A gadget to compute the second term of the sum in section 5, step 5

src/folding/circuits/utils.rs

@@ -3,7 +3,7 @@ use ark_r1cs_std::fields::{fp::FpVar, FieldVar};
 use ark_relations::r1cs::SynthesisError;
 use std::marker::PhantomData;
 
-/// EqEval is a gadget for computing $\tilde{eq}(a, b) = \Pi_{i=1}^{l}(a_i \cdot b_i + (1 - a_i)(1 - b_i))$
+/// EqEval is a gadget for computing $\tilde{eq}(a, b) = \prod_{i=1}^{l}(a_i \cdot b_i + (1 - a_i)(1 - b_i))$
 /// :warning: This is not the ark_r1cs_std::eq::EqGadget
 pub struct EqEvalGadget<F: PrimeField> {
     _f: PhantomData<F>,

src/folding/hypernova/circuit.rs

@@ -15,13 +15,13 @@ pub struct SumMulsGammaPowsEqSigmaGadget<F: PrimeField> {
 }
 
 impl<F: PrimeField> SumMulsGammaPowsEqSigmaGadget<F> {
-    /// Computes the sum $\Sigma_{j}^{j + n} \gamma^{j} \cdot eq_eval \cdot \sigma_{j}$, where $n$ is the length of the `sigmas` vector
+    /// Computes the sum $\sum_{j}^{j + n} \gamma^{j} \cdot eq_eval \cdot \sigma_{j}$, where $n$ is the length of the `sigmas` vector
     /// It corresponds to the first term of the sum that $\mathcal{V}$ has to compute at section 5, step 5 of "A multi-folding scheme for CCS".
     ///
     /// # Arguments
     /// - `sigmas`: vector of $\sigma_j$ values
     /// - `eq_eval`: the value of $\tilde{eq}(x_j, x^{\prime})$
-    /// - `gamma`: a `GammaVar`, which supports a `pow` method, representing $\gamma$
+    /// - `gamma`: value $\gamma$
     /// - `j`: the power at which we start to compute $\gamma^{j}$. This is needed in the contexxt of multifolding.
     ///
     /// # Notes
@@ -42,15 +42,44 @@ impl<F: PrimeField> SumMulsGammaPowsEqSigmaGadget<F> {
     }
 }
 
+/// Gadget to compute $\sum_{i=1}^{q} c_i * \prod_{j \in S_i} theta_j$.
+pub struct SumCiMulProdThetajGadget<F: PrimeField> {
+    _f: PhantomData<F>,
+}
+
+impl<F: PrimeField> SumCiMulProdThetajGadget<F> {
+    /// Computes $\sum_{i=1}^{q} c_i * \prod_{j \in S_i} theta_j$
+    ///
+    /// # Arguments
+    /// - `c_i`: vector of $c_i$ values
+    /// - `thetas`: vector of pre-processed $\thetas[j]$ values corresponding to a particular `ccs.S[i]`
+    ///
+    /// # Notes
+    /// This is a part of the second term of the sum that $\mathcal{V}$ has to compute at section 5, step 5 of "A multi-folding scheme for CCS".
+    /// The first term is computed by `SumMulsGammaPowsEqSigmaGadget::sum_muls_gamma_pows_eq_sigma`.
+    /// This is a doct product between a vector of c_i values and a vector of pre-processed $\theta_j$ values, where $j$ is a value from $S_i$.
+    /// Hence, this requires some pre-processing of the $\theta_j$ values, before running this gadget.
+    pub fn sum_ci_mul_prod_thetaj(c_i: Vec<FpVar<F>>, thetas: Vec<Vec<FpVar<F>>>) -> FpVar<F> {
+        let mut result = FpVar::<F>::zero();
+        for (i, c_i) in c_i.iter().enumerate() {
+            let prod = &thetas[i].iter().fold(FpVar::one(), |acc, e| acc * e);
+            result += c_i * prod;
+        }
+        result
+    }
+}
+
 #[cfg(test)]
 mod tests {
-    use super::SumMulsGammaPowsEqSigmaGadget;
+    use super::{SumCiMulProdThetajGadget, SumMulsGammaPowsEqSigmaGadget};
     use crate::{
         ccs::{
             tests::{get_test_ccs, get_test_z},
             CCS,
         },
-        folding::hypernova::utils::{compute_sigmas_and_thetas, sum_muls_gamma_pows_eq_sigma},
+        folding::hypernova::utils::{
+            compute_sigmas_and_thetas, sum_ci_mul_prod_thetaj, sum_muls_gamma_pows_eq_sigma,
+        },
         pedersen::Pedersen,
         utils::virtual_polynomial::eq_eval,
     };
@@ -102,4 +131,44 @@ mod tests {
             assert_eq!(expected, computed.value().unwrap());
         }
     }
+
+    #[test]
+    pub fn test_sum_ci_mul_prod_thetaj_gadget() {
+        let mut rng = test_rng();
+        let ccs: CCS<Projective> = get_test_ccs();
+        let z1 = get_test_z(3);
+        let z2 = get_test_z(4);
+
+        let r_x_prime: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
+
+        // Initialize a multifolding object
+        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let (lcccs_instance, _) = ccs.to_lcccs(&mut rng, &pedersen_params, &z1).unwrap();
+        let sigmas_thetas =
+            compute_sigmas_and_thetas(&ccs, &[z1.clone()], &[z2.clone()], &r_x_prime);
+
+        let mut e_lcccs = Vec::new();
+        for r_x in &vec![lcccs_instance.r_x] {
+            e_lcccs.push(eq_eval(r_x, &r_x_prime).unwrap());
+        }
+
+        // Initialize cs
+        let cs = ConstraintSystem::<Fr>::new_ref();
+        let vec_thetas = sigmas_thetas.1;
+        for (_, thetas) in vec_thetas.iter().enumerate() {
+            // sum c_i * prod theta_j
+            let expected = sum_ci_mul_prod_thetaj(&ccs, thetas); // from `compute_c_from_sigmas_and_thetas`
+            let mut prepared_thetas = Vec::new();
+            for i in 0..ccs.q {
+                let prepared: Vec<Fr> = ccs.S[i].iter().map(|j| thetas[*j]).collect();
+                prepared_thetas
+                    .push(Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(prepared)).unwrap());
+            }
+            let computed = SumCiMulProdThetajGadget::sum_ci_mul_prod_thetaj(
+                Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(ccs.c.clone())).unwrap(),
+                prepared_thetas,
+            );
+            assert_eq!(expected, computed.value().unwrap());
+        }
+    }
 }

src/folding/hypernova/utils.rs

@@ -86,7 +86,7 @@ pub fn compute_sigmas_and_thetas<C: CurveGroup>(
     SigmasThetas(sigmas, thetas)
 }
 
-/// Computes the sum $\Sigma_{j = 0}^{n} \gamma^{\text{pow} + j} \cdot eq_eval \cdot \sigma_{j}$
+/// Computes the sum $\sum_{j = 0}^{n} \gamma^{\text{pow} + j} \cdot eq_eval \cdot \sigma_{j}$
 /// `pow` corresponds to `i * ccs.t` in `compute_c_from_sigmas_and_thetas`
 pub fn sum_muls_gamma_pows_eq_sigma<F: PrimeField>(
     gamma: F,
@@ -102,6 +102,22 @@ pub fn sum_muls_gamma_pows_eq_sigma<F: PrimeField>(
     result
 }
 
+/// Computes $\sum_{i=1}^{q} c_i * \prod_{j \in S_i} theta_j$
+pub fn sum_ci_mul_prod_thetaj<C: CurveGroup>(
+    ccs: &CCS<C>,
+    thetas: &[C::ScalarField],
+) -> C::ScalarField {
+    let mut result = C::ScalarField::zero();
+    for i in 0..ccs.q {
+        let mut prod = C::ScalarField::one();
+        for j in ccs.S[i].clone() {
+            prod *= thetas[j];
+        }
+        result += ccs.c[i] * prod;
+    }
+    result
+}
+
 /// Compute the right-hand-side of step 5 of the multifolding scheme
 pub fn compute_c_from_sigmas_and_thetas<C: CurveGroup>(
     ccs: &CCS<C>,
@@ -127,14 +143,7 @@ pub fn compute_c_from_sigmas_and_thetas<C: CurveGroup>(
     let e2 = eq_eval(beta, r_x_prime).unwrap();
     for (k, thetas) in vec_thetas.iter().enumerate() {
         // + gamma^{t+1} * e2 * sum c_i * prod theta_j
-        let mut lhs = C::ScalarField::zero();
-        for i in 0..ccs.q {
-            let mut prod = C::ScalarField::one();
-            for j in ccs.S[i].clone() {
-                prod *= thetas[j];
-            }
-            lhs += ccs.c[i] * prod;
-        }
+        let lhs = sum_ci_mul_prod_thetaj(ccs, thetas);
         let gamma_t1 = gamma.pow([(mu * ccs.t + k) as u64]);
         c += gamma_t1 * e2 * lhs;
     }