Merge b120f9e111 into 38821bbf14

src/groups/curves/short_weierstrass/mod.rs

@@ -350,23 +350,38 @@ where
         // We can convert to projective safely because the result is guaranteed to be
         // non-zero by the condition on `affine_bits.len()`, and by the fact
         // that `accumulator` is non-zero
-        let result = accumulator.into_projective();
+        *mul_result += accumulator.into_projective();
         // If bits[0] is 0, then we have to subtract `self`; else, we subtract zero.
-        let subtrahend = bits[0].select(&Self::zero(), &initial_acc_value)?;
-        *mul_result += result - subtrahend;
+        let neg = NonZeroAffineVar::new(initial_acc_value.x, initial_acc_value.y.negate()?);
+        *mul_result = bits[0].select(mul_result, &mul_result.add_mixed(&neg)?)?;
 
         // Now, let's finish off the rest of the bits using our complete formulae
-        for bit in proj_bits {
+        for bit in proj_bits.iter().rev().skip(1).rev() {
             if bit.is_constant() {
                 if *bit == &Boolean::TRUE {
-                    *mul_result += &multiple_of_power_of_two.into_projective();
+                    *mul_result = mul_result.add_mixed(&multiple_of_power_of_two)?;
                 }
             } else {
-                let temp = &*mul_result + &multiple_of_power_of_two.into_projective();
+                let temp = mul_result.add_mixed(&multiple_of_power_of_two)?;
                 *mul_result = bit.select(&temp, &mul_result)?;
             }
             multiple_of_power_of_two.double_in_place()?;
         }
+
+        // last bit
+        // we don't need the last doubling of multiple_of_power_of_two
+        let n = proj_bits.len();
+        if n >= 1 {
+            if proj_bits[n - 1].is_constant() {
+                if proj_bits[n - 1] == &Boolean::TRUE {
+                    *mul_result = mul_result.add_mixed(&multiple_of_power_of_two)?;
+                }
+            } else {
+                let temp = mul_result.add_mixed(&multiple_of_power_of_two)?;
+                *mul_result = proj_bits[n - 1].select(&temp, &mul_result)?;
+            }
+        }
+
         Ok(())
     }
 }
@@ -498,6 +513,78 @@ where
         Ok(Self::new(self.x.clone(), self.y.negate()?, self.z.clone()))
     }
 
+    /// Computes `bits * self`, where `bits` is a little-endian
+    /// `Boolean` representation of a scalar.
+    ///
+    /// [Joye07](<https://www.iacr.org/archive/ches2007/47270135/47270135.pdf>), Alg.1.
+    #[tracing::instrument(target = "r1cs", skip(bits))]
+    fn scalar_mul_joye_le<'a>(
+        &self,
+        bits: impl Iterator<Item = &'a Boolean<<P::BaseField as Field>::BasePrimeField>>,
+    ) -> Result<Self, SynthesisError> {
+        if self.is_constant() {
+            if self.value().unwrap().is_zero() {
+                return Ok(self.clone());
+            }
+        }
+        let self_affine = self.to_affine()?;
+        let (x, y, infinity) = (self_affine.x, self_affine.y, self_affine.infinity);
+        // We first handle the non-zero case, and then later will conditionally select
+        // zero if `self` was zero. However, we also want to make sure that generated
+        // constraints are satisfiable in both cases.
+        //
+        // In particular, using non-sensible values for `x` and `y` in zero-case may
+        // cause `unchecked` operations to generate constraints that can never
+        // be satisfied, depending on the curve equation coefficients.
+        //
+        // The safest approach is to use coordinates of some point from the curve, thus
+        // not violating assumptions of `NonZeroAffine`. For instance, generator
+        // point.
+        let x = infinity.select(&F::constant(P::GENERATOR.x), &x)?;
+        let y = infinity.select(&F::constant(P::GENERATOR.y), &y)?;
+        let non_zero_self = NonZeroAffineVar::new(x, y);
+
+        let mut bits = bits.collect::<Vec<_>>();
+        if bits.len() == 0 {
+            return Ok(Self::zero());
+        }
+        // Remove unnecessary constant zeros in the most-significant positions.
+        bits = bits
+            .into_iter()
+            // We iterate from the MSB down.
+            .rev()
+            // Skip leading zeros, if they are constants.
+            .skip_while(|b| b.is_constant() && (b.value().unwrap() == false))
+            .collect();
+        // After collecting we are in big-endian form; we have to reverse to get back to
+        // little-endian.
+        bits.reverse();
+
+        // second bit
+        let mut acc = non_zero_self.triple()?;
+        let mut acc0 = bits[1].select(&acc, &non_zero_self)?;
+        let mut acc1 = bits[1].select(&non_zero_self, &acc)?;
+
+        for bit in bits.iter().skip(2).rev().skip(1).rev() {
+            acc = acc0.double_and_select_add_unchecked(bit, &acc1)?;
+            acc0 = bit.select(&acc, &acc0)?;
+            acc1 = bit.select(&acc1, &acc)?;
+        }
+
+        // last bit
+        let n = bits.len() - 1;
+        acc = acc0.double_and_select_add_unchecked(bits[n], &acc1)?;
+        acc0 = bits[n].select(&acc, &acc0)?;
+
+        // first bit
+        let temp = NonZeroAffineVar::new(non_zero_self.x, non_zero_self.y.negate()?);
+        acc1 = acc0.add_unchecked(&temp)?;
+        acc0 = bits[0].select(&acc0, &acc1)?;
+
+        let mul_result = acc0.into_projective();
+        infinity.select(&Self::zero(), &mul_result)
+    }
+
     /// Computes `bits * self`, where `bits` is a little-endian
     /// `Boolean` representation of a scalar.
     #[tracing::instrument(target = "r1cs", skip(bits))]
@@ -516,12 +603,13 @@ where
         // zero if `self` was zero. However, we also want to make sure that generated
         // constraints are satisfiable in both cases.
         //
-        // In particular, using non-sensible values for `x` and `y` in zero-case may cause
-        // `unchecked` operations to generate constraints that can never be satisfied, depending
-        // on the curve equation coefficients.
+        // In particular, using non-sensible values for `x` and `y` in zero-case may
+        // cause `unchecked` operations to generate constraints that can never
+        // be satisfied, depending on the curve equation coefficients.
         //
-        // The safest approach is to use coordinates of some point from the curve, thus not
-        // violating assumptions of `NonZeroAffine`. For instance, generator point.
+        // The safest approach is to use coordinates of some point from the curve, thus
+        // not violating assumptions of `NonZeroAffine`. For instance, generator
+        // point.
         let x = infinity.select(&F::constant(P::GENERATOR.x), &x)?;
         let y = infinity.select(&F::constant(P::GENERATOR.y), &y)?;
         let non_zero_self = NonZeroAffineVar::new(x, y);
@@ -558,6 +646,73 @@ where
         infinity.select(&Self::zero(), &mul_result)
     }
 
+    /// Computes `bits1 * self + bits2 * p`, where `bits1` and `bits2` are
+    /// big-endian `Boolean` representation of the scalars.
+    ///
+    /// `self` and `p` are non-zero and `self` ≠ `-p`.
+    #[tracing::instrument(target = "r1cs", skip(bits1, bits2))]
+    fn joint_scalar_mul_be<'a>(
+        &self,
+        p: &Self,
+        bits1: impl Iterator<Item = &'a Boolean<<P::BaseField as Field>::BasePrimeField>>,
+        bits2: impl Iterator<Item = &'a Boolean<<P::BaseField as Field>::BasePrimeField>>,
+    ) -> Result<Self, SynthesisError> {
+        // prepare bits decomposition
+        let mut bits1 = bits1.collect::<Vec<_>>();
+        if bits1.len() == 0 {
+            return Ok(Self::zero());
+        }
+        // Remove unnecessary constant zeros in the most-significant positions.
+        bits1 = bits1
+            .into_iter()
+            // We iterate from the MSB down.
+            .rev()
+            // Skip leading zeros, if they are constants.
+            .skip_while(|b| b.is_constant() && (b.value().unwrap() == false))
+            .collect();
+
+        let mut bits2 = bits2.collect::<Vec<_>>();
+        if bits2.len() == 0 {
+            return Ok(Self::zero());
+        }
+        // Remove unnecessary constant zeros in the most-significant positions.
+        bits2 = bits2
+            .into_iter()
+            // We iterate from the MSB down.
+            .rev()
+            // Skip leading zeros, if they are constants.
+            .skip_while(|b| b.is_constant() && (b.value().unwrap() == false))
+            .collect();
+
+        // precompute points
+        let aff1 = self.to_affine()?;
+        let nz_aff1 = NonZeroAffineVar::new(aff1.x, aff1.y);
+
+        let aff2 = p.to_affine()?;
+        let nz_aff2 = NonZeroAffineVar::new(aff2.x, aff2.y);
+
+        let mut aff1_neg = NonZeroAffineVar::new(nz_aff1.x.clone(), nz_aff1.y.negate()?);
+        let mut aff2_neg = NonZeroAffineVar::new(nz_aff2.x.clone(), nz_aff2.y.negate()?);
+        let mut acc = nz_aff1.add_unchecked(&nz_aff2.clone())?;
+
+        // double-and-add loop
+        for (bit1, bit2) in (bits1.iter().rev().skip(1).rev()).zip(bits2.iter().rev().skip(1).rev())
+        {
+            let mut b = bit1.select(&nz_aff1, &aff1_neg)?;
+            acc = acc.double_and_add_unchecked(&b)?;
+            b = bit2.select(&nz_aff2, &aff2_neg)?;
+            acc = acc.add_unchecked(&b)?;
+        }
+
+        // last bit
+        aff1_neg = aff1_neg.add_unchecked(&acc)?;
+        acc = bits1[bits1.len() - 1].select(&acc, &aff1_neg)?;
+        aff2_neg = aff2_neg.add_unchecked(&acc)?;
+        acc = bits2[bits1.len() - 1].select(&acc, &aff2_neg)?;
+
+        Ok(acc.into_projective())
+    }
+
     #[tracing::instrument(target = "r1cs", skip(scalar_bits_with_bases))]
     fn precomputed_base_scalar_mul_le<'a, I, B>(
         &mut self,
@@ -978,61 +1133,3 @@ where
         Ok(bytes)
     }
 }
-
-#[cfg(test)]
-mod test_sw_curve {
-    use crate::{
-        alloc::AllocVar,
-        eq::EqGadget,
-        fields::{fp::FpVar, nonnative::NonNativeFieldVar},
-        groups::{curves::short_weierstrass::ProjectiveVar, CurveVar},
-        ToBitsGadget,
-    };
-    use ark_ec::{
-        short_weierstrass::{Projective, SWCurveConfig},
-        CurveGroup,
-    };
-    use ark_ff::PrimeField;
-    use ark_relations::r1cs::{ConstraintSystem, Result};
-    use ark_std::UniformRand;
-    use num_traits::Zero;
-
-    fn zero_point_scalar_mul_satisfied<G>() -> Result<bool>
-    where
-        G: CurveGroup,
-        G::BaseField: PrimeField,
-        G::Config: SWCurveConfig,
-    {
-        let mut rng = ark_std::test_rng();
-
-        let cs = ConstraintSystem::new_ref();
-        let point_in = Projective::<G::Config>::zero();
-        let point_out = Projective::<G::Config>::zero();
-        let scalar = G::ScalarField::rand(&mut rng);
-
-        let point_in =
-            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_witness(cs.clone(), || {
-                Ok(point_in)
-            })?;
-        let point_out =
-            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_input(cs.clone(), || {
-                Ok(point_out)
-            })?;
-        let scalar = NonNativeFieldVar::new_input(cs.clone(), || Ok(scalar))?;
-
-        let mul = point_in.scalar_mul_le(scalar.to_bits_le().unwrap().iter())?;
-
-        point_out.enforce_equal(&mul)?;
-
-        cs.is_satisfied()
-    }
-
-    #[test]
-    fn test_zero_point_scalar_mul() {
-        assert!(zero_point_scalar_mul_satisfied::<ark_bls12_381::G1Projective>().unwrap());
-        assert!(zero_point_scalar_mul_satisfied::<ark_pallas::Projective>().unwrap());
-        assert!(zero_point_scalar_mul_satisfied::<ark_mnt4_298::G1Projective>().unwrap());
-        assert!(zero_point_scalar_mul_satisfied::<ark_mnt6_298::G1Projective>().unwrap());
-        assert!(zero_point_scalar_mul_satisfied::<ark_bn254::G1Projective>().unwrap());
-    }
-}

src/groups/curves/short_weierstrass/non_zero_affine.rs

@@ -130,6 +130,79 @@ where
         }
     }
 
+    /// Conditionally computes `(self + other) + self` or `(self + other) +
+    /// other` depending on the value of `cond`.
+    ///
+    /// This follows the formulae from [\[ELM03\]](https://arxiv.org/abs/math/0208038).
+    #[tracing::instrument(target = "r1cs", skip(self))]
+    pub fn double_and_select_add_unchecked(
+        &self,
+        cond: &Boolean<<P::BaseField as Field>::BasePrimeField>,
+        other: &Self,
+    ) -> Result<Self, SynthesisError> {
+        if [self].is_constant() || other.is_constant() {
+            // /!\ TODO: correct constant case /!\
+            self.double()?.add_unchecked(other)
+        } else {
+            // It's okay to use `unchecked` the precondition is that `self != ±other` (i.e.
+            // same logic as in `add_unchecked`)
+            let (x1, y1) = (&self.x, &self.y);
+            let (x2, y2) = (&other.x, &other.y);
+
+            // Calculate self + other:
+            // slope lambda := (y2 - y1)/(x2 - x1);
+            // x3 = lambda^2 - x1 - x2;
+            // y3 = lambda * (x1 - x3) - y1
+            let numerator = y2 - y1;
+            let denominator = x2 - x1;
+            let lambda_1 = numerator.mul_by_inverse_unchecked(&denominator)?;
+
+            let x3 = lambda_1.square()? - x1 - x2;
+
+            let x = &F::conditionally_select(&cond, &x1, &x2)?;
+            let y = &F::conditionally_select(&cond, &y1, &y2)?;
+
+            // Calculate final addition slope:
+            let lambda_2 =
+                (lambda_1 + y.double()?.mul_by_inverse_unchecked(&(&x3 - x))?).negate()?;
+
+            // Calculate (self + other) + (self or other):
+            let x4 = lambda_2.square()? - x - x3;
+            let y4 = lambda_2 * &(x - &x4) - y;
+            Ok(Self::new(x4, y4))
+        }
+    }
+
+    /// Triples `self`.
+    ///
+    /// This follows the formulae from [\[ELM03\]](https://arxiv.org/abs/math/0208038).
+    #[tracing::instrument(target = "r1cs", skip(self))]
+    pub fn triple(&self) -> Result<Self, SynthesisError> {
+        if [self].is_constant() {
+            self.double()?.add_unchecked(self)
+        } else {
+            let (x1, y1) = (&self.x, &self.y);
+            let x1_sqr = x1.square()?;
+            // tangent lambda_1 := (3 * x1^2 + a) / (2 * y1);
+            // x3 = lambda_1^2 - 2x1
+            // y3 = lambda_1 * (x1 - x3) - y1
+            let numerator = x1_sqr.double()? + &x1_sqr + P::COEFF_A;
+            let denominator = y1.double()?;
+            // It's okay to use `unchecked` here, because the precondition of `double` is
+            // that self != zero.
+            let lambda_1 = numerator.mul_by_inverse_unchecked(&denominator)?;
+            let x3 = lambda_1.square()? - x1.double()?;
+
+            // Calculate final addition slope:
+            let lambda_2 =
+                (lambda_1 + y1.double()?.mul_by_inverse_unchecked(&(&x3 - x1))?).negate()?;
+
+            let x4 = lambda_2.square()? - x1 - x3;
+            let y4 = lambda_2 * &(x1 - &x4) - y1;
+            Ok(Self::new(x4, y4))
+        }
+    }
+
     /// Doubles `self` in place.
     #[tracing::instrument(target = "r1cs", skip(self))]
     pub fn double_in_place(&mut self) -> Result<(), SynthesisError> {
@@ -390,4 +463,50 @@ mod test_non_zero_affine {
 
