Verifier circuit (#23)

* ECC scalar multiplication (first draft)

* fix clippy nits

* start implementing the ro gadget: 1st design Poseidon + truncate

* truncate to 128 bits

* implement add + double in constraints

* finish implementing constraints for ecc

* cargo fmt

* input of smul should be an array of bits

* cleanup ro a bit. Make the challenge returned be a vec of allocated bits

* switch to neptune 6.0

* start implementing high level circuit

* incomplete version of the verifier circuit with many TODOS

* optimize ecc ops. add i ==0 case to the circuit

* fix 0/1 constants at the circuit

* wrap CompressedGroupElement of Pallas and Vesta

* cargo fmt

* generate poseidon constants once instead of every time we call get_challenge

* Implement RO-based poseidon to use outside of circuit. Reorganize the repo

* add inner circuit to verification circuit

* start adding folding of the io. there is an error in the first call to  mult_mod

* add test to check that bellperson-nonnative is compatible with nova

* remove swap file

* add another test that fails

* add inputs to the circuits in tests

* rename q to m in circuit.rs. add more tests in test_bellperson_non_native. change a in test_mult_mod to expose error

* push test for equal_with_carried. fix the issue is src/r1cs.rs

* cargo fmt + update the verifier circuit: add folding of X and update all hashes with X

