diff --git a/ceno_zkvm/src/scheme/prover.rs b/ceno_zkvm/src/scheme/prover.rs
index b0ff7b656..2c8cae8bc 100644
--- a/ceno_zkvm/src/scheme/prover.rs
+++ b/ceno_zkvm/src/scheme/prover.rs
@@ -216,7 +216,7 @@ impl<E: ExtensionField, PCS: PolynomialCommitmentScheme<E>> ZKVMProver<E, PCS> {
         pp: &PCS::ProverParam,
         circuit_pk: &ProvingKey<E, PCS>,
         witnesses: Vec<ArcMultilinearExtension<'_, E>>,
-        wits_commit: PCS::CommitmentWithData,
+        wits_commit: PCS::CommitmentWithWitness,
         pi: &[ArcMultilinearExtension<'_, E>],
         num_instances: usize,
         transcript: &mut impl Transcript<E>,
@@ -661,7 +661,7 @@ impl<E: ExtensionField, PCS: PolynomialCommitmentScheme<E>> ZKVMProver<E, PCS> {
         pp: &PCS::ProverParam,
         circuit_pk: &ProvingKey<E, PCS>,
         witnesses: Vec<ArcMultilinearExtension<'_, E>>,
-        wits_commit: PCS::CommitmentWithData,
+        wits_commit: PCS::CommitmentWithWitness,
         pi: &[ArcMultilinearExtension<'_, E>],
         transcript: &mut impl Transcript<E>,
         challenges: &[E; 2],
diff --git a/ceno_zkvm/src/structs.rs b/ceno_zkvm/src/structs.rs
index af41851f9..b006b10e1 100644
--- a/ceno_zkvm/src/structs.rs
+++ b/ceno_zkvm/src/structs.rs
@@ -103,7 +103,7 @@ impl<F: Clone> PointAndEval<F> {
 #[derive(Clone, Debug)]
 pub struct ProvingKey<E: ExtensionField, PCS: PolynomialCommitmentScheme<E>> {
     pub fixed_traces: Option<Vec<DenseMultilinearExtension<E>>>,
-    pub fixed_commit_wd: Option<PCS::CommitmentWithData>,
+    pub fixed_commit_wd: Option<PCS::CommitmentWithWitness>,
     pub vk: VerifyingKey<E, PCS>,
 }
 
diff --git a/mpcs/src/basefold.rs b/mpcs/src/basefold.rs
index a3657422e..1aec0fb9d 100644
--- a/mpcs/src/basefold.rs
+++ b/mpcs/src/basefold.rs
@@ -58,7 +58,7 @@ type SumCheck<F> = ClassicSumCheck<CoefficientsProver<F>>;
 
 mod structure;
 pub use structure::{
-    Basefold, BasefoldBasecodeParams, BasefoldCommitment, BasefoldCommitmentWithData,
+    Basefold, BasefoldBasecodeParams, BasefoldCommitment, BasefoldCommitmentWithWitness,
     BasefoldDefault, BasefoldParams, BasefoldProverParams, BasefoldRSParams,
     BasefoldVerifierParams,
 };
@@ -273,7 +273,7 @@ where
     type Param = BasefoldParams<E, Spec>;
     type ProverParam = BasefoldProverParams<E, Spec>;
     type VerifierParam = BasefoldVerifierParams<E, Spec>;
-    type CommitmentWithData = BasefoldCommitmentWithData<E>;
+    type CommitmentWithWitness = BasefoldCommitmentWithWitness<E>;
     type Commitment = BasefoldCommitment<E>;
     type CommitmentChunk = Digest<E::BaseField>;
     type Proof = BasefoldProof<E>;
@@ -307,7 +307,7 @@ where
     fn commit(
         pp: &Self::ProverParam,
         poly: &DenseMultilinearExtension<E>,
-    ) -> Result<Self::CommitmentWithData, Error> {
+    ) -> Result<Self::CommitmentWithWitness, Error> {
         let timer = start_timer!(|| "Basefold::commit");
 
         let is_base = match poly.evaluations {
@@ -325,9 +325,9 @@ where
             PolyEvalsCodeword::Normal((bh_evals, codeword)) => {
                 let codeword_tree = MerkleTree::<E>::from_leaves(codeword);
 
-                // All these values are stored in the `CommitmentWithData` because
+                // All these values are stored in the `CommitmentWithWitness` because
                 // they are useful in opening, and we don't want to recompute them.
-                Ok(Self::CommitmentWithData {
+                Ok(Self::CommitmentWithWitness {
                     codeword_tree,
                     polynomials_bh_evals: vec![bh_evals],
                     num_vars: poly.num_vars,
@@ -338,9 +338,9 @@ where
             PolyEvalsCodeword::TooSmall(evals) => {
                 let codeword_tree = MerkleTree::<E>::from_leaves(evals.clone());
 
-                // All these values are stored in the `CommitmentWithData` because
+                // All these values are stored in the `CommitmentWithWitness` because
                 // they are useful in opening, and we don't want to recompute them.
-                Ok(Self::CommitmentWithData {
+                Ok(Self::CommitmentWithWitness {
                     codeword_tree,
                     polynomials_bh_evals: vec![evals],
                     num_vars: poly.num_vars,
@@ -359,7 +359,7 @@ where
     fn batch_commit(
         pp: &Self::ProverParam,
         polys: &[DenseMultilinearExtension<E>],
-    ) -> Result<Self::CommitmentWithData, Error> {
+    ) -> Result<Self::CommitmentWithWitness, Error> {
         // assumptions
         // 1. there must be at least one polynomial
         // 2. all polynomials must exist in the same field type
@@ -413,7 +413,7 @@ where
                     })
                     .collect::<(Vec<_>, Vec<_>)>();
                 let codeword_tree = MerkleTree::<E>::from_batch_leaves(codewords);
-                Self::CommitmentWithData {
+                Self::CommitmentWithWitness {
                     codeword_tree,
                     polynomials_bh_evals: bh_evals,
                     num_vars: polys[0].num_vars,
@@ -433,7 +433,7 @@ where
                     })
                     .collect::<Vec<_>>();
                 let codeword_tree = MerkleTree::<E>::from_batch_leaves(bh_evals.clone());
-                Self::CommitmentWithData {
+                Self::CommitmentWithWitness {
                     codeword_tree,
                     polynomials_bh_evals: bh_evals,
                     num_vars: polys[0].num_vars,
@@ -457,7 +457,7 @@ where
         Ok(())
     }
 
-    fn get_pure_commitment(comm: &Self::CommitmentWithData) -> Self::Commitment {
+    fn get_pure_commitment(comm: &Self::CommitmentWithWitness) -> Self::Commitment {
         comm.to_commitment()
     }
 
@@ -467,7 +467,7 @@ where
     fn open(
         pp: &Self::ProverParam,
         poly: &DenseMultilinearExtension<E>,
-        comm: &Self::CommitmentWithData,
+        comm: &Self::CommitmentWithWitness,
         point: