diff --git a/Cargo.toml b/Cargo.toml
index ba7ae624..89d112f6 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -56,3 +56,6 @@ lto = "thin"
 incremental = true
 debug-assertions = true
 debug = true
+
+[patch.crates-io]
+ark-r1cs-std = { git = "https://github.com/arkworks-rs/r1cs-std", branch = "reduce-generics", optional = true, default-features = false }
\ No newline at end of file
diff --git a/bls12_377/Cargo.toml b/bls12_377/Cargo.toml
index 63ac9fab..19086a47 100644
--- a/bls12_377/Cargo.toml
+++ b/bls12_377/Cargo.toml
@@ -31,4 +31,4 @@ std = [ "ark-std/std", "ark-ff/std", "ark-ec/std" ]
 curve = [ "scalar_field", "base_field" ]
 scalar_field = []
 base_field = []
-r1cs = [ "base_field", "ark-r1cs-std" ]
\ No newline at end of file
+r1cs = [ "base_field", "ark-r1cs-std" ]
diff --git a/bls12_377/src/constraints/curves.rs b/bls12_377/src/constraints/curves.rs
index 5441eba0..3b4cedb7 100644
--- a/bls12_377/src/constraints/curves.rs
+++ b/bls12_377/src/constraints/curves.rs
@@ -16,14 +16,8 @@ pub type G2PreparedVar = bls12::G2PreparedVar<Parameters>;
 #[test]
 fn test() {
     use ark_ec::models::bls12::Bls12Parameters;
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as Bls12Parameters>::G1Parameters,
-        G1Var,
-    >()
-    .unwrap();
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as Bls12Parameters>::G2Parameters,
-        G2Var,
-    >()
-    .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as Bls12Parameters>::G1Parameters>()
+        .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as Bls12Parameters>::G2Parameters>()
+        .unwrap();
 }
diff --git a/bls12_377/src/constraints/mod.rs b/bls12_377/src/constraints/mod.rs
index a67838f5..24b9a587 100644
--- a/bls12_377/src/constraints/mod.rs
+++ b/bls12_377/src/constraints/mod.rs
@@ -130,17 +130,17 @@
 //! let pairing_result_native = Bls12_377::pairing(a_native, b_native);
 //!
 //! // Prepare `a` and `b` for pairing.
-//! let a_prep = constraints::PairingVar::prepare_g1(&a)?;
-//! let b_prep = constraints::PairingVar::prepare_g2(&b)?;
-//! let pairing_result = constraints::PairingVar::pairing(a_prep, b_prep)?;
+//! let a_prep = Bls12_377::prepare_g1(&a)?;
+//! let b_prep = Bls12_377::prepare_g2(&b)?;
+//! let pairing_result = Bls12_377::pairing_gadget(a_prep, b_prep)?;
 //!
 //! // Check that the value of &a + &b is correct.
 //! assert_eq!(pairing_result.value()?, pairing_result_native);
 //!
 //! // Check that operations on variables and constants are equivalent.
-//! let a_prep_const = constraints::PairingVar::prepare_g1(&a_const)?;
-//! let b_prep_const = constraints::PairingVar::prepare_g2(&b_const)?;
-//! let pairing_result_const = constraints::PairingVar::pairing(a_prep_const, b_prep_const)?;
+//! let a_prep_const = Bls12_377::prepare_g1(&a_const)?;
+//! let b_prep_const = Bls12_377::prepare_g2(&b_const)?;
+//! let pairing_result_const = Bls12_377::pairing_gadget(a_prep_const, b_prep_const)?;
 //! println!("Done here 3");
 //!
 //! pairing_result.enforce_equal(&pairing_result_const)?;
diff --git a/bls12_377/src/constraints/pairing.rs b/bls12_377/src/constraints/pairing.rs
index f659f294..cd664aae 100644
--- a/bls12_377/src/constraints/pairing.rs
+++ b/bls12_377/src/constraints/pairing.rs
@@ -1,10 +1,8 @@
-use crate::Parameters;
-
 /// Specifies the constraints for computing a pairing in the BLS12-377 bilinear group.
-pub type PairingVar = ark_r1cs_std::pairing::bls12::PairingVar<Parameters>;
+pub use crate::Bls12_377;
 