* make limb_width and n_limbs parameters

* make params part of h1

* allocate the field order as constant. add check that z0 == zi when i == 0

* fix error in test_poseidon_ro

* remove merge error

* small fixes

* small fixes to comments

* clippy lints

* small edits; rename tests

* move inputize before from_num

* _limbs --> _bn

* _limbs --> _bn

Co-authored-by: Ioanna <iontzialla@gmail.com>

src/gadgets/ecc.rs

@@ -0,0 +1,219 @@
+#![allow(non_snake_case)]
+use ff::{PrimeField, PrimeFieldBits};
+use rand::rngs::OsRng;
+use std::marker::PhantomData;
+
+#[derive(Debug, Clone)]
+pub struct Point<Fp, Fq>
+where
+  Fp: PrimeField,
+  Fq: PrimeField + PrimeFieldBits,
+{
+  pub(crate) x: Fp, //TODO: Make this not public
+  pub(crate) y: Fp,
+  is_infinity: bool,
+  _p: PhantomData<Fq>,
+}
+
+impl<Fp, Fq> Point<Fp, Fq>
+where
+  Fp: PrimeField,
+  Fq: PrimeField + PrimeFieldBits,
+{
+  #[allow(dead_code)]
+  pub fn new(x: Fp, y: Fp, is_infinity: bool) -> Self {
+    Self {
+      x,
+      y,
+      is_infinity,
+      _p: Default::default(),
+    }
+  }
+
+  #[allow(dead_code)]
+  pub fn random_vartime() -> Self {
+    loop {
+      let x = Fp::random(&mut OsRng);
+      let y = (x * x * x + Fp::one() + Fp::one() + Fp::one() + Fp::one() + Fp::one()).sqrt();
+      if y.is_some().unwrap_u8() == 1 {
+        return Self {
+          x,
+          y: y.unwrap(),
+          is_infinity: false,
+          _p: Default::default(),
+        };
+      }
+    }
+  }
+
+  pub fn add(&self, other: &Point<Fp, Fq>) -> Self {
+    if self.is_infinity {
+      return other.clone();
+    }
+
+    if other.is_infinity {
+      return self.clone();
+    }
+
+    let lambda = (other.y - self.y) * (other.x - self.x).invert().unwrap();
+    let x = lambda * lambda - self.x - other.x;
+    let y = lambda * (self.x - x) - self.y;
+    Self {
+      x,
+      y,
+      is_infinity: false,
+      _p: Default::default(),
+    }
+  }
+
+  pub fn double(&self) -> Self {
+    if self.is_infinity {
+      return Self {
+        x: Fp::zero(),
+        y: Fp::zero(),
+        is_infinity: true,
+        _p: Default::default(),
+      };
+    }
+
+    let lambda = (Fp::one() + Fp::one() + Fp::one())
+      * self.x
+      * self.x
+      * ((Fp::one() + Fp::one()) * self.y).invert().unwrap();
+    let x = lambda * lambda - self.x - self.x;
+    let y = lambda * (self.x - x) - self.y;
+    Self {
+      x,
+      y,
+      is_infinity: false,
+      _p: Default::default(),
+    }
+  }
+
+  #[allow(dead_code)]
+  pub fn scalar_mul_mont(&self, scalar: &Fq) -> Self {
+    let mut R0 = Self {
+      x: Fp::zero(),
+      y: Fp::zero(),
+      is_infinity: true,
+      _p: Default::default(),
+    };
+
+    let mut R1 = self.clone();
+    let bits = scalar.to_le_bits();
+    for i in (0..bits.len()).rev() {
+      if bits[i] {
+        R0 = R0.add(&R1);
+        R1 = R1.double();
+      } else {
+        R1 = R0.add(&R1);
+        R0 = R0.double();
+      }
+    }
+    R0
+  }
+
+  #[allow(dead_code)]
+  pub fn scalar_mul(&self, scalar: &Fq) -> Self {
+    let mut res = Self {
+      x: Fp::zero(),
+      y: Fp::zero(),
+      is_infinity: true,
+      _p: Default::default(),
+    };
+
+    let bits = scalar.to_le_bits();
+    for i in (0..bits.len()).rev() {
+      res = res.double();
+      if bits[i] {
+        res = self.add(&res);
+      }
+    }
+    res
+  }
+}
+
+#[cfg(test)]
+#[allow(clippy::too_many_arguments)]
+mod fp {
+  use ff::PrimeField;
+
+  #[derive(PrimeField)]
+  #[PrimeFieldModulus = "28948022309329048855892746252171976963363056481941560715954676764349967630337"]
+  #[PrimeFieldGenerator = "5"]
+  #[PrimeFieldReprEndianness = "little"]
+  pub struct Fp([u64; 4]);
+}
+
+#[cfg(test)]
+#[allow(clippy::too_many_arguments)]
+mod fq {
+  use ff::PrimeField;
+
+  #[derive(PrimeField)]
+  #[PrimeFieldModulus = "28948022309329048855892746252171976963363056481941647379679742748393362948097"]
+  #[PrimeFieldGenerator = "5"]
+  #[PrimeFieldReprEndianness = "little"]
+  pub struct Fq([u64; 4]);
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+  use super::{fp::Fp, fq::Fq};
+  use ff::Field;
+  use pasta_curves::arithmetic::CurveAffine;
+  use pasta_curves::group::Curve;
+  use pasta_curves::EpAffine;
+  use std::ops::Mul;
+
+  #[test]
+  fn test_ecc_ops() {
+    // perform some curve arithmetic
+    let a = Point::<Fp, Fq>::random_vartime();
+    let b = Point::<Fp, Fq>::random_vartime();
+    let c = a.add(&b);
+    let d = a.double();
+    let s = Fq::random(&mut OsRng);
+    let e = a.scalar_mul(&s);
+
+    // perform the same computation by translating to pasta_curve types
+    let a_pasta = EpAffine::from_xy(
+      pasta_curves::Fp::from_repr(a.x.to_repr().0).unwrap(),
+      pasta_curves::Fp::from_repr(a.y.to_repr().0).unwrap(),
+    )
+    .unwrap();
+    let b_pasta = EpAffine::from_xy(
+      pasta_curves::Fp::from_repr(b.x.to_repr().0).unwrap(),
+      pasta_curves::Fp::from_repr(b.y.to_repr().0).unwrap(),
+    )
+    .unwrap();
+    let c_pasta = (a_pasta + b_pasta).to_affine();
+    let d_pasta = (a_pasta + a_pasta).to_affine();
+    let e_pasta = a_pasta
+      .mul(pasta_curves::Fq::from_repr(s.to_repr().0).unwrap())
+      .to_affine();
+
+    // transform c, d, and e into pasta_curve types
+    let c_pasta_2 = EpAffine::from_xy(
+      pasta_curves::Fp::from_repr(c.x.to_repr().0).unwrap(),
+      pasta_curves::Fp::from_repr(c.y.to_repr().0).unwrap(),
+    )
+    .unwrap();
+    let d_pasta_2 = EpAffine::from_xy(
+      pasta_curves::Fp::from_repr(d.x.to_repr().0).unwrap(),
+      pasta_curves::Fp::from_repr(d.y.to_repr().0).unwrap(),
+    )
+    .unwrap();
+    let e_pasta_2 = EpAffine::from_xy(
+      pasta_curves::Fp::from_repr(e.x.to_repr().0).unwrap(),
+      pasta_curves::Fp::from_repr(e.y.to_repr().0).unwrap(),
+    )
+    .unwrap();
+
+    // check that we have the same outputs
+    assert_eq!(c_pasta, c_pasta_2);
+    assert_eq!(d_pasta, d_pasta_2);
+    assert_eq!(e_pasta, e_pasta_2);
+  }
+}

src/gadgets/ecc_circuit.rs

@@ -0,0 +1,552 @@
+#![allow(non_snake_case)]
+use crate::gadgets::utils::{
+  alloc_one, alloc_zero, conditionally_select, conditionally_select2, select_one_or, select_zero_or,
+};
+use bellperson::{
+  gadgets::{
+    boolean::{AllocatedBit, Boolean},
+    num::AllocatedNum,
+    Assignment,
+  },
+  ConstraintSystem, SynthesisError,
+};
+use ff::PrimeField;
+use rand::rngs::OsRng;
+
+#[derive(Clone)]
+pub struct AllocatedPoint<Fp>
+where
+  Fp: PrimeField,
+{
+  pub(crate) x: AllocatedNum<Fp>,
+  pub(crate) y: AllocatedNum<Fp>,
+  pub(crate) is_infinity: AllocatedNum<Fp>,
+}
+
+impl<Fp> AllocatedPoint<Fp>
+where
+  Fp: PrimeField,
+{
+  // Creates a new allocated point from allocated nums.
+  pub fn new(x: AllocatedNum<Fp>, y: AllocatedNum<Fp>, is_infinity: AllocatedNum<Fp>) -> Self {
+    Self { x, y, is_infinity }
+  }
+
+  // Check that is infinity is 0/1
+  #[allow(dead_code)]
+  pub fn check_is_infinity<CS: ConstraintSystem<Fp>>(
+    &self,
+    mut cs: CS,
+  ) -> Result<(), SynthesisError> {
+    // Check that is_infinity * ( 1 - is_infinity ) = 0
+    cs.enforce(
+      || "is_infinity is bit",
+      |lc| lc + self.is_infinity.get_variable(),
+      |lc| lc + CS::one() - self.is_infinity.get_variable(),
+      |lc| lc,
+    );
+    Ok(())
+  }
+
+  #[allow(dead_code)]
+  // Allocate a random point. Only used for testing
+  pub fn random_vartime<CS: ConstraintSystem<Fp>>(mut cs: CS) -> Result<Self, SynthesisError> {
+    loop {
+      let x = Fp::random(&mut OsRng);
+      let y = (x * x * x + Fp::one() + Fp::one() + Fp::one() + Fp::one() + Fp::one()).sqrt();
+      if y.is_some().unwrap_u8() == 1 {
+        let x_alloc = AllocatedNum::alloc(cs.namespace(|| "x"), || Ok(x))?;
+        let y_alloc = AllocatedNum::alloc(cs.namespace(|| "y"), || Ok(y.unwrap()))?;
+        let is_infinity = alloc_zero(cs.namespace(|| "Is Infinity"))?;
+        return Ok(Self::new(x_alloc, y_alloc, is_infinity));
+      }
+    }
+  }
+
+  // Make the point io
+  #[allow(dead_code)]
+  pub fn inputize<CS: ConstraintSystem<Fp>>(&self, mut cs: CS) -> Result<(), SynthesisError> {
+    let _ = self.x.inputize(cs.namespace(|| "Input point.x"));
+    let _ = self.y.inputize(cs.namespace(|| "Input point.y"));
+    let _ = self
+      .is_infinity
+      .inputize(cs.namespace(|| "Input point.is_infinity"));
+    Ok(())
+  }
+
+  // Adds other point to this point and returns the result
+  // Assumes that both other.is_infinity and this.is_infinty are bits
+  pub fn add<CS: ConstraintSystem<Fp>>(
+    &self,
+    mut cs: CS,
+    other: &AllocatedPoint<Fp>,
+  ) -> Result<Self, SynthesisError> {
+    // Allocate the boolean variables that check if either of the points is infinity
+
+    //************************************************************************/
+    // lambda = (other.y - self.y) * (other.x - self.x).invert().unwrap();
+    //************************************************************************/
+    // First compute (other.x - self.x).inverse()
+    // If either self or other are 1 then compute bogus values
+
+    // x_diff = other != inf && self != inf ? (other.x - self.x) : 1
+    let x_diff_actual = AllocatedNum::alloc(cs.namespace(|| "actual x diff"), || {
+      Ok(*other.x.get_value().get()? - *self.x.get_value().get()?)
+    })?;
+    cs.enforce(
+      || "actual x_diff is correct",
+      |lc| lc + other.x.get_variable() - self.x.get_variable(),
+      |lc| lc + CS::one(),
+      |lc| lc + x_diff_actual.get_variable(),
+    );
+
+    // Compute self.is_infinity OR other.is_infinity
+    let at_least_one_inf = AllocatedNum::alloc(cs.namespace(|| "at least one inf"), || {
+      Ok(*self.is_infinity.get_value().get()? * *other.is_infinity.get_value().get()?)
+    })?;
+    cs.enforce(
+      || "at least one inf = self.is_infinity * other.is_infinity",
+      |lc| lc + self.is_infinity.get_variable(),
+      |lc| lc + other.is_infinity.get_variable(),
+      |lc| lc + at_least_one_inf.get_variable(),
+    );
+
+    // x_diff = 1 if either self.is_infinity or other.is_infinity else x_diff_actual
+    let x_diff = select_one_or(
+      cs.namespace(|| "Compute x_diff"),
+      &x_diff_actual,
+      &at_least_one_inf,
+    )?;
+
+    let x_diff_inv = AllocatedNum::alloc(cs.namespace(|| "x diff inverse"), || {
+      if *at_least_one_inf.get_value().get()? == Fp::one() {
+        // Set to default
+        Ok(Fp::one())
+      } else {
+        // Set to the actual inverse
+        let inv = (*other.x.get_value().get()? - *self.x.get_value().get()?).invert();
+        if inv.is_some().unwrap_u8() == 1 {
+          Ok(inv.unwrap())
+        } else {
+          Err(SynthesisError::DivisionByZero)
+        }
+      }
+    })?;
+
+    cs.enforce(
+      || "Check inverse",
+      |lc| lc + x_diff.get_variable(),
+      |lc| lc + x_diff_inv.get_variable(),
+      |lc| lc + CS::one(),
+    );
+
+    let lambda = AllocatedNum::alloc(cs.namespace(|| "lambda"), || {
+      Ok(
+        (*other.y.get_value().get()? - *self.y.get_value().get()?)
+          * x_diff_inv.get_value().get()?,
+      )
+    })?;
+    cs.enforce(
+      || "Check that lambda is correct",
+      |lc| lc + other.y.get_variable() - self.y.get_variable(),
+      |lc| lc + x_diff_inv.get_variable(),
+      |lc| lc + lambda.get_variable(),
+    );
+
+    //************************************************************************/
+    // x = lambda * lambda - self.x - other.x;
+    //************************************************************************/
+    let x = AllocatedNum::alloc(cs.namespace(|| "x"), || {
+      Ok(
+        *lambda.get_value().get()? * lambda.get_value().get()?
+          - *self.x.get_value().get()?
+          - *other.x.get_value().get()?,
+      )
+    })?;
+    cs.enforce(
+      || "check that x is correct",
+      |lc| lc + lambda.get_variable(),
+      |lc| lc + lambda.get_variable(),
+      |lc| lc + x.get_variable() + self.x.get_variable() + other.x.get_variable(),
+    );
+
+    //************************************************************************/
+    // y = lambda * (self.x - x) - self.y;
+    //************************************************************************/
+    let y = AllocatedNum::alloc(cs.namespace(|| "y"), || {
+      Ok(
+        *lambda.get_value().get()? * (*self.x.get_value().get()? - *x.get_value().get()?)
+          - *self.y.get_value().get()?,
+      )
+    })?;
+
+    cs.enforce(
+      || "Check that y is correct",
+      |lc| lc + lambda.get_variable(),
+      |lc| lc + self.x.get_variable() - x.get_variable(),
+      |lc| lc + y.get_variable() + self.y.get_variable(),
+    );
+
+    let is_infinity = AllocatedNum::alloc(cs.namespace(|| "is infinity"), || Ok(Fp::zero()))?;
+
+    //************************************************************************/
+    // We only return the computed x, y if neither of the points is infinity.
+    // if self.is_infinity return other.clone()
+    // elif other.is_infinity return self.clone()
+    // Otherwise return the computed points.
+    //************************************************************************/
+    // Now compute the output x
+    let inner_x = conditionally_select2(
+      cs.namespace(|| "final x: inner if"),
+      &self.x,
+      &x,
+      &other.is_infinity,
+    )?;
+    let final_x = conditionally_select2(
+      cs.namespace(|| "final x: outer if"),
+      &other.x,
+      &inner_x,
+      &self.is_infinity,
+    )?;
+
+    // The output y
+    let inner_y = conditionally_select2(
+      cs.namespace(|| "final y: inner if"),
+      &self.y,
+      &y,
+      &other.is_infinity,
+    )?;
+    let final_y = conditionally_select2(
+      cs.namespace(|| "final y: outer if"),
+      &other.y,
+      &inner_y,
+      &self.is_infinity,
+    )?;
+
+    // The output is_infinity
+    let inner_is_infinity = conditionally_select2(
+      cs.namespace(|| "final is infinity: inner if"),
+      &self.is_infinity,
+      &is_infinity,
+      &other.is_infinity,
+    )?;
+    let final_is_infinity = conditionally_select2(
+      cs.namespace(|| "final is infinity: outer if"),
+      &other.is_infinity,
+      &inner_is_infinity,
+      &self.is_infinity,
+    )?;
+    Ok(Self::new(final_x, final_y, final_is_infinity))
+  }
+
+  pub fn double<CS: ConstraintSystem<Fp>>(&self, mut cs: CS) -> Result<Self, SynthesisError> {
+    //*************************************************************/
+    // lambda = (Fp::one() + Fp::one() + Fp::one())
+    //  * self.x
+    //  * self.x
+    //  * ((Fp::one() + Fp::one()) * self.y).invert().unwrap();
+    /*************************************************************/
+
+    // Compute tmp = (Fp::one() + Fp::one())* self.y ? self != inf : 1
+    let tmp_actual = AllocatedNum::alloc(cs.namespace(|| "tmp_actual"), || {
+      Ok(*self.y.get_value().get()? + *self.y.get_value().get()?)
+    })?;
+    cs.enforce(
+      || "check tmp_actual",
+      |lc| lc + CS::one() + CS::one(),
+      |lc| lc + self.y.get_variable(),
+      |lc| lc + tmp_actual.get_variable(),
+    );
+
+    let tmp = select_one_or(cs.namespace(|| "tmp"), &tmp_actual, &self.is_infinity)?;
+
+    // Compute inv = tmp.invert
+    let tmp_inv = AllocatedNum::alloc(cs.namespace(|| "tmp inverse"), || {
+      if *self.is_infinity.get_value().get()? == Fp::one() {
+        // Return default value 1
+        Ok(Fp::one())
+      } else {
+        // Return the actual inverse
+        let inv = (*tmp.get_value().get()?).invert();
+        if inv.is_some().unwrap_u8() == 1 {
+          Ok(inv.unwrap())
+        } else {
+          Err(SynthesisError::DivisionByZero)
+        }
+      }
+    })?;
+    cs.enforce(
+      || "Check inverse",
+      |lc| lc + tmp.get_variable(),
+      |lc| lc + tmp_inv.get_variable(),
+      |lc| lc + CS::one(),
+    );
+
+    // Now compute lambda as (Fp::one() + Fp::one + Fp::one()) * self.x * self.x * tmp_inv
+    let prod_1 = AllocatedNum::alloc(cs.namespace(|| "alloc prod 1"), || {
+      Ok(*tmp_inv.get_value().get()? * self.x.get_value().get()?)
+    })?;
+    cs.enforce(
+      || "Check prod 1",
+      |lc| lc + self.x.get_variable(),
+      |lc| lc + tmp_inv.get_variable(),
+      |lc| lc + prod_1.get_variable(),
+    );
+
+    let prod_2 = AllocatedNum::alloc(cs.namespace(|| "alloc prod 2"), || {
+      Ok(*prod_1.get_value().get()? * self.x.get_value().get()?)
+    })?;
+    cs.enforce(
+      || "Check prod 2",
+      |lc| lc + self.x.get_variable(),
+      |lc| lc + prod_1.get_variable(),
+      |lc| lc + prod_2.get_variable(),
+    );
+
+    let lambda = AllocatedNum::alloc(cs.namespace(|| "lambda"), || {
+      Ok(*prod_2.get_value().get()? * (Fp::one() + Fp::one() + Fp::one()))
+    })?;
+    cs.enforce(
+      || "Check lambda",
+      |lc| lc + CS::one() + CS::one() + CS::one(),
+      |lc| lc + prod_2.get_variable(),
+      |lc| lc + lambda.get_variable(),
+    );
+
+    /*************************************************************/
+    //          x = lambda * lambda - self.x - self.x;
+    /*************************************************************/
+
+    let x = AllocatedNum::alloc(cs.namespace(|| "x"), || {
+      Ok(
+        ((*lambda.get_value().get()?) * (*lambda.get_value().get()?))
+          - *self.x.get_value().get()?
+          - self.x.get_value().get()?,
+      )
+    })?;
+    cs.enforce(
+      || "Check x",
+      |lc| lc + lambda.get_variable(),
+      |lc| lc + lambda.get_variable(),
+      |lc| lc + x.get_variable() + self.x.get_variable() + self.x.get_variable(),
+    );
+
+    /*************************************************************/
+    //        y = lambda * (self.x - x) - self.y;
+    /*************************************************************/
+
+    let y = AllocatedNum::alloc(cs.namespace(|| "y"), || {
+      Ok(
+        (*lambda.get_value().get()?) * (*self.x.get_value().get()? - x.get_value().get()?)
+          - self.y.get_value().get()?,
+      )
+    })?;
+    cs.enforce(
+      || "Check y",
+      |lc| lc + lambda.get_variable(),
+      |lc| lc + self.x.get_variable() - x.get_variable(),
+      |lc| lc + y.get_variable() + self.y.get_variable(),
+    );
+
+    /*************************************************************/
+    // Only return the computed x and y if the point is not infinity
+    /*************************************************************/
+
+    // x
+    let final_x = select_zero_or(cs.namespace(|| "final x"), &x, &self.is_infinity)?;
+
+    // y
+    let final_y = select_zero_or(cs.namespace(|| "final y"), &y, &self.is_infinity)?;
+
+    // is_infinity
+    let final_is_infinity = self.is_infinity.clone();
+
+    Ok(Self::new(final_x, final_y, final_is_infinity))
+  }
+
+  #[allow(dead_code)]
+  pub fn scalar_mul_mont<CS: ConstraintSystem<Fp>>(
+    &self,
+    mut cs: CS,
+    scalar: Vec<AllocatedBit>,
+  ) -> Result<Self, SynthesisError> {
+    /*************************************************************/
+    // Initialize RO = Self {
+    //  x: Fp::zero(),
+    //  y: Fp::zero(),
+    //  is_infinity: true,
+    //  _p: Default::default(),
+    //};
+    /*************************************************************/
+
+    let zero = alloc_zero(cs.namespace(|| "Allocate zero"))?;
+    let one = alloc_one(cs.namespace(|| "Allocate one"))?;
+    let mut R0 = Self::new(zero.clone(), zero, one);
+
+    /*************************************************************/
+    // Initialize R1 and the bits of the scalar
+    /*************************************************************/
+
+    let mut R1 = self.clone();
+
+    for i in (0..scalar.len()).rev() {
+      /*************************************************************/
+      //if bits[i] {
+      //  R0 = R0.add(&R1);
+      //  R1 = R1.double();
+      //} else {
+      //  R0 = R0.double();
+      //  R1 = R0.add(&R1);
+      //}
+      /*************************************************************/
+
+      let R0_and_R1 = R0.add(cs.namespace(|| format!("{}: R0 + R1", i)), &R1)?;
+      let R0_double = R0.double(cs.namespace(|| format!("{}: 2 * R0", i)))?;
+      let R1_double = R1.double(cs.namespace(|| format!("{}: 2 * R1", i)))?;
+
+      R0 = Self::conditionally_select(
+        cs.namespace(|| format!("{}: Update R0", i)),
+        &R0_and_R1,
+        &R0_double,
+        &Boolean::from(scalar[i].clone()),
+      )?;
+
+      R1 = Self::conditionally_select(
+        cs.namespace(|| format!("{}: Update R1", i)),
+        &R1_double,
+        &R0_and_R1,
+        &Boolean::from(scalar[i].clone()),
+      )?;
+    }
+    Ok(R0)
+  }
+
+  #[allow(dead_code)]
+  pub fn scalar_mul<CS: ConstraintSystem<Fp>>(
+    &self,
+    mut cs: CS,
+    scalar: Vec<AllocatedBit>,
+  ) -> Result<Self, SynthesisError> {
+    /*************************************************************/
+    // Initialize res = Self {
+    //  x: Fp::zero(),
+    //  y: Fp::zero(),
+    //  is_infinity: true,
+    //  _p: Default::default(),
+    //};
+    /*************************************************************/
+
+    let zero = alloc_zero(cs.namespace(|| "Allocate zero"))?;
+    let one = alloc_one(cs.namespace(|| "Allocate one"))?;
+    let mut res = Self::new(zero.clone(), zero, one);
+
+    for i in (0..scalar.len()).rev() {
+      /*************************************************************/
+      //  res = res.double();
+      /*************************************************************/
+
+      res = res.double(cs.namespace(|| format!("{}: double", i)))?;
+
+      /*************************************************************/
+      //  if scalar[i] {
+      //    res = self.add(&res);
+      //  }
+      /*************************************************************/
+
+      let self_and_res = self.add(cs.namespace(|| format!("{}: add", i)), &res)?;
+      res = Self::conditionally_select(
+        cs.namespace(|| format!("{}: Update res", i)),
+        &self_and_res,
+        &res,
+        &Boolean::from(scalar[i].clone()),
+      )?;
+    }
+    Ok(res)
+  }
+
+  /// If condition outputs a otherwise outputs b
+  pub fn conditionally_select<CS: ConstraintSystem<Fp>>(
+    mut cs: CS,
+    a: &Self,
+    b: &Self,
+    condition: &Boolean,
+  ) -> Result<Self, SynthesisError> {
+    let x = conditionally_select(cs.namespace(|| "select x"), &a.x, &b.x, condition)?;
+
+    let y = conditionally_select(cs.namespace(|| "select y"), &a.y, &b.y, condition)?;
+
+    let is_infinity = conditionally_select(
+      cs.namespace(|| "select is_infinity"),
+      &a.is_infinity,
+      &b.is_infinity,
+      condition,
+    )?;
+
+    Ok(Self::new(x, y, is_infinity))
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use super::*;
+  use crate::bellperson::shape_cs::ShapeCS;
+  use crate::bellperson::solver::SatisfyingAssignment;
+  type G = pasta_curves::pallas::Point;
+  type Fp = pasta_curves::pallas::Scalar;
+  type Fq = pasta_curves::vesta::Scalar;
+  use crate::bellperson::r1cs::{NovaShape, NovaWitness};
+  use crate::gadgets::ecc::Point;
+  use ff::PrimeFieldBits;
+
+  fn synthesize_smul<Fp, Fq, CS>(mut cs: CS) -> (AllocatedPoint<Fp>, AllocatedPoint<Fp>, Fq)
+  where
+    Fp: PrimeField,
+    Fq: PrimeField + PrimeFieldBits,
+    CS: ConstraintSystem<Fp>,
+  {
+    let a = AllocatedPoint::<Fp>::random_vartime(cs.namespace(|| "a")).unwrap();
+    let _ = a.inputize(cs.namespace(|| "inputize a")).unwrap();
+    let s = Fq::random(&mut OsRng);
+    // Allocate random bits and only keep 128 bits
+    let bits: Vec<AllocatedBit> = s
+      .to_le_bits()
+      .into_iter()
+      .enumerate()
+      .map(|(i, bit)| AllocatedBit::alloc(cs.namespace(|| format!("bit {}", i)), Some(bit)))
+      .collect::<Result<Vec<AllocatedBit>, SynthesisError>>()
+      .unwrap();
+    let e = a.scalar_mul(cs.namespace(|| "Scalar Mul"), bits).unwrap();
+    let _ = e.inputize(cs.namespace(|| "inputize e")).unwrap();
+    (a, e, s)
+  }
+
+  #[test]
+  fn test_ecc_circuit_ops() {
+    // First create the shape
+    let mut cs: ShapeCS<G> = ShapeCS::new();
+    let _ = synthesize_smul::<Fp, Fq, _>(cs.namespace(|| "synthesize"));
+    println!("Number of constraints: {}", cs.num_constraints());
+    let shape = cs.r1cs_shape();
+    let gens = cs.r1cs_gens();
+
+    // Then the satisfying assignment
+    let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
+    let (a, e, s) = synthesize_smul::<Fp, Fq, _>(cs.namespace(|| "synthesize"));
+    let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
+
+    let a_p: Point<Fp, Fq> = Point::new(
+      a.x.get_value().unwrap(),
+      a.y.get_value().unwrap(),
+      a.is_infinity.get_value().unwrap() == Fp::one(),
+    );
+    let e_p: Point<Fp, Fq> = Point::new(
+      e.x.get_value().unwrap(),
+      e.y.get_value().unwrap(),
+      e.is_infinity.get_value().unwrap() == Fp::one(),
+    );
+    let e_new = a_p.scalar_mul(&s);
+    assert!(e_p.x == e_new.x && e_p.y == e_new.y);
+    // Make sure that this is satisfiable
+    assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
+  }
+}

src/gadgets/mod.rs

@@ -0,0 +1,3 @@
+mod ecc;
+pub mod ecc_circuit;
+pub mod utils;

src/gadgets/utils.rs

@@ -0,0 +1,338 @@
+use bellperson::{
+  gadgets::{
+    boolean::{AllocatedBit, Boolean},
+    num::AllocatedNum,
+    Assignment,
+  },
+  ConstraintSystem, LinearCombination, SynthesisError,
+};
+use bellperson_nonnative::mp::bignat::{nat_to_limbs, BigNat};
+use ff::{PrimeField, PrimeFieldBits};
+use rug::Integer;
+
+/// Gets as input the little indian representation of a number and spits out the number
+#[allow(dead_code)]
+pub fn le_bits_to_num<Scalar, CS>(
+  mut cs: CS,
+  bits: Vec<AllocatedBit>,
+) -> Result<AllocatedNum<Scalar>, SynthesisError>
+where
+  Scalar: PrimeField + PrimeFieldBits,
+  CS: ConstraintSystem<Scalar>,
+{
+  // We loop over the input bits and construct the constraint
+  // and the field element that corresponds to the result
+  let mut lc = LinearCombination::zero();
+  let mut coeff = Scalar::one();
+  let mut fe = Some(Scalar::zero());
+  for bit in bits.iter() {
+    lc = lc + (coeff, bit.get_variable());
+    fe = bit.get_value().map(|val| {
+      if val {
+        fe.unwrap() + coeff
+      } else {
+        fe.unwrap()
+      }
+    });
+    coeff = coeff.double();
+  }
+  let num = AllocatedNum::alloc(cs.namespace(|| "Field element"), || {
+    fe.ok_or(SynthesisError::AssignmentMissing)
+  })?;
+  lc = lc - num.get_variable();
+  cs.enforce(|| "compute number from bits", |lc| lc, |lc| lc, |_| lc);
+  Ok(num)
+}
+
+/// Allocate a variable that is set to zero
+pub fn alloc_zero<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+) -> Result<AllocatedNum<F>, SynthesisError> {
+  let zero = AllocatedNum::alloc(cs.namespace(|| "alloc"), || Ok(F::zero()))?;
+  cs.enforce(
+    || "check zero is valid",
+    |lc| lc,
+    |lc| lc,
+    |lc| lc + zero.get_variable(),
+  );
+  Ok(zero)
+}
+
+/// Allocate a variable that is set to one
+pub fn alloc_one<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+) -> Result<AllocatedNum<F>, SynthesisError> {
+  let one = AllocatedNum::alloc(cs.namespace(|| "alloc"), || Ok(F::one()))?;
+  cs.enforce(
+    || "check one is valid",
+    |lc| lc + CS::one(),
+    |lc| lc + CS::one(),
+    |lc| lc + one.get_variable(),
+  );
+
+  Ok(one)
+}
+
+/// Allocate bignat a constant
+pub fn alloc_bignat_constant<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  val: &Integer,
+  limb_width: usize,
+  n_limbs: usize,
+) -> Result<BigNat<F>, SynthesisError> {
+  let limbs = nat_to_limbs(val, limb_width, n_limbs).unwrap();
+  let bignat = BigNat::alloc_from_limbs(
+    cs.namespace(|| "alloc bignat"),
+    || Ok(limbs.clone()),
+    None,
+    limb_width,
+    n_limbs,
+  )?;
+  // Now enforce that the limbs are all equal to the constants
+  #[allow(clippy::needless_range_loop)]
+  for i in 0..n_limbs {
+    cs.enforce(
+      || format!("check limb {}", i),
+      |lc| lc + &bignat.limbs[i],
+      |lc| lc + CS::one(),
+      |lc| lc + (limbs[i], CS::one()),
+    );
+  }
+  Ok(bignat)
+}
+
+/// Check that two numbers are equal and return a bit
+pub fn alloc_num_equals<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  a: AllocatedNum<F>,
+  b: AllocatedNum<F>,
+) -> Result<AllocatedBit, SynthesisError> {
+  // Allocate and constrain `r`: result boolean bit.
+  // It equals `true` if `a` equals `b`, `false` otherwise
+
+  let r_value = match (a.get_value(), b.get_value()) {
+    (Some(a), Some(b)) => Some(a == b),
+    _ => None,
+  };
+
+  let r = AllocatedBit::alloc(cs.namespace(|| "r"), r_value)?;
+
+  let delta = AllocatedNum::alloc(cs.namespace(|| "delta"), || {
+    let a_value = *a.get_value().get()?;
+    let b_value = *b.get_value().get()?;
+
+    let mut delta = a_value;
+    delta.sub_assign(&b_value);
+
+    Ok(delta)
+  })?;
+
+  cs.enforce(
+    || "delta = (a - b)",
+    |lc| lc + a.get_variable() - b.get_variable(),
+    |lc| lc + CS::one(),
+    |lc| lc + delta.get_variable(),
+  );
+
+  let delta_inv = AllocatedNum::alloc(cs.namespace(|| "delta_inv"), || {
+    let delta = *delta.get_value().get()?;
+
+    if delta.is_zero().unwrap_u8() == 1 {
+      Ok(F::one()) // we can return any number here, it doesn't matter
+    } else {
+      Ok(delta.invert().unwrap())
+    }
+  })?;
+
+  // Allocate `t = delta * delta_inv`
+  // If `delta` is non-zero (a != b), `t` will equal 1
+  // If `delta` is zero (a == b), `t` cannot equal 1
+
+  let t = AllocatedNum::alloc(cs.namespace(|| "t"), || {
+    let mut tmp = *delta.get_value().get()?;
+    tmp.mul_assign(&(*delta_inv.get_value().get()?));
+
+    Ok(tmp)
+  })?;
+
+  // Constrain allocation:
+  // t = (a - b) * delta_inv
+  cs.enforce(
+    || "t = (a - b) * delta_inv",
+    |lc| lc + a.get_variable() - b.get_variable(),
+    |lc| lc + delta_inv.get_variable(),
+    |lc| lc + t.get_variable(),
+  );
+
+  // Constrain:
+  // (a - b) * (t - 1) == 0
+  // This enforces that correct `delta_inv` was provided,
+  // and thus `t` is 1 if `(a - b)` is non zero (a != b )
+  cs.enforce(
+    || "(a - b) * (t - 1) == 0",
+    |lc| lc + a.get_variable() - b.get_variable(),
+    |lc| lc + t.get_variable() - CS::one(),
+    |lc| lc,
+  );
+
+  // Constrain:
+  // (a - b) * r == 0
+  // This enforces that `r` is zero if `(a - b)` is non-zero (a != b)
+  cs.enforce(
+    || "(a - b) * r == 0",
+    |lc| lc + a.get_variable() - b.get_variable(),
+    |lc| lc + r.get_variable(),
+    |lc| lc,
+  );
+
+  // Constrain:
+  // (t - 1) * (r - 1) == 0
+  // This enforces that `r` is one if `t` is not one (a == b)
+  cs.enforce(
+    || "(t - 1) * (r - 1) == 0",
+    |lc| lc + t.get_variable() - CS::one(),
+    |lc| lc + r.get_variable() - CS::one(),
+    |lc| lc,
+  );
+
+  Ok(r)
+}
+
+/// If condition return a otherwise b
+pub fn conditionally_select<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  a: &AllocatedNum<F>,
+  b: &AllocatedNum<F>,
+  condition: &Boolean,
+) -> Result<AllocatedNum<F>, SynthesisError> {
+  let c = AllocatedNum::alloc(cs.namespace(|| "conditional select result"), || {
+    if *condition.get_value().get()? {
+      Ok(*a.get_value().get()?)
+    } else {
+      Ok(*b.get_value().get()?)
+    }
+  })?;
+
+  // a * condition + b*(1-condition) = c ->
+  // a * condition - b*condition = c - b
+
+  cs.enforce(
+    || "conditional select constraint",
+    |lc| lc + a.get_variable() - b.get_variable(),
+    |_| condition.lc(CS::one(), F::one()),
+    |lc| lc + c.get_variable() - b.get_variable(),
+  );
+
+  Ok(c)
+}
+
+/// If condition return a otherwise b where a and b are BigNats
+pub fn conditionally_select_bignat<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  a: &BigNat<F>,
+  b: &BigNat<F>,
+  condition: &Boolean,
+) -> Result<BigNat<F>, SynthesisError> {
+  assert!(a.limbs.len() == b.limbs.len());
+  let c = BigNat::alloc_from_nat(
+    cs.namespace(|| "conditional select result"),
+    || {
+      if *condition.get_value().get()? {
+        Ok(a.value.get()?.clone())
+      } else {
+        Ok(b.value.get()?.clone())
+      }
+    },
+    a.params.limb_width,
+    a.params.n_limbs,
+  )?;
+
+  // a * condition + b*(1-condition) = c ->
+  // a * condition - b*condition = c - b
+  for i in 0..c.limbs.len() {
+    cs.enforce(
+      || format!("conditional select constraint {}", i),
+      |lc| lc + &a.limbs[i] - &b.limbs[i],
+      |_| condition.lc(CS::one(), F::one()),
+      |lc| lc + &c.limbs[i] - &b.limbs[i],
+    );
+  }
+  Ok(c)
+}
+
+/// Same as the above but Condition is an AllocatedNum that needs to be
+/// 0 or 1. 1 => True, 0 => False
+pub fn conditionally_select2<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  a: &AllocatedNum<F>,
+  b: &AllocatedNum<F>,
+  condition: &AllocatedNum<F>,
+) -> Result<AllocatedNum<F>, SynthesisError> {
+  let c = AllocatedNum::alloc(cs.namespace(|| "conditional select result"), || {
+    if *condition.get_value().get()? == F::one() {
+      Ok(*a.get_value().get()?)
+    } else {
+      Ok(*b.get_value().get()?)
+    }
+  })?;
+
+  // a * condition + b*(1-condition) = c ->
+  // a * condition - b*condition = c - b
+
+  cs.enforce(
+    || "conditional select constraint",
+    |lc| lc + a.get_variable() - b.get_variable(),
+    |lc| lc + condition.get_variable(),
+    |lc| lc + c.get_variable() - b.get_variable(),
+  );
+
+  Ok(c)
+}
+
+/// If condition set to 0 otherwise a
+pub fn select_zero_or<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  a: &AllocatedNum<F>,
+  condition: &AllocatedNum<F>,
+) -> Result<AllocatedNum<F>, SynthesisError> {
+  let c = AllocatedNum::alloc(cs.namespace(|| "conditional select result"), || {
+    if *condition.get_value().get()? == F::one() {
+      Ok(F::zero())
+    } else {
+      Ok(*a.get_value().get()?)
+    }
+  })?;
+
+  // a * (1 - condition) = c
+  cs.enforce(
+    || "conditional select constraint",
+    |lc| lc + a.get_variable(),
+    |lc| lc + CS::one() - condition.get_variable(),
+    |lc| lc + c.get_variable(),
+  );
+
+  Ok(c)
+}
+
+/// If condition set to 1 otherwise a
+pub fn select_one_or<F: PrimeField, CS: ConstraintSystem<F>>(
+  mut cs: CS,
+  a: &AllocatedNum<F>,
+  condition: &AllocatedNum<F>,
+) -> Result<AllocatedNum<F>, SynthesisError> {
+  let c = AllocatedNum::alloc(cs.namespace(|| "conditional select result"), || {
+    if *condition.get_value().get()? == F::one() {
+      Ok(F::one())
+    } else {
+      Ok(*a.get_value().get()?)
+    }
+  })?;
+
+  cs.enforce(
+    || "conditional select constraint",
+    |lc| lc + CS::one() - a.get_variable(),
+    |lc| lc + condition.get_variable(),
+    |lc| lc + c.get_variable() - a.get_variable(),
+  );
+  Ok(c)
+}