diff --git a/src/pairing/bls12/mod.rs b/src/pairing/bls12/mod.rs
index 33454c41..d9f22352 100644
--- a/src/pairing/bls12/mod.rs
+++ b/src/pairing/bls12/mod.rs
@@ -1,9 +1,9 @@
 use ark_relations::r1cs::SynthesisError;
 
-use super::PairingVar as PG;
+use super::{PairingGadget as PG, PairingWithGadget};
 
 use crate::{
-    fields::{fp::FpVar, fp12::Fp12Var, fp2::Fp2Var, FieldVar},
+    fields::{fp::FpVar, fp12::Fp12Var, fp2::Fp2Var, FieldVar, FieldWithVar},
     groups::bls12::{G1AffineVar, G1PreparedVar, G1Var, G2PreparedVar, G2Var},
 };
 use ark_ec::bls12::{Bls12, Bls12Parameters, TwistType};
@@ -11,11 +11,14 @@ use ark_ff::fields::BitIteratorBE;
 use core::marker::PhantomData;
 
 /// Specifies the constraints for computing a pairing in a BLS12 bilinear group.
-pub struct PairingVar<P: Bls12Parameters>(PhantomData<P>);
+pub struct Bls12Gadget<P: Bls12Parameters>(PhantomData<P>);
 
 type Fp2V<P> = Fp2Var<<P as Bls12Parameters>::Fp2Params>;
 
-impl<P: Bls12Parameters> PairingVar<P> {
+impl<P: Bls12Parameters> Bls12Gadget<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
     // Evaluate the line function at point p.
     #[tracing::instrument(target = "r1cs")]
     fn ell(
@@ -59,7 +62,17 @@ impl<P: Bls12Parameters> PairingVar<P> {
     }
 }
 
-impl<P: Bls12Parameters> PG<Bls12<P>, P::Fp> for PairingVar<P> {
+impl<P: Bls12Parameters> PairingWithGadget for Bls12<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
+    type Gadget = Bls12Gadget<P>;
+}
+
+impl<P: Bls12Parameters> PG<Bls12<P>> for Bls12Gadget<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
     type G1Var = G1Var<P>;
     type G2Var = G2Var<P>;
     type G1PreparedVar = G1PreparedVar<P>;
diff --git a/src/pairing/mnt4/mod.rs b/src/pairing/mnt4/mod.rs
index 0ddbec59..f9024058 100644
--- a/src/pairing/mnt4/mod.rs
+++ b/src/pairing/mnt4/mod.rs
@@ -1,9 +1,9 @@
 use ark_relations::r1cs::SynthesisError;
 
-use super::PairingVar as PG;
+use super::{PairingGadget as PG, PairingWithGadget};
 
 use crate::{
-    fields::{fp::FpVar, fp2::Fp2Var, fp4::Fp4Var, FieldVar},
+    fields::{fp::FpVar, fp2::Fp2Var, fp4::Fp4Var, FieldVar, FieldWithVar},
     groups::mnt4::{
         AteAdditionCoefficientsVar, AteDoubleCoefficientsVar, G1PreparedVar, G1Var, G2PreparedVar,
         G2ProjectiveExtendedVar, G2Var,
@@ -15,14 +15,17 @@ use ark_ff::BitIteratorBE;
 use core::marker::PhantomData;
 
 /// Specifies the constraints for computing a pairing in a MNT4 bilinear group.
-pub struct PairingVar<P: MNT4Parameters>(PhantomData<P>);
+pub struct MNT4Gadget<P: MNT4Parameters>(PhantomData<P>);
 
 type Fp2G<P> = Fp2Var<<P as MNT4Parameters>::Fp2Params>;
 type Fp4G<P> = Fp4Var<<P as MNT4Parameters>::Fp4Params>;
 /// A variable corresponding to `ark_ec::mnt4::GT`.
 pub type GTVar<P> = Fp4G<P>;
 
-impl<P: MNT4Parameters> PairingVar<P> {
+impl<P: MNT4Parameters> MNT4Gadget<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
     #[tracing::instrument(target = "r1cs", skip(r))]
     pub(crate) fn doubling_step_for_flipped_miller_loop(
         r: &G2ProjectiveExtendedVar<P>,
@@ -186,7 +189,17 @@ impl<P: MNT4Parameters> PairingVar<P> {
     }
 }
 
-impl<P: MNT4Parameters> PG<MNT4<P>, P::Fp> for PairingVar<P> {
+impl<P: MNT4Parameters> PairingWithGadget for MNT4<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
+    type Gadget = MNT4Gadget<P>;
+}
+
+impl<P: MNT4Parameters> PG<MNT4<P>> for MNT4Gadget<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
     type G1Var = G1Var<P>;
     type G2Var = G2Var<P>;
     type G1PreparedVar = G1PreparedVar<P>;
diff --git a/src/pairing/mnt6/mod.rs b/src/pairing/mnt6/mod.rs
index bb849b29..4711d304 100644
--- a/src/pairing/mnt6/mod.rs
+++ b/src/pairing/mnt6/mod.rs
@@ -1,9 +1,9 @@
 use ark_relations::r1cs::SynthesisError;
 