         assert!(cs.is_satisfied().unwrap());
     }
+
+    #[test]
+    fn correctness_double_and_add_select() {
+        let cs = ConstraintSystem::<Fq>::new_ref();
+
+        let x = FpVar::Var(
+            AllocatedFp::<Fq>::new_witness(cs.clone(), || Ok(G1Config::GENERATOR.x)).unwrap(),
+        );
+        let y = FpVar::Var(
+            AllocatedFp::<Fq>::new_witness(cs.clone(), || Ok(G1Config::GENERATOR.y)).unwrap(),
+        );
+
+        // The following code tests `double_and_add`.
+        let sum_a = {
+            let a = ProjectiveVar::<G1Config, FpVar<Fq>>::new(
+                x.clone(),
+                y.clone(),
+                FpVar::Constant(Fq::one()),
+            );
+
+            let mut cur = a.clone();
+            cur.double_in_place().unwrap();
+            for _ in 1..10 {
+                cur.double_in_place().unwrap();
+                cur = cur + &a;
+            }
+
+            let sum = cur.value().unwrap().into_affine();
+            (sum.x, sum.y)
+        };
+
+        let sum_b = {
+            let a = NonZeroAffineVar::<G1Config, FpVar<Fq>>::new(x, y);
+
+            let mut cur = a.double().unwrap();
+            for _ in 1..10 {
+                cur = cur.double_and_add_unchecked(&a).unwrap();
+            }
+
+            (cur.x.value().unwrap(), cur.y.value().unwrap())
+        };
+
+        assert!(cs.is_satisfied().unwrap());
+        assert_eq!(sum_a.0, sum_b.0);
+        assert_eq!(sum_a.1, sum_b.1);
+    }
 }

src/groups/mod.rs

@@ -107,6 +107,44 @@ pub trait CurveVar:
         Ok(res)
     }
 
+    /// Computes `bits * self`, where `bits` is a little-endian
+    /// `Boolean` representation of a scalar.
+    ///
+    /// [Joye07](<https://www.iacr.org/archive/ches2007/47270135/47270135.pdf>), Alg.1.
+    #[tracing::instrument(target = "r1cs", skip(bits))]
+    fn scalar_mul_joye_le<'a>(
+        &self,
+        bits: impl Iterator<Item = &'a Boolean<ConstraintF>>,
+    ) -> Result<Self, SynthesisError> {
+        // TODO: in the constant case we should call precomputed_scalar_mul_le,
+        // but rn there's a bug when doing this with TE curves.
+
+        // Computes the standard little-endian double-and-add algorithm
+        // (Algorithm 3.26, Guide to Elliptic Curve Cryptography)
+        let mut res = Self::zero();
+        let mut multiple = self.clone();
+        for bit in bits {
+            let tmp = res.clone() + &multiple;
+            res = bit.select(&tmp, &res)?;
+            multiple.double_in_place()?;
+        }
+        Ok(res)
+    }
+
+    /// Computes a `I1 * self + I2 * p` in place, where `I1` and `I2` are
+    /// `Boolean` *big-endian* representation of the scalars.
+    #[tracing::instrument(target = "r1cs", skip(bits1, bits2))]
+    fn joint_scalar_mul_be<'a>(
+        &self,
+        p: &Self,
+        bits1: impl Iterator<Item = &'a Boolean<ConstraintF>>,
+        bits2: impl Iterator<Item = &'a Boolean<ConstraintF>>,
+    ) -> Result<Self, SynthesisError> {
+        let res1 = self.scalar_mul_le(bits1)?;
+        let res2 = p.scalar_mul_le(bits2)?;
+        Ok(res1 + res2)
+    }
+
     /// Computes a `I * self` in place, where `I` is a `Boolean` *little-endian*
     /// representation of the scalar.
     ///
@@ -161,3 +199,141 @@ pub trait CurveVar:
         Ok(result)
     }
 }
+
+#[cfg(test)]
+mod test_sw_arithmetic {
+    use crate::{
+        alloc::AllocVar,
+        eq::EqGadget,
+        fields::{fp::FpVar, nonnative::NonNativeFieldVar},
+        groups::{curves::short_weierstrass::ProjectiveVar, CurveVar},
+        ToBitsGadget,
+    };
+    use ark_ec::{
+        short_weierstrass::{Projective, SWCurveConfig},
+        CurveGroup,
+    };
+    use ark_ff::PrimeField;
+    use ark_relations::r1cs::{ConstraintSystem, Result};
+    use ark_std::UniformRand;
+
+    fn point_scalar_mul_satisfied<G>() -> Result<bool>
+    where
+        G: CurveGroup,
+        G::BaseField: PrimeField,
+        G::Config: SWCurveConfig,
+    {
+        let mut rng = ark_std::test_rng();
+
+        let cs = ConstraintSystem::new_ref();
+        let point_in = Projective::<G::Config>::rand(&mut rng);
+        let scalar = G::ScalarField::rand(&mut rng);
+        let point_out = point_in * scalar;
+
+        let point_in =
+            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_witness(cs.clone(), || {
+                Ok(point_in)
+            })?;
+        let point_out =
+            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_input(cs.clone(), || {
+                Ok(point_out)
+            })?;
+        let scalar = NonNativeFieldVar::new_input(cs.clone(), || Ok(scalar))?;
+
+        let mul = point_in.scalar_mul_le(scalar.to_bits_le().unwrap().iter())?;
+
+        point_out.enforce_equal(&mul)?;
+
+        println!("#r1cs for scalar_mul_le: {}", cs.num_constraints());
+
+        cs.is_satisfied()
+    }
+
+    fn point_scalar_mul_joye_satisfied<G>() -> Result<bool>
+    where
+        G: CurveGroup,
+        G::BaseField: PrimeField,
+        G::Config: SWCurveConfig,
+    {
+        let mut rng = ark_std::test_rng();
+
+        let cs = ConstraintSystem::new_ref();
+        let point_in = Projective::<G::Config>::rand(&mut rng);
+        let scalar = G::ScalarField::rand(&mut rng);
+        let point_out = point_in * scalar;
+
+        let point_in =
+            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_witness(cs.clone(), || {
+                Ok(point_in)
+            })?;
+        let point_out =
+            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_input(cs.clone(), || {
+                Ok(point_out)
+            })?;
+        let scalar = NonNativeFieldVar::new_input(cs.clone(), || Ok(scalar))?;
+
+        let mul = point_in.scalar_mul_joye_le(scalar.to_bits_le().unwrap().iter())?;
+
+        point_out.enforce_equal(&mul)?;
+
+        println!("#r1cs for scalar_mul_joye_le: {}", cs.num_constraints());
+
+        cs.is_satisfied()
+    }
+
+    fn point_joint_scalar_mul_satisfied<G>() -> Result<bool>
+    where
+        G: CurveGroup,
+        G::BaseField: PrimeField,
+        G::Config: SWCurveConfig,
+    {
+        let mut rng = ark_std::test_rng();
+
+        let cs = ConstraintSystem::new_ref();
+        let point_in1 = Projective::<G::Config>::rand(&mut rng);
+        let point_in2 = Projective::<G::Config>::rand(&mut rng);
+        let scalar1 = G::ScalarField::rand(&mut rng);
+        let scalar2 = G::ScalarField::rand(&mut rng);
+        let point_out = point_in1 * scalar1 + point_in2 * scalar2;
+
+        let point_in1 =
+            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_witness(cs.clone(), || {
+                Ok(point_in1)
+            })?;
+        let point_in2 =
+            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_witness(cs.clone(), || {
+                Ok(point_in2)
+            })?;
+        let point_out =
+            ProjectiveVar::<G::Config, FpVar<G::BaseField>>::new_input(cs.clone(), || {
+                Ok(point_out)
+            })?;
+        let scalar1 = NonNativeFieldVar::new_input(cs.clone(), || Ok(scalar1))?;
+        let scalar2 = NonNativeFieldVar::new_input(cs.clone(), || Ok(scalar2))?;
+
+        let res = point_in1.joint_scalar_mul_be(
+            &point_in2,
+            scalar1.to_bits_le().unwrap().iter(),
+            scalar2.to_bits_le().unwrap().iter(),
+        )?;
+
+        point_out.enforce_equal(&res)?;
+
+        println!("#r1cs for joint_scalar_mul: {}", cs.num_constraints());
+
+        cs.is_satisfied()
+    }
+
+    #[test]
+    fn test_point_scalar_mul() {
+        assert!(point_scalar_mul_satisfied::<ark_bn254::G1Projective>().unwrap());
+    }
+    #[test]
+    fn test_point_scalar_mul_joye() {
+        assert!(point_scalar_mul_joye_satisfied::<ark_bn254::G1Projective>().unwrap());
+    }
+    #[test]
+    fn test_point_joint_scalar_mul() {
+        assert!(point_joint_scalar_mul_satisfied::<ark_bn254::G1Projective>().unwrap());
+    }
+}