
     input.extend_from_slice(pt_bytes.as_slice());
 
-    let r1 = RegistrationRequest::<RistrettoPoint>::deserialize(input.as_slice()).unwrap();
+    let r1 = RegistrationRequest::<Default>::deserialize(input.as_slice()).unwrap();
     let r1_bytes = r1.serialize();
     assert_eq!(input, r1_bytes);
 }
@@ -123,7 +123,7 @@ fn register_second_message_roundtrip() {
     input.extend_from_slice(&pubkey_length.to_be_bytes()[std::mem::size_of::<usize>() - 2..]);
     input.extend_from_slice(&pubkey_bytes.as_slice());
 
-    let r2 = RegistrationResponse::<RistrettoPoint>::deserialize(input.as_slice()).unwrap();
+    let r2 = RegistrationResponse::<Default>::deserialize(input.as_slice()).unwrap();
     let r2_bytes = r2.serialize();
     assert_eq!(input, r2_bytes);
 }
@@ -157,7 +157,7 @@ fn register_third_message_roundtrip() {
     input.extend_from_slice(&pubkey_bytes[..]);
     input.extend_from_slice(&envelope_bytes);
 
-    let r3 = RegistrationUpload::<sha2::Sha512, RistrettoPoint>::deserialize(&input[..]).unwrap();
+    let r3 = RegistrationUpload::<Default>::deserialize(&input[..]).unwrap();
     let r3_bytes = r3.serialize();
     assert_eq!(input, r3_bytes);
 }
@@ -359,17 +359,17 @@ fn test_i2osp_os2ip(bytes in vec(any::<u8>(), 0..std::mem::size_of::<usize>()))
 
 #[test]
 fn test_nocrash_register_first_message(bytes in vec(any::<u8>(), 0..200)) {
-    RegistrationRequest::<RistrettoPoint>::try_from(&bytes[..]).map_or(true, |_| true);
+    RegistrationRequest::<Default>::try_from(&bytes[..]).map_or(true, |_| true);
 }
 
 #[test]
 fn test_nocrash_register_second_message(bytes in vec(any::<u8>(), 0..200)) {
-    RegistrationResponse::<RistrettoPoint>::try_from(&bytes[..]).map_or(true, |_| true);
+    RegistrationResponse::<Default>::try_from(&bytes[..]).map_or(true, |_| true);
 }
 
 #[test]
 fn test_nocrash_register_third_message(bytes in vec(any::<u8>(), 0..200)) {
-    RegistrationUpload::<sha2::Sha512, RistrettoPoint>::try_from(&bytes[..]).map_or(true, |_| true);
+    RegistrationUpload::<Default>::try_from(&bytes[..]).map_or(true, |_| true);
 }
 
 #[test]