-use super::PairingVar as PG;
+use super::{PairingGadget as PG, PairingWithGadget};
 
 use crate::{
-    fields::{fp::FpVar, fp3::Fp3Var, fp6_2over3::Fp6Var, FieldVar},
+    fields::{fp::FpVar, fp3::Fp3Var, fp6_2over3::Fp6Var, FieldVar, FieldWithVar},
     groups::mnt6::{
         AteAdditionCoefficientsVar, AteDoubleCoefficientsVar, G1PreparedVar, G1Var, G2PreparedVar,
         G2ProjectiveExtendedVar, G2Var,
@@ -14,14 +14,17 @@ use ark_ff::fields::BitIteratorBE;
 use core::marker::PhantomData;
 
 /// Specifies the constraints for computing a pairing in a MNT6 bilinear group.
-pub struct PairingVar<P: MNT6Parameters>(PhantomData<P>);
+pub struct MNT6Gadget<P: MNT6Parameters>(PhantomData<P>);
 
 type Fp3G<P> = Fp3Var<<P as MNT6Parameters>::Fp3Params>;
 type Fp6G<P> = Fp6Var<<P as MNT6Parameters>::Fp6Params>;
 /// A variable corresponding to `ark_ec::mnt6::GT`.
 pub type GTVar<P> = Fp6G<P>;
 
-impl<P: MNT6Parameters> PairingVar<P> {
+impl<P: MNT6Parameters> MNT6Gadget<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
     #[tracing::instrument(target = "r1cs", skip(r))]
     pub(crate) fn doubling_step_for_flipped_miller_loop(
         r: &G2ProjectiveExtendedVar<P>,
@@ -181,7 +184,17 @@ impl<P: MNT6Parameters> PairingVar<P> {
     }
 }
 
-impl<P: MNT6Parameters> PG<MNT6<P>, P::Fp> for PairingVar<P> {
+impl<P: MNT6Parameters> PairingWithGadget for MNT6<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
+    type Gadget = MNT6Gadget<P>;
+}
+
+impl<P: MNT6Parameters> PG<MNT6<P>> for MNT6Gadget<P>
+where
+    P::Fp: FieldWithVar<Var = FpVar<P::Fp>>,
+{
     type G1Var = G1Var<P>;
     type G2Var = G2Var<P>;
     type G1PreparedVar = G1PreparedVar<P>;
diff --git a/src/pairing/mod.rs b/src/pairing/mod.rs
index 157f26d8..7ac4d797 100644
--- a/src/pairing/mod.rs
+++ b/src/pairing/mod.rs
@@ -1,6 +1,5 @@
 use crate::prelude::*;
 use ark_ec::PairingEngine;
-use ark_ff::Field;
 use ark_relations::r1cs::SynthesisError;
 use core::fmt::Debug;
 
@@ -11,39 +10,37 @@ pub mod mnt4;
 /// This module implements pairings for MNT6 bilinear groups.
 pub mod mnt6;
 
+pub trait PairingWithGadget: PairingEngine {
+    type Gadget: PairingGadget<Self>;
+}
+
 /// Specifies the constraints for computing a pairing in the yybilinear group
 /// `E`.
-pub trait PairingVar<E: PairingEngine, ConstraintF: Field = <E as PairingEngine>::Fq> {
+pub trait PairingGadget<E: PairingEngine> {
     /// An variable representing an element of `G1`.
     /// This is the R1CS equivalent of `E::G1Projective`.
-    type G1Var: CurveVar<E::G1Projective, ConstraintF>
-        + AllocVar<E::G1Projective, ConstraintF>
-        + AllocVar<E::G1Affine, ConstraintF>;
+    type G1Var: CurveVar<E::G1Projective, E::Fq>
+        + AllocVar<E::G1Projective, E::Fq>
+        + AllocVar<E::G1Affine, E::Fq>;
 
     /// An variable representing an element of `G2`.
     /// This is the R1CS equivalent of `E::G2Projective`.
-    type G2Var: CurveVar<E::G2Projective, ConstraintF>
-        + AllocVar<E::G2Projective, ConstraintF>
-        + AllocVar<E::G2Affine, ConstraintF>;
+    type G2Var: CurveVar<E::G2Projective, E::Fq>
+        + AllocVar<E::G2Projective, E::Fq>
+        + AllocVar<E::G2Affine, E::Fq>;
 
     /// An variable representing an element of `GT`.
     /// This is the R1CS equivalent of `E::GT`.
-    type GTVar: FieldVar<E::Fqk, ConstraintF>;
+    type GTVar: FieldVar<E::Fqk, E::Fq>;
 
     /// An variable representing cached precomputation  that can speed up
     /// pairings computations. This is the R1CS equivalent of
     /// `E::G1Prepared`.
-    type G1PreparedVar: ToBytesGadget<ConstraintF>
-        + AllocVar<E::G1Prepared, ConstraintF>
-        + Clone
-        + Debug;
+    type G1PreparedVar: ToBytesGadget<E::Fq> + AllocVar<E::G1Prepared, E::Fq> + Clone + Debug;
     /// An variable representing cached precomputation  that can speed up
     /// pairings computations. This is the R1CS equivalent of
     /// `E::G2Prepared`.
-    type G2PreparedVar: ToBytesGadget<ConstraintF>
-        + AllocVar<E::G2Prepared, ConstraintF>
-        + Clone
-        + Debug;
+    type G2PreparedVar: ToBytesGadget<E::Fq> + AllocVar<E::G2Prepared, E::Fq> + Clone + Debug;
 
     /// Computes a multi-miller loop between elements
     /// of `p` and `q`.