diff --git a/src/polynomial/commitment.rs b/src/polynomial/commitment.rs
index df92a31b..4b2bf872 100644
--- a/src/polynomial/commitment.rs
+++ b/src/polynomial/commitment.rs
@@ -4,7 +4,6 @@ use rayon::prelude::*;
 use crate::field::extension_field::Extendable;
 use crate::field::extension_field::{FieldExtension, Frobenius};
 use crate::field::field::Field;
-use crate::field::interpolation::interpolate2;
 use crate::fri::{prover::fri_proof, verifier::verify_fri_proof, FriConfig};
 use crate::merkle_tree::MerkleTree;
 use crate::plonk_challenger::Challenger;
@@ -139,7 +138,7 @@ impl<F: Field> ListPolynomialCommitment<F> {
         let zs_composition_poly = alpha.reduce_polys(zs_polys);
 
         let zs_quotient = Self::compute_quotient([zeta, g * zeta], zs_composition_poly);
-        final_poly = alpha.shift_poly(final_poly);
+        alpha.shift_poly(&mut final_poly);
         final_poly += zs_quotient;
 
         // When working in an extension field, need to check that wires are in the base field.
@@ -153,7 +152,7 @@ impl<F: Field> ListPolynomialCommitment<F> {
 
         let wires_quotient =
             Self::compute_quotient([zeta, zeta.frobenius()], wire_composition_poly);
-        final_poly = alpha.shift_poly(final_poly);
+        alpha.shift_poly(&mut final_poly);
         final_poly += wires_quotient;
 
         let lde_final_poly = final_poly.lde(config.rate_bits);
diff --git a/src/polynomial/polynomial.rs b/src/polynomial/polynomial.rs
index aefcc0c6..02f66684 100644
--- a/src/polynomial/polynomial.rs
+++ b/src/polynomial/polynomial.rs
@@ -1,6 +1,6 @@
 use std::cmp::max;
 use std::iter::Sum;
-use std::ops::{Add, AddAssign, Mul, Sub, SubAssign};
+use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
 
 use crate::field::extension_field::Extendable;
 use crate::field::fft::{fft, ifft};
@@ -253,6 +253,16 @@ impl<F: Field> AddAssign for PolynomialCoeffs<F> {
     }
 }
 
+impl<F: Field> AddAssign<&Self> for PolynomialCoeffs<F> {
+    fn add_assign(&mut self, rhs: &Self) {
+        let len = max(self.len(), rhs.len());
+        self.coeffs.resize(len, F::ZERO);
+        for (l, &r) in self.coeffs.iter_mut().zip(&rhs.coeffs) {
+            *l += r;
+        }
+    }
+}
+
 impl<F: Field> SubAssign for PolynomialCoeffs<F> {
     fn sub_assign(&mut self, rhs: Self) {
         let len = max(self.len(), rhs.len());
@@ -263,6 +273,16 @@ impl<F: Field> SubAssign for PolynomialCoeffs<F> {
     }
 }
 
+impl<F: Field> SubAssign<&Self> for PolynomialCoeffs<F> {
+    fn sub_assign(&mut self, rhs: &Self) {
+        let len = max(self.len(), rhs.len());
+        self.coeffs.resize(len, F::ZERO);
+        for (l, &r) in self.coeffs.iter_mut().zip(&rhs.coeffs) {
+            *l -= r;
+        }
+    }
+}
+
 impl<F: Field> Mul<F> for &PolynomialCoeffs<F> {
     type Output = PolynomialCoeffs<F>;
 
@@ -272,6 +292,12 @@ impl<F: Field> Mul<F> for &PolynomialCoeffs<F> {
     }
 }
 
+impl<F: Field> MulAssign<F> for PolynomialCoeffs<F> {
+    fn mul_assign(&mut self, rhs: F) {
+        self.coeffs.iter_mut().for_each(|x| *x *= rhs);
+    }
+}
+
 impl<F: Field> Mul for &PolynomialCoeffs<F> {
     type Output = PolynomialCoeffs<F>;
 
diff --git a/src/util/scaling.rs b/src/util/scaling.rs
index 7155cb37..25da24c7 100644
--- a/src/util/scaling.rs
+++ b/src/util/scaling.rs
@@ -20,9 +20,9 @@ impl<F: Field> ReducingFactor<F> {
         self.base * x
     }
 
-    fn mul_poly(&mut self, p: PolynomialCoeffs<F>) -> PolynomialCoeffs<F> {
+    fn mul_poly(&mut self, p: &mut PolynomialCoeffs<F>) {
         self.count += 1;
-        &p * self.base
+        *p *= self.base;
     }
 
     pub fn reduce(&mut self, iter: impl DoubleEndedIterator<Item = impl Borrow<F>>) -> F {
@@ -34,8 +34,10 @@ impl<F: Field> ReducingFactor<F> {
         &mut self,
         polys: impl DoubleEndedIterator<Item = impl Borrow<PolynomialCoeffs<F>>>,
     ) -> PolynomialCoeffs<F> {
-        polys.rev().fold(PolynomialCoeffs::empty(), |acc, x| {
-            &self.mul_poly(acc) + x.borrow()
+        polys.rev().fold(PolynomialCoeffs::empty(), |mut acc, x| {
+            self.mul_poly(&mut acc);
+            acc += x.borrow();
+            acc
         })
     }
 
@@ -45,10 +47,9 @@ impl<F: Field> ReducingFactor<F> {
         tmp
     }
 
-    pub fn shift_poly(&mut self, p: PolynomialCoeffs<F>) -> PolynomialCoeffs<F> {
-        let tmp = &p * self.base.exp(self.count);
+    pub fn shift_poly(&mut self, p: &mut PolynomialCoeffs<F>) {
+        *p *= self.base.exp(self.count);
         self.count = 0;
-        tmp
     }
 
     pub fn reset(&mut self) {