@@ -7,9 +7,8 @@
 pub use errors::ArithErrors;
 pub use multilinear_polynomial::{
     evaluate_no_par, evaluate_opt, fix_last_variables, fix_last_variables_no_par, fix_variables,
-    identity_permutation_mle, identity_permutation_mles, merge_polynomials, random_mle_list,
-    random_permutation_mle, random_permutation_mles, random_zero_mle_list,
-    DenseMultilinearExtension,
+    identity_permutation_mles, merge_polynomials, random_mle_list, random_permutation_mles,
+    random_zero_mle_list, DenseMultilinearExtension,
 };
 pub use univariate_polynomial::{build_l, get_uni_domain};
 pub use util::{bit_decompose, gen_eval_point, get_batched_nv, get_index};

@@ -72,35 +72,6 @@ end_timer!(start);
     list
 }
 
-/// An MLE that represent an identity permutation: `f(index) \mapto index`
-/// TODO(binyi): remove
-pub fn identity_permutation_mle<F: PrimeField>(
-    num_vars: usize,
-) -> Rc<DenseMultilinearExtension<F>> {
-    let s_id_vec = (0..1u64 << num_vars).map(F::from).collect();
-    Rc::new(DenseMultilinearExtension::from_evaluations_vec(
-        num_vars, s_id_vec,
-    ))
-}
-
-/// An MLE that represent a random permutation
-/// TODO(binyi): remove
-pub fn random_permutation_mle<F: PrimeField, R: RngCore>(
-    num_vars: usize,
-    rng: &mut R,
-) -> Rc<DenseMultilinearExtension<F>> {
-    let len = 1u64 << num_vars;
-    let mut s_id_vec: Vec<F> = (0..len).map(F::from).collect();
-    let mut s_perm_vec = vec![];
-    for _ in 0..len {
-        let index = rng.next_u64() as usize % s_id_vec.len();
-        s_perm_vec.push(s_id_vec.remove(index));
-    }
-    Rc::new(DenseMultilinearExtension::from_evaluations_vec(
-        num_vars, s_perm_vec,
-    ))
-}
-
 /// A list of MLEs that represents an identity permutation
 pub fn identity_permutation_mles<F: PrimeField>(
     num_vars: usize,

@@ -1,4 +1,4 @@
-use arithmetic::identity_permutation_mle;
+use arithmetic::identity_permutation_mles;
 use ark_ff::PrimeField;
 use ark_poly::MultilinearExtension;
 use ark_std::{log2, test_rng};
@@ -93,7 +93,7 @@ num_pub_input: num_constraints,
             gate_func: gate.clone(),
         };
 
-        let permutation = identity_permutation_mle(merged_nv as usize).to_evaluations();
+        let permutation = identity_permutation_mles(merged_nv as usize, 1)[0].to_evaluations();
         let index = HyperPlonkIndex {
             params,
             permutation,

@@ -5,7 +5,7 @@ utils::{build_f, eval_f, prover_sanity_check, PcsAccumulator},
     witness::WitnessColumn,
     HyperPlonkSNARK,
 };
-use arithmetic::{evaluate_opt, identity_permutation_mle, merge_polynomials, VPAuxInfo};
+use arithmetic::{evaluate_opt, identity_permutation_mles, merge_polynomials, VPAuxInfo};
 use ark_ec::PairingEngine;
 use ark_poly::DenseMultilinearExtension;
 use ark_std::{end_timer, log2, start_timer, One, Zero};
@@ -526,8 +526,8 @@ // - alpha, beta, gamma from challenge
 
         // we evaluate MLE directly instead of using s_id/s_perm PCS verify
         // Verification takes n pairings while evaluate takes 2^n field ops.
-        let s_id = identity_permutation_mle::<E::Fr>(perm_check_point.len());
-        let s_id_eval = evaluate_opt(&s_id, perm_check_point);
+        let s_ids = identity_permutation_mles::<E::Fr>(perm_check_point.len(), 1);
+        let s_id_eval = evaluate_opt(&s_ids[0], perm_check_point);
         let s_perm_eval = evaluate_opt(&vk.permutation_oracle, perm_check_point);
 
         let q_x_rec = prod_evals[1] - prod_evals[2] * prod_evals[3]
@@ -629,7 +629,7 @@ use crate::{
         custom_gate::CustomizedGates, selectors::SelectorColumn, structs::HyperPlonkParams,
         witness::WitnessColumn,
     };
-    use arithmetic::random_permutation_mle;
+    use arithmetic::random_permutation_mles;
     use ark_bls12_381::Bls12_381;
     use ark_std::test_rng;
     use subroutines::pcs::prelude::MultilinearKzgPCS;
@@ -672,7 +672,9 @@ num_constraints,
             num_pub_input,
             gate_func,
         };
-        let permutation = identity_permutation_mle(merged_nv).evaluations.clone();
+        let permutation = identity_permutation_mles(merged_nv, 1)[0]
+            .evaluations
+            .clone();
         let q1 = SelectorColumn(vec![E::Fr::one(), E::Fr::one(), E::Fr::one(), E::Fr::one()]);
         let index = HyperPlonkIndex {
             params,
@@ -714,7 +716,7 @@ &vk, &pi.0, &proof,
         )?;
 
         // bad path 1: wrong permutation
-        let rand_perm: Vec<E::Fr> = random_permutation_mle(merged_nv, &mut rng)
+        let rand_perm: Vec<E::Fr> = random_permutation_mles(merged_nv, 1, &mut rng)[0]
             .evaluations
             .clone();
         let mut bad_index = index.clone();

@@ -1,4 +1,4 @@
-use arithmetic::{identity_permutation_mle, VPAuxInfo, VirtualPolynomial};
+use arithmetic::{identity_permutation_mles, VPAuxInfo, VirtualPolynomial};
 use ark_bls12_381::{Bls12_381, Fr};
 use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
 use ark_std::test_rng;
@@ -144,10 +144,10 @@ } else {
             10
         };
 
-        let w = Rc::new(DenseMultilinearExtension::rand(nv, &mut rng));
+        let ws = vec![Rc::new(DenseMultilinearExtension::rand(nv, &mut rng))];
 
-        // s_perm is the identity map
-        let s_perm = identity_permutation_mle(nv);
+        // identity map
+        let perms = identity_permutation_mles(nv, 1);
 
         let proof = {
             let start = Instant::now();
@@ -157,9 +157,9 @@ transcript.append_message(b"testing", b"initializing transcript for testing")?;
 
             let (proof, _q_x) = <PolyIOP<Fr> as PermutationCheck<Bls12_381, KZG>>::prove(
                 &pcs_param,
-                &w,
-                &w,
-                &s_perm,
+                &ws,
+                &ws,
+                &perms,
                 &mut transcript,
             )?;
 
@@ -218,20 +218,21 @@
         let f: DenseMultilinearExtension<Fr> = DenseMultilinearExtension::rand(nv, &mut rng);
         let mut g = f.clone();
         g.evaluations.reverse();
-        let f = Rc::new(f);
-        let g = Rc::new(g);
+        let fs = vec![Rc::new(f)];
+        let gs = vec![Rc::new(g)];
 
         let proof = {
             let start = Instant::now();
             let mut transcript = <PolyIOP<Fr> as ProductCheck<Bls12_381, KZG>>::init_transcript();
             transcript.append_message(b"testing", b"initializing transcript for testing")?;
 
-            let (proof, _prod_x) = <PolyIOP<Fr> as ProductCheck<Bls12_381, KZG>>::prove(
-                &pcs_param,
-                &f,
-                &g,
-                &mut transcript,
-            )?;
+            let (proof, _prod_x, _frac_poly) =
+                <PolyIOP<Fr> as ProductCheck<Bls12_381, KZG>>::prove(
+                    &pcs_param,
+                    &fs,
+                    &gs,
+                    &mut transcript,
+                )?;
 
             println!(
                 "product check proving time for {} variables: {} ns",