diff --git a/src/constraints.rs b/src/constraints.rs
index f2b962d..125bdc8 100644
--- a/src/constraints.rs
+++ b/src/constraints.rs
@@ -1,32 +1,33 @@
-use crate::{Parameters, Signature};
+use crate::{Parameters, PublicKey, Signature};
 
 use ark_ec::{AffineCurve, ProjectiveCurve};
 use ark_ed_on_bn254::{constraints::EdwardsVar, EdwardsParameters, FqParameters};
 use ark_ff::{
     fields::{Field, Fp256},
-    to_bytes, ToConstraintField,
+    to_bytes, PrimeField, ToConstraintField,
 };
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     bits::uint8::UInt8,
     boolean::Boolean,
-    fields::fp::FpVar,
+    eq::EqGadget,
+    fields::{fp::FpVar, FieldVar},
     groups::{curves::twisted_edwards::AffineVar, GroupOpsBounds},
     prelude::CurveVar,
     ToBitsGadget,
 };
 use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, Namespace, SynthesisError};
-use ark_std::ops::Mul;
+use ark_std::ops::{Add, Mul};
 
 use core::{borrow::Borrow, marker::PhantomData};
 use derivative::Derivative;
 
 // hash
+use arkworks_native_gadgets::poseidon as poseidon_native;
 use arkworks_r1cs_gadgets::poseidon;
-use arkworks_r1cs_gadgets::poseidon::{FieldHasherGadget, PoseidonGadget};
+use arkworks_r1cs_gadgets::poseidon::{FieldHasherGadget, PoseidonGadget, PoseidonParametersVar};
 
-// type ConstraintF<C> = <<C as ProjectiveCurve>::BaseField as Field>::BasePrimeField;
-type ConstraintF<C> = <C as ProjectiveCurve>::ScalarField; // Fr
+use crate::ConstraintF;
 
 #[derive(Derivative)]
 #[derivative(
@@ -42,6 +43,25 @@ where
     _group: PhantomData<*const C>,
 }
 
+impl<C, GC> AllocVar<PublicKey<C>, ConstraintF<C>> for PublicKeyVar<C, GC>
+where
+    C: ProjectiveCurve,
+    GC: CurveVar<C, ConstraintF<C>>,
+    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
+{
+    fn new_variable<T: Borrow<PublicKey<C>>>(
+        cs: impl Into<Namespace<ConstraintF<C>>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        let pub_key = GC::new_variable(cs, f, mode)?;
+        Ok(Self {
+            pub_key,
+            _group: PhantomData,
+        })
+    }
+}
+
 #[derive(Derivative)]
 #[derivative(
     Debug(bound = "C: ProjectiveCurve, GC: CurveVar<C, ConstraintF<C>>"),
@@ -51,8 +71,6 @@ pub struct SignatureVar<C: ProjectiveCurve, GC: CurveVar<C, ConstraintF<C>>>
 where
     for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
-    // s: FpVar<ConstraintF>,
-    // s: C::ScalarField,
     s: Vec<UInt8<ConstraintF<C>>>,
     r: GC,
     _curve: PhantomData<C>,
@@ -61,8 +79,7 @@ where
 impl<C, GC> AllocVar<Signature<C>, ConstraintF<C>> for SignatureVar<C, GC>
 where
     C: ProjectiveCurve,
-    // TODO not sure on '+ AllocVarar'
-    GC: CurveVar<C, ConstraintF<C>> + AllocVar<GC, ConstraintF<C>>,
+    GC: CurveVar<C, ConstraintF<C>>,
     for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
     fn new_variable<T: Borrow<Signature<C>>>(
@@ -86,8 +103,8 @@ where
             let r = GC::new_variable(cs.clone(), || Ok(val.borrow().r), mode)?;
 
             Ok(Self {
-                s: s, // TODO not sure of FpVar::Constant
-                r: r,
+                s,
+                r,
                 _curve: PhantomData,
             })
         })
@@ -129,8 +146,7 @@ pub struct BlindSigVerifyGadget<C: ProjectiveCurve, GC: CurveVar<C, ConstraintF<
 where
     for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
-    params: Parameters<C>,
-    // sig: Signature<C>,
+    _params: Parameters<C>, // TODO review if needed, maybe delete
     _gc: PhantomData<GC>,
 }
 
@@ -143,8 +159,9 @@ where
         EdwardsParameters,
         FpVar<Fp256<FqParameters>>,
     >: From<GC>,
-    FpVar<<C as ProjectiveCurve>::ScalarField>: Mul<FpVar<Fp256<FqParameters>>>,
-    FpVar<<C as ProjectiveCurve>::ScalarField>: From<<C as ProjectiveCurve>::ScalarField>,
+    <C as ProjectiveCurve>::BaseField: PrimeField,
+    FpVar<<C as ProjectiveCurve>::BaseField>: Mul<FpVar<Fp256<FqParameters>>>,
+    FpVar<<C as ProjectiveCurve>::BaseField>: From<FpVar<Fp256<FqParameters>>>,
 {
     fn verify(
         parameters: &ParametersVar<C, GC>,
@@ -152,32 +169,85 @@ where
         m: FpVar<ConstraintF<C>>,
         s: &SignatureVar<C, GC>,
         q: &PublicKeyVar<C, GC>,
-    ) -> Result<Boolean<ConstraintF<C>>, SynthesisError>
-    where
-        <C as ProjectiveCurve>::ScalarField: Iterator, // WIP
-        <C as ProjectiveCurve>::ScalarField: From<
-            <FpVar<<C as ProjectiveCurve>::ScalarField> as Mul<FpVar<Fp256<FqParameters>>>>::Output,
-        >,
-    {
+    ) -> Result<Boolean<ConstraintF<C>>, SynthesisError> {
         let s_s = s.s.clone();
 
         let sG = parameters
             .generator
             .scalar_mul_le(s_s.to_bits_le()?.iter())?;
 
-        // G * s == R + Q * (R.x * H(m))
         // Note: in a circuit that aggregates multiple verifications, the hashing step could be
         // done outside the signature verification, once for all 1 votes and once for all 0 votes,
         // saving lots of constraints
         let hm = poseidon_hash.hash(&[m])?;
         let r = EdwardsVar::from(s.r.clone()); // WIP
+        let rx_fpvar: FpVar<ConstraintF<C>> = r.x.into();
 
-        let rx_hm: ConstraintF<C> = ConstraintF::<C>::from(hm * r.x);
-        let rx_hm_fp: FpVar<ConstraintF<C>> = FpVar::<ConstraintF<C>>::from(rx_hm);
-
-        let Q_rx_hm = q.pub_key.scalar_mul_le(rx_hm_fp.to_bits_le()?.iter())?;
+        // G * s == R + Q * (R.x * H(m))
+        let Q_rx_hm_0 = q.pub_key.scalar_mul_le(rx_fpvar.to_bits_le()?.iter())?;
+        let Q_rx_hm = Q_rx_hm_0.scalar_mul_le(hm.to_bits_le()?.iter())?;
         let RHS = s.r.clone() + Q_rx_hm;
 
         sG.is_eq(&RHS)
     }
 }
+
+#[cfg(test)]
+mod test {
+    use super::*;
+    use crate::{poseidon_setup_params, BlindSigScheme};
+    use ark_ed_on_bn254::constraints::EdwardsVar as BabyJubJubVar;
+    use ark_ed_on_bn254::EdwardsProjective as BabyJubJub;
+
+    use arkworks_native_gadgets::poseidon;
+    use arkworks_utils::Curve;
+
+    use ark_relations::r1cs::ConstraintSystem;
+
+    type Fq = <BabyJubJub as ProjectiveCurve>::BaseField;
+    // type Fr = <BabyJubJub as ProjectiveCurve>::ScalarField;
+    type S = BlindSigScheme<BabyJubJub>;
+
+    #[test]
+    fn test_verify() {
+        let poseidon_params = poseidon_setup_params::<Fq>(Curve::Bn254, 5, 3);
+        let poseidon_hash = poseidon::Poseidon::new(poseidon_params);
+        let mut rng = ark_std::test_rng();
+
+        // create signature using native-rust lib
+        let params = S::setup();
+        let (pk, sk) = S::keygen(&params, &mut rng);
+        let (k, signer_r) = S::new_request_params(&params, &mut rng);
+        let m = Fq::from(1234);
+        let (m_blinded, u) = S::blind(&params, &mut rng, &poseidon_hash, m, signer_r).unwrap();
+        let s_blinded = S::blind_sign(sk, k, m_blinded);
+        let s = S::unblind(s_blinded, u);
+        let verified = S::verify(&params, &poseidon_hash, m, s.clone(), pk);
+        assert!(verified);
+
+        // use the constraint system to verify the signature
+        type SG = BlindSigVerifyGadget<BabyJubJub, BabyJubJubVar>;
+        let cs = ConstraintSystem::<Fq>::new_ref();
+
+        let params_var =
+            ParametersVar::<BabyJubJub, BabyJubJubVar>::new_constant(cs.clone(), params).unwrap();
+        let signature_var =
+            SignatureVar::<BabyJubJub, BabyJubJubVar>::new_witness(cs.clone(), || Ok(&s)).unwrap();
+        let pk_var =
+            PublicKeyVar::<BabyJubJub, BabyJubJubVar>::new_witness(cs.clone(), || Ok(&pk)).unwrap();
+        let m_var = FpVar::<Fq>::new_witness(cs.clone(), || Ok(&m)).unwrap();
+        let poseidon_hash_var =
+            PoseidonGadget::<Fq>::from_native(&mut cs.clone(), poseidon_hash).unwrap();
+
+        let valid_sig = SG::verify(
+            &params_var,
+            &poseidon_hash_var,
+            m_var,
+            &signature_var,
+            &pk_var,
+        )
+        .unwrap();
+        valid_sig.enforce_equal(&Boolean::<Fq>::TRUE).unwrap();
+        assert!(cs.is_satisfied().unwrap());
+    }
+}
diff --git a/src/lib.rs b/src/lib.rs
index eac07f2..c4b303b 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -30,7 +30,7 @@ use ark_ed_on_bn254::{
     EdwardsAffine, EdwardsParameters, EdwardsProjective, FqParameters, Fr, FrParameters,
 };
 
-type ConstraintF<C> = <<C as ProjectiveCurve>::BaseField as Field>::BasePrimeField;
+pub type ConstraintF<C> = <<C as ProjectiveCurve>::BaseField as Field>::BasePrimeField;
 pub type SecretKey<C> = <C as ProjectiveCurve>::ScalarField;
 pub type PublicKey<C> = <C as ProjectiveCurve>::Affine;
 pub type BlindedSignature<C> = <C as ProjectiveCurve>::ScalarField;
@@ -177,7 +177,7 @@ where
     {
         let sG = parameters.generator.mul(s.s.into_repr());
 
-        // TODO hash(m) must be \in Fr
+        // TODO the output of hash(m) must be \in Fr
         let hm = poseidon_hash.hash(&[m]).unwrap();
         // let hm_fr = C::ScalarField::from_repr(hm.into_repr()).unwrap();
         let hm_fr = C::ScalarField::from_le_bytes_mod_order(&to_bytes!(hm).unwrap()); // WIP TMP