diff --git a/ferveo/src/lib.rs b/ferveo/src/lib.rs
index a6b1703d..b5cbc990 100644
--- a/ferveo/src/lib.rs
+++ b/ferveo/src/lib.rs
@@ -499,7 +499,7 @@ mod test_dkg_full {
         // Now, we're going to refresh the shares and check that the shared secret is the same
 
         // Dealer computes a new random polynomial with constant term x_r = 0
-        let polynomial = make_random_polynomial_at::<E>(
+        let polynomial = make_random_polynomial_with_root::<E>(
             dkg.dkg_params.security_threshold as usize,
             &Fr::zero(),
             rng,
diff --git a/ferveo/src/refresh.rs b/ferveo/src/refresh.rs
index 9493f8af..dc104e54 100644
--- a/ferveo/src/refresh.rs
+++ b/ferveo/src/refresh.rs
@@ -61,7 +61,7 @@ pub fn recover_share_from_updated_private_shares<E: Pairing>(
     }
 }
 
-pub fn make_random_polynomial_at<E: Pairing>(
+pub fn make_random_polynomial_with_root<E: Pairing>(
     threshold: usize,
     root: &E::ScalarField,
     rng: &mut impl RngCore,
@@ -78,6 +78,7 @@ pub fn make_random_polynomial_at<E: Pairing>(
     let d_i_0 = E::ScalarField::zero() - threshold_poly.evaluate(root);
     threshold_poly[0] = d_i_0;
 
+    // Evaluating the polynomial at the root should result in 0
     debug_assert!(threshold_poly.evaluate(root) == E::ScalarField::zero());
     debug_assert!(threshold_poly.coeffs.len() == threshold);
 
@@ -120,7 +121,7 @@ mod tests_refresh {
     type ScalarField = <E as Pairing>::ScalarField;
 
     use crate::{
-        make_random_polynomial_at, prepare_share_updates_for_recovery,
+        make_random_polynomial_with_root, prepare_share_updates_for_recovery,
         recover_share_from_updated_private_shares, refresh_private_key_share,
         update_share_for_recovery,
     };
@@ -359,7 +360,7 @@ mod tests_refresh {
         // Now, we're going to refresh the shares and check that the shared secret is the same
 
         // Dealer computes a new random polynomial with constant term x_r
-        let polynomial = make_random_polynomial_at::<E>(
+        let polynomial = make_random_polynomial_with_root::<E>(
             threshold,
             &ScalarField::zero(),
             rng,
diff --git a/tpke/benches/arkworks.rs b/tpke/benches/arkworks.rs
index a1f6a50f..957f5fc4 100644
--- a/tpke/benches/arkworks.rs
+++ b/tpke/benches/arkworks.rs
@@ -14,7 +14,7 @@ use ark_ff::{BigInteger256, Field, One, UniformRand, Zero};
 use criterion::{
     black_box, criterion_group, criterion_main, BenchmarkId, Criterion,
 };
-use group_threshold_cryptography_pre_release::make_random_polynomial_at;
+use group_threshold_cryptography_pre_release::make_random_polynomial_with_root;
 use itertools::izip;
 use rand::prelude::StdRng;
 use rand_core::{RngCore, SeedableRng};
@@ -219,7 +219,7 @@ pub fn bench_random_poly(c: &mut Criterion) {
         result
     }
 
-    pub fn naive_make_random_polynomial_at<E: Pairing>(
+    pub fn naive_make_random_polynomial_with_root<E: Pairing>(
         threshold: usize,
         root: &Fr,
         rng: &mut impl RngCore,
@@ -248,7 +248,7 @@ pub fn bench_random_poly(c: &mut Criterion) {
         let mut ark = {
             let mut rng = rng.clone();
             move || {
-                black_box(make_random_polynomial_at::<E>(
+                black_box(make_random_polynomial_with_root::<E>(
                     threshold,
                     &Fr::zero(),
                     &mut rng,
@@ -258,7 +258,7 @@ pub fn bench_random_poly(c: &mut Criterion) {
         let mut naive = {
             let mut rng = rng.clone();
             move || {
-                black_box(naive_make_random_polynomial_at::<E>(
+                black_box(naive_make_random_polynomial_with_root::<E>(
                     threshold,
                     &Fr::zero(),
                     &mut rng,
diff --git a/tpke/benches/tpke.rs b/tpke/benches/tpke.rs
index c6ad85ae..69ddb848 100644
--- a/tpke/benches/tpke.rs
+++ b/tpke/benches/tpke.rs
@@ -552,7 +552,7 @@ pub fn bench_refresh_shares(c: &mut Criterion) {
         let setup = SetupSimple::new(shares_num, msg_size, rng);
         let threshold = setup.shared.threshold;
         let polynomial =
-            make_random_polynomial_at::<E>(threshold, &Fr::zero(), rng);
+            make_random_polynomial_with_root::<E>(threshold, &Fr::zero(), rng);
         let p = setup.contexts[0].clone();
         group.bench_function(
             BenchmarkId::new("refresh_private_key_share", shares_num),