 #[test]
 fn test() {
     use crate::Bls12_377;
-    ark_curve_constraint_tests::pairing::bilinearity_test::<Bls12_377, PairingVar>().unwrap()
+    ark_curve_constraint_tests::pairing::bilinearity_test::<Bls12_377>().unwrap()
 }
diff --git a/curve-constraint-tests/src/lib.rs b/curve-constraint-tests/src/lib.rs
index ed2c59d3..797425ba 100644
--- a/curve-constraint-tests/src/lib.rs
+++ b/curve-constraint-tests/src/lib.rs
@@ -233,7 +233,8 @@ pub mod fields {
 pub mod curves {
     use ark_ec::{
         short_weierstrass_jacobian::GroupProjective as SWProjective,
-        twisted_edwards_extended::GroupProjective as TEProjective, ProjectiveCurve,
+        twisted_edwards_extended::GroupProjective as TEProjective, ModelParameters,
+        ProjectiveCurve,
     };
     use ark_ff::{BitIteratorLE, Field, FpParameters, One, PrimeField};
     use ark_relations::r1cs::{ConstraintSystem, SynthesisError};
@@ -241,12 +242,13 @@ pub mod curves {
 
     use ark_r1cs_std::prelude::*;
 
-    pub fn group_test<C, ConstraintF, GG>() -> Result<(), SynthesisError>
+    type ConstraintF<P> = <<P as ModelParameters>::BaseField as Field>::BasePrimeField;
+
+    pub fn group_test<C, ConstraintF>() -> Result<(), SynthesisError>
     where
-        C: ProjectiveCurve,
+        C: CurveWithVar<ConstraintF>,
         ConstraintF: Field,
-        GG: CurveVar<C, ConstraintF>,
-        for<'a> &'a GG: GroupOpsBounds<'a, C, GG>,
+        for<'a> &'a C::Var: GroupOpsBounds<'a, C, C::Var>,
     {
         let modes = [
             AllocationMode::Input,
@@ -259,12 +261,14 @@ pub mod curves {
             let mut rng = test_rng();
             let a_native = C::rand(&mut rng);
             let b_native = C::rand(&mut rng);
-            let a = GG::new_variable(ark_relations::ns!(cs, "generate_a"), || Ok(a_native), mode)
-                .unwrap();
-            let b = GG::new_variable(ark_relations::ns!(cs, "generate_b"), || Ok(b_native), mode)
-                .unwrap();
-
-            let zero = GG::zero();
+            let a =
+                C::Var::new_variable(ark_relations::ns!(cs, "generate_a"), || Ok(a_native), mode)
+                    .unwrap();
+            let b =
+                C::Var::new_variable(ark_relations::ns!(cs, "generate_b"), || Ok(b_native), mode)
+                    .unwrap();
+
+            let zero = C::Var::zero();
             assert_eq!(zero.value()?, zero.value()?);
 
             // a == a
@@ -273,14 +277,14 @@ pub mod curves {
             assert_eq!((&a + &zero).value()?, a.value()?);
             // a - 0 = a
             assert_eq!((&a - &zero).value()?, a.value()?);
-            // a - a = 0
-            assert_eq!((&a - &a).value()?, zero.value()?);
             // a + b = b + a
             let a_b = &a + &b;
             let b_a = &b + &a;
             assert_eq!(a_b.value()?, b_a.value()?);
             a_b.enforce_equal(&b_a)?;
             assert!(cs.is_satisfied().unwrap());
+            // a - a = 0
+            assert_eq!((&a - &a).value()?, zero.value()?);
 
             // (a + b) + a = a + (b + a)
             let ab_a = &a_b + &a;
@@ -380,13 +384,15 @@ pub mod curves {
         Ok(())
     }
 
-    pub fn sw_test<P, GG>() -> Result<(), SynthesisError>
+    type SWVar<P> = <SWProjective<P> as CurveWithVar<ConstraintF<P>>>::Var;
+
+    pub fn sw_test<P>() -> Result<(), SynthesisError>
     where
         P: ark_ec::SWModelParameters,
-        GG: CurveVar<SWProjective<P>, <P::BaseField as Field>::BasePrimeField>,
-        for<'a> &'a GG: GroupOpsBounds<'a, SWProjective<P>, GG>,
+        SWProjective<P>: CurveWithVar<ConstraintF<P>> + ProjectiveCurve,
+        for<'a> &'a SWVar<P>: GroupOpsBounds<'a, SWProjective<P>, SWVar<P>>,
     {
-        group_test::<SWProjective<P>, _, GG>()?;
+        group_test::<SWProjective<P>, _>()?;
         let modes = [
             AllocationMode::Input,
             AllocationMode::Witness,
@@ -405,14 +411,12 @@ pub mod curves {
             let b_affine = b.into_affine();
 
             let ns = ark_relations::ns!(cs, "allocating variables");
-            let mut gadget_a = GG::new_variable(cs.clone(), || Ok(a), mode)?;
-            let gadget_b = GG::new_variable(cs.clone(), || Ok(b), mode)?;
-            let zero = GG::zero();
+            let mut gadget_a = SWVar::<P>::new_variable(cs.clone(), || Ok(a), mode)?;
+            let gadget_b = SWVar::<P>::new_variable(cs.clone(), || Ok(b), mode)?;
+            let zero = SWVar::<P>::zero();
             drop(ns);
-            assert_eq!(gadget_a.value()?.into_affine().x, a_affine.x);
-            assert_eq!(gadget_a.value()?.into_affine().y, a_affine.y);
-            assert_eq!(gadget_b.value()?.into_affine().x, b_affine.x);
-            assert_eq!(gadget_b.value()?.into_affine().y, b_affine.y);
+            assert_eq!(gadget_a.value()?.into_affine(), a_affine);
+            assert_eq!(gadget_b.value()?.into_affine(), b_affine);
             assert_eq!(cs.which_is_unsatisfied().unwrap(), None);
 
             // Check addition
@@ -453,13 +457,15 @@ pub mod curves {
         Ok(())
     }
 
-    pub fn te_test<P, GG>() -> Result<(), SynthesisError>
+    type TEVar<P> = <TEProjective<P> as CurveWithVar<ConstraintF<P>>>::Var;
+
+    pub fn te_test<P>() -> Result<(), SynthesisError>
     where
         P: ark_ec::TEModelParameters,
-        GG: CurveVar<TEProjective<P>, <P::BaseField as Field>::BasePrimeField>,
-        for<'a> &'a GG: GroupOpsBounds<'a, TEProjective<P>, GG>,
+        TEProjective<P>: CurveWithVar<ConstraintF<P>> + ProjectiveCurve,
+        for<'a> &'a TEVar<P>: GroupOpsBounds<'a, TEProjective<P>, TEVar<P>>,
     {
-        group_test::<TEProjective<P>, _, GG>()?;
+        group_test::<TEProjective<P>, _>()?;
         let modes = [
             AllocationMode::Input,
             AllocationMode::Witness,
@@ -478,14 +484,12 @@ pub mod curves {
             let b_affine = b.into_affine();
 
             let ns = ark_relations::ns!(cs, "allocating variables");
-            let mut gadget_a = GG::new_variable(cs.clone(), || Ok(a), mode)?;
-            let gadget_b = GG::new_variable(cs.clone(), || Ok(b), mode)?;
+            let mut gadget_a = TEVar::<P>::new_variable(cs.clone(), || Ok(a), mode)?;
+            let gadget_b = TEVar::<P>::new_variable(cs.clone(), || Ok(b), mode)?;
             drop(ns);
 
-            assert_eq!(gadget_a.value()?.into_affine().x, a_affine.x);
-            assert_eq!(gadget_a.value()?.into_affine().y, a_affine.y);
-            assert_eq!(gadget_b.value()?.into_affine().x, b_affine.x);
-            assert_eq!(gadget_b.value()?.into_affine().y, b_affine.y);
+            assert_eq!(gadget_a.value()?.into_affine(), a_affine);
+            assert_eq!(gadget_b.value()?.into_affine(), b_affine);
             assert_eq!(cs.which_is_unsatisfied()?, None);
 
             // Check addition
@@ -527,16 +531,21 @@ pub mod curves {
 pub mod pairing {
     use ark_ec::{PairingEngine, ProjectiveCurve};
     use ark_ff::{BitIteratorLE, Field, PrimeField};
-    use ark_r1cs_std::prelude::*;
+    use ark_r1cs_std::{fields::fp::FpVar, prelude::*};
     use ark_relations::r1cs::{ConstraintSystem, SynthesisError};
     use ark_std::{test_rng, vec::Vec, UniformRand};
 
     #[allow(dead_code)]
-    pub fn bilinearity_test<E: PairingEngine, P: PairingVar<E>>() -> Result<(), SynthesisError>
+    pub fn bilinearity_test<P: PairingGadget>() -> Result<(), SynthesisError>
     where
-        for<'a> &'a P::G1Var: GroupOpsBounds<'a, E::G1Projective, P::G1Var>,
-        for<'a> &'a P::G2Var: GroupOpsBounds<'a, E::G2Projective, P::G2Var>,
-        for<'a> &'a P::GTVar: FieldOpsBounds<'a, E::Fqk, P::GTVar>,
+        for<'a> &'a P::G1Var: GroupOpsBounds<'a, P::G1Projective, P::G1Var>,
+        for<'a> &'a P::G2Var: GroupOpsBounds<'a, P::G2Projective, P::G2Var>,
+        for<'a> &'a P::GTVar: FieldOpsBounds<'a, P::Fqk, P::GTVar>,
+        P::Fq: FieldWithVar<Var = FpVar<P::Fq>>,
+        P::Fqe: FieldWithVar,
+        P::Fqk: FieldWithVar<Var = P::GTVar>,
+        P::G1Projective: CurveWithVar<P::Fq, Var = P::G1Var>,
+        P::G2Projective: CurveWithVar<P::Fq, Var = P::G2Var>,
     {
         let modes = [
             AllocationMode::Input,
@@ -544,12 +553,12 @@ pub mod pairing {
             AllocationMode::Constant,
         ];
         for &mode in &modes {
-            let cs = ConstraintSystem::<E::Fq>::new_ref();
+            let cs = ConstraintSystem::<P::Fq>::new_ref();
 
             let mut rng = test_rng();
-            let a = E::G1Projective::rand(&mut rng);
-            let b = E::G2Projective::rand(&mut rng);
-            let s = E::Fr::rand(&mut rng);
+            let a = P::G1Projective::rand(&mut rng);
+            let b = P::G2Projective::rand(&mut rng);
+            let s = P::Fr::rand(&mut rng);
 
             let mut sa = a;
             sa *= s;
@@ -571,16 +580,16 @@ pub mod pairing {
 
             let (ans1_g, ans1_n) = {
                 let _ml_constraints = cs.num_constraints();
-                let ml_g = P::miller_loop(&[sa_prep_g], &[b_prep_g.clone()])?;
+                let ml_g = P::miller_loop_gadget(&[sa_prep_g], &[b_prep_g.clone()])?;
                 let _fe_constraints = cs.num_constraints();
-                let ans_g = P::final_exponentiation(&ml_g)?;
-                let ans_n = E::pairing(sa, b);
+                let ans_g = P::final_exponentiation_gadget(&ml_g)?;
+                let ans_n = <P as PairingEngine>::pairing(sa, b);
                 (ans_g, ans_n)
             };
 
             let (ans2_g, ans2_n) = {
-                let ans_g = P::pairing(a_prep_g.clone(), sb_prep_g)?;
-                let ans_n = E::pairing(a, sb);
+                let ans_g = P::pairing_gadget(a_prep_g.clone(), sb_prep_g)?;
+                let ans_n = <P as PairingEngine>::pairing(a, sb);
                 (ans_g, ans_n)
             };
 
@@ -589,8 +598,8 @@ pub mod pairing {
                     .map(Boolean::constant)
                     .collect::<Vec<_>>();
 
-                let mut ans_g = P::pairing(a_prep_g, b_prep_g)?;
-                let mut ans_n = E::pairing(a, b);
+                let mut ans_g = P::pairing_gadget(a_prep_g, b_prep_g)?;
+                let mut ans_n = <P as PairingEngine>::pairing(a, b);
                 ans_n = ans_n.pow(s.into_repr());
                 ans_g = ans_g.pow_le(&s_iter)?;
 
diff --git a/ed_on_bls12_377/src/constraints/curves.rs b/ed_on_bls12_377/src/constraints/curves.rs
index 4b15106d..a6586102 100644
--- a/ed_on_bls12_377/src/constraints/curves.rs
+++ b/ed_on_bls12_377/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
 
-use crate::constraints::FqVar;
-
 /// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
-pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+pub type EdwardsVar = AffineVar<EdwardsParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters, EdwardsVar>().unwrap();
+    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters>().unwrap();
 }
diff --git a/ed_on_bls12_381/src/constraints/curves.rs b/ed_on_bls12_381/src/constraints/curves.rs
index 9c9f7837..a6586102 100644
--- a/ed_on_bls12_381/src/constraints/curves.rs
+++ b/ed_on_bls12_381/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
 
-use crate::constraints::FqVar;
-
 /// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
-pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+pub type EdwardsVar = AffineVar<EdwardsParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::te_test::<_, EdwardsVar>().unwrap();
+    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters>().unwrap();
 }
diff --git a/ed_on_bls12_381_bandersnatch/src/constraints/curves.rs b/ed_on_bls12_381_bandersnatch/src/constraints/curves.rs
index 9c9f7837..a6586102 100644
--- a/ed_on_bls12_381_bandersnatch/src/constraints/curves.rs
+++ b/ed_on_bls12_381_bandersnatch/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
 
-use crate::constraints::FqVar;
-
 /// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
-pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+pub type EdwardsVar = AffineVar<EdwardsParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::te_test::<_, EdwardsVar>().unwrap();
+    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters>().unwrap();
 }
diff --git a/ed_on_bn254/src/constraints/curves.rs b/ed_on_bn254/src/constraints/curves.rs
index 9c9f7837..a6586102 100644
--- a/ed_on_bn254/src/constraints/curves.rs
+++ b/ed_on_bn254/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
 
-use crate::constraints::FqVar;
-
 /// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
-pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+pub type EdwardsVar = AffineVar<EdwardsParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::te_test::<_, EdwardsVar>().unwrap();
+    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters>().unwrap();
 }
diff --git a/ed_on_cp6_782/src/constraints/curves.rs b/ed_on_cp6_782/src/constraints/curves.rs
index 4b15106d..a6586102 100644
--- a/ed_on_cp6_782/src/constraints/curves.rs
+++ b/ed_on_cp6_782/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
 
-use crate::constraints::FqVar;
-
 /// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
-pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+pub type EdwardsVar = AffineVar<EdwardsParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters, EdwardsVar>().unwrap();
+    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters>().unwrap();
 }
diff --git a/ed_on_mnt4_298/src/constraints/curves.rs b/ed_on_mnt4_298/src/constraints/curves.rs
index 3ceded42..a6586102 100644
--- a/ed_on_mnt4_298/src/constraints/curves.rs
+++ b/ed_on_mnt4_298/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
 
-use crate::constraints::fields::FqVar;
-
 /// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
-pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+pub type EdwardsVar = AffineVar<EdwardsParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters, EdwardsVar>().unwrap();
+    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters>().unwrap();
 }
diff --git a/ed_on_mnt4_753/src/constraints/curves.rs b/ed_on_mnt4_753/src/constraints/curves.rs
index 3ceded42..a6586102 100644
--- a/ed_on_mnt4_753/src/constraints/curves.rs
+++ b/ed_on_mnt4_753/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
 
-use crate::constraints::fields::FqVar;
-
 /// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
-pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+pub type EdwardsVar = AffineVar<EdwardsParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters, EdwardsVar>().unwrap();
+    ark_curve_constraint_tests::curves::te_test::<EdwardsParameters>().unwrap();
 }
diff --git a/mnt4_298/src/constraints/curves.rs b/mnt4_298/src/constraints/curves.rs
index 46401164..6a49f0dd 100644
--- a/mnt4_298/src/constraints/curves.rs
+++ b/mnt4_298/src/constraints/curves.rs
@@ -16,14 +16,8 @@ pub type G2PreparedVar = mnt4::G2PreparedVar<Parameters>;
 #[test]
 fn test() {
     use ark_ec::models::mnt4::MNT4Parameters;
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT4Parameters>::G1Parameters,
-        G1Var,
-    >()
-    .unwrap();
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT4Parameters>::G2Parameters,
-        G2Var,
-    >()
-    .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT4Parameters>::G1Parameters>()
+        .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT4Parameters>::G2Parameters>()
+        .unwrap();
 }
diff --git a/mnt4_298/src/constraints/mod.rs b/mnt4_298/src/constraints/mod.rs
index 48919084..2feb7ac2 100644
--- a/mnt4_298/src/constraints/mod.rs
+++ b/mnt4_298/src/constraints/mod.rs
@@ -130,17 +130,17 @@
 //! let pairing_result_native = MNT4_298::pairing(a_native, b_native);
 //!
 //! // Prepare `a` and `b` for pairing.
-//! let a_prep = constraints::PairingVar::prepare_g1(&a)?;
-//! let b_prep = constraints::PairingVar::prepare_g2(&b)?;
-//! let pairing_result = constraints::PairingVar::pairing(a_prep, b_prep)?;
+//! let a_prep = MNT4_298::prepare_g1(&a)?;
+//! let b_prep = MNT4_298::prepare_g2(&b)?;
+//! let pairing_result = MNT4_298::pairing_gadget(a_prep, b_prep)?;
 //!
 //! // Check that the value of &a + &b is correct.
 //! assert_eq!(pairing_result.value()?, pairing_result_native);
 //!
 //! // Check that operations on variables and constants are equivalent.
-//! let a_prep_const = constraints::PairingVar::prepare_g1(&a_const)?;
-//! let b_prep_const = constraints::PairingVar::prepare_g2(&b_const)?;
-//! let pairing_result_const = constraints::PairingVar::pairing(a_prep_const, b_prep_const)?;
+//! let a_prep_const = MNT4_298::prepare_g1(&a_const)?;
+//! let b_prep_const = MNT4_298::prepare_g2(&b_const)?;
+//! let pairing_result_const = MNT4_298::pairing_gadget(a_prep_const, b_prep_const)?;
 //! println!("Done here 3");
 //!
 //! pairing_result.enforce_equal(&pairing_result_const)?;
diff --git a/mnt4_298/src/constraints/pairing.rs b/mnt4_298/src/constraints/pairing.rs
index 7941ad05..5b5c195a 100644
--- a/mnt4_298/src/constraints/pairing.rs
+++ b/mnt4_298/src/constraints/pairing.rs
@@ -1,10 +1,8 @@
-use crate::Parameters;
-
 /// Specifies the constraints for computing a pairing in the MNT4-298 bilinear group.
-pub type PairingVar = ark_r1cs_std::pairing::mnt4::PairingVar<Parameters>;
+pub use crate::MNT4_298;
 
 #[test]
 fn test() {
     use crate::MNT4_298;
-    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT4_298, PairingVar>().unwrap()
+    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT4_298>().unwrap()
 }
diff --git a/mnt4_753/src/constraints/curves.rs b/mnt4_753/src/constraints/curves.rs
index febfeeda..1ceba87e 100644
--- a/mnt4_753/src/constraints/curves.rs
+++ b/mnt4_753/src/constraints/curves.rs
@@ -16,14 +16,8 @@ pub type G2PreparedVar = mnt4::G2PreparedVar<Parameters>;
 #[test]
 fn test() {
     use ark_ec::models::mnt4::MNT4Parameters;
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT4Parameters>::G1Parameters,
-        G1Var,
-    >()
-    .unwrap();
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT4Parameters>::G2Parameters,
-        G2Var,
-    >()
-    .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT4Parameters>::G1Parameters>()
+        .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT4Parameters>::G2Parameters>()
+        .unwrap();
 }
diff --git a/mnt4_753/src/constraints/mod.rs b/mnt4_753/src/constraints/mod.rs
index 4744e4c9..25a83235 100644
--- a/mnt4_753/src/constraints/mod.rs
+++ b/mnt4_753/src/constraints/mod.rs
@@ -130,17 +130,17 @@
 //! let pairing_result_native = MNT4_753::pairing(a_native, b_native);
 //!
 //! // Prepare `a` and `b` for pairing.
-//! let a_prep = constraints::PairingVar::prepare_g1(&a)?;
-//! let b_prep = constraints::PairingVar::prepare_g2(&b)?;
-//! let pairing_result = constraints::PairingVar::pairing(a_prep, b_prep)?;
+//! let a_prep = MNT4_753::prepare_g1(&a)?;
+//! let b_prep = MNT4_753::prepare_g2(&b)?;
+//! let pairing_result = MNT4_753::pairing_gadget(a_prep, b_prep)?;
 //!
 //! // Check that the value of &a + &b is correct.
 //! assert_eq!(pairing_result.value()?, pairing_result_native);
 //!
 //! // Check that operations on variables and constants are equivalent.
-//! let a_prep_const = constraints::PairingVar::prepare_g1(&a_const)?;
-//! let b_prep_const = constraints::PairingVar::prepare_g2(&b_const)?;
-//! let pairing_result_const = constraints::PairingVar::pairing(a_prep_const, b_prep_const)?;
+//! let a_prep_const = MNT4_753::prepare_g1(&a_const)?;
+//! let b_prep_const = MNT4_753::prepare_g2(&b_const)?;
+//! let pairing_result_const = MNT4_753::pairing_gadget(a_prep_const, b_prep_const)?;
 //! println!("Done here 3");
 //!
 //! pairing_result.enforce_equal(&pairing_result_const)?;
diff --git a/mnt4_753/src/constraints/pairing.rs b/mnt4_753/src/constraints/pairing.rs
index 3e00df45..24af61c7 100644
--- a/mnt4_753/src/constraints/pairing.rs
+++ b/mnt4_753/src/constraints/pairing.rs
@@ -1,10 +1,8 @@
-use crate::Parameters;
-
 /// Specifies the constraints for computing a pairing in the MNT4-753 bilinear group.
-pub type PairingVar = ark_r1cs_std::pairing::mnt4::PairingVar<Parameters>;
+pub use crate::MNT4_753;
 
 #[test]
 fn test() {
     use crate::MNT4_753;
-    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT4_753, PairingVar>().unwrap()
+    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT4_753>().unwrap()
 }
diff --git a/mnt6_298/src/constraints/curves.rs b/mnt6_298/src/constraints/curves.rs
index 069ee9e2..9c35417e 100644
--- a/mnt6_298/src/constraints/curves.rs
+++ b/mnt6_298/src/constraints/curves.rs
@@ -16,14 +16,8 @@ pub type G2PreparedVar = mnt6::G2PreparedVar<Parameters>;
 #[test]
 fn test() {
     use ark_ec::models::mnt6::MNT6Parameters;
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT6Parameters>::G1Parameters,
-        G1Var,
-    >()
-    .unwrap();
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT6Parameters>::G2Parameters,
-        G2Var,
-    >()
-    .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT6Parameters>::G1Parameters>()
+        .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT6Parameters>::G2Parameters>()
+        .unwrap();
 }
diff --git a/mnt6_298/src/constraints/mod.rs b/mnt6_298/src/constraints/mod.rs
index 00925676..6221424b 100644
--- a/mnt6_298/src/constraints/mod.rs
+++ b/mnt6_298/src/constraints/mod.rs
@@ -130,17 +130,17 @@
 //! let pairing_result_native = MNT6_298::pairing(a_native, b_native);
 //!
 //! // Prepare `a` and `b` for pairing.
-//! let a_prep = constraints::PairingVar::prepare_g1(&a)?;
-//! let b_prep = constraints::PairingVar::prepare_g2(&b)?;
-//! let pairing_result = constraints::PairingVar::pairing(a_prep, b_prep)?;
+//! let a_prep = MNT6_298::prepare_g1(&a)?;
+//! let b_prep = MNT6_298::prepare_g2(&b)?;
+//! let pairing_result = MNT6_298::pairing_gadget(a_prep, b_prep)?;
 //!
 //! // Check that the value of &a + &b is correct.
 //! assert_eq!(pairing_result.value()?, pairing_result_native);
 //!
 //! // Check that operations on variables and constants are equivalent.
-//! let a_prep_const = constraints::PairingVar::prepare_g1(&a_const)?;
-//! let b_prep_const = constraints::PairingVar::prepare_g2(&b_const)?;
-//! let pairing_result_const = constraints::PairingVar::pairing(a_prep_const, b_prep_const)?;
+//! let a_prep_const = MNT6_298::prepare_g1(&a_const)?;
+//! let b_prep_const = MNT6_298::prepare_g2(&b_const)?;
+//! let pairing_result_const = MNT6_298::pairing_gadget(a_prep_const, b_prep_const)?;
 //! println!("Done here 3");
 //!
 //! pairing_result.enforce_equal(&pairing_result_const)?;
diff --git a/mnt6_298/src/constraints/pairing.rs b/mnt6_298/src/constraints/pairing.rs
index 14641bea..1aaa4d3a 100644
--- a/mnt6_298/src/constraints/pairing.rs
+++ b/mnt6_298/src/constraints/pairing.rs
@@ -1,10 +1,8 @@
-use crate::Parameters;
-
 /// Specifies the constraints for computing a pairing in the MNT6-298 bilinear group.
-pub type PairingVar = ark_r1cs_std::pairing::mnt6::PairingVar<Parameters>;
+pub use crate::MNT6_298;
 
 #[test]
 fn test() {
     use crate::MNT6_298;
-    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT6_298, PairingVar>().unwrap()
+    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT6_298>().unwrap()
 }
diff --git a/mnt6_753/src/constraints/curves.rs b/mnt6_753/src/constraints/curves.rs
index e747fd23..c7126d6e 100644
--- a/mnt6_753/src/constraints/curves.rs
+++ b/mnt6_753/src/constraints/curves.rs
@@ -16,14 +16,8 @@ pub type G2PreparedVar = mnt6::G2PreparedVar<Parameters>;
 #[test]
 fn test() {
     use ark_ec::models::mnt6::MNT6Parameters;
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT6Parameters>::G1Parameters,
-        G1Var,
-    >()
-    .unwrap();
-    ark_curve_constraint_tests::curves::sw_test::<
-        <Parameters as MNT6Parameters>::G2Parameters,
-        G2Var,
-    >()
-    .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT6Parameters>::G1Parameters>()
+        .unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<<Parameters as MNT6Parameters>::G2Parameters>()
+        .unwrap();
 }
diff --git a/mnt6_753/src/constraints/mod.rs b/mnt6_753/src/constraints/mod.rs
index c540ac7f..29f025a8 100644
--- a/mnt6_753/src/constraints/mod.rs
+++ b/mnt6_753/src/constraints/mod.rs
@@ -130,17 +130,17 @@
 //! let pairing_result_native = MNT6_753::pairing(a_native, b_native);
 //!
 //! // Prepare `a` and `b` for pairing.
-//! let a_prep = constraints::PairingVar::prepare_g1(&a)?;
-//! let b_prep = constraints::PairingVar::prepare_g2(&b)?;
-//! let pairing_result = constraints::PairingVar::pairing(a_prep, b_prep)?;
+//! let a_prep = MNT6_753::prepare_g1(&a)?;
+//! let b_prep = MNT6_753::prepare_g2(&b)?;
+//! let pairing_result = MNT6_753::pairing_gadget(a_prep, b_prep)?;
 //!
 //! // Check that the value of &a + &b is correct.
 //! assert_eq!(pairing_result.value()?, pairing_result_native);
 //!
 //! // Check that operations on variables and constants are equivalent.
-//! let a_prep_const = constraints::PairingVar::prepare_g1(&a_const)?;
-//! let b_prep_const = constraints::PairingVar::prepare_g2(&b_const)?;
-//! let pairing_result_const = constraints::PairingVar::pairing(a_prep_const, b_prep_const)?;
+//! let a_prep_const = MNT6_753::prepare_g1(&a_const)?;
+//! let b_prep_const = MNT6_753::prepare_g2(&b_const)?;
+//! let pairing_result_const = MNT6_753::pairing_gadget(a_prep_const, b_prep_const)?;
 //! println!("Done here 3");
 //!
 //! pairing_result.enforce_equal(&pairing_result_const)?;
diff --git a/mnt6_753/src/constraints/pairing.rs b/mnt6_753/src/constraints/pairing.rs
index 8979aec8..3ef1b35c 100644
--- a/mnt6_753/src/constraints/pairing.rs
+++ b/mnt6_753/src/constraints/pairing.rs
@@ -1,10 +1,8 @@
-use crate::Parameters;
-
 /// Specifies the constraints for computing a pairing in the MNT6-753 bilinear group.
-pub type PairingVar = ark_r1cs_std::pairing::mnt6::PairingVar<Parameters>;
+pub use crate::MNT6_753;
 
 #[test]
 fn test() {
     use crate::MNT6_753;
-    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT6_753, PairingVar>().unwrap()
+    ark_curve_constraint_tests::pairing::bilinearity_test::<MNT6_753>().unwrap()
 }
diff --git a/pallas/src/constraints/curves.rs b/pallas/src/constraints/curves.rs
index 4f839b16..cf7faff9 100644
--- a/pallas/src/constraints/curves.rs
+++ b/pallas/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::short_weierstrass::ProjectiveVar;
 
-use crate::constraints::FBaseVar;
-
 /// A group element in the Pallas prime-order group.
-pub type GVar = ProjectiveVar<PallasParameters, FBaseVar>;
+pub type GVar = ProjectiveVar<PallasParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::sw_test::<PallasParameters, GVar>().unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<PallasParameters>().unwrap();
 }
diff --git a/vesta/src/constraints/curves.rs b/vesta/src/constraints/curves.rs
index fac522a5..ed432b31 100644
--- a/vesta/src/constraints/curves.rs
+++ b/vesta/src/constraints/curves.rs
@@ -1,12 +1,10 @@
 use crate::*;
 use ark_r1cs_std::groups::curves::short_weierstrass::ProjectiveVar;
 
-use crate::constraints::FBaseVar;
-
 /// A group element in the Vesta prime-order group.
-pub type GVar = ProjectiveVar<VestaParameters, FBaseVar>;
+pub type GVar = ProjectiveVar<VestaParameters>;
 
 #[test]
 fn test() {
-    ark_curve_constraint_tests::curves::sw_test::<VestaParameters, GVar>().unwrap();
+    ark_curve_constraint_tests::curves::sw_test::<VestaParameters>().unwrap();
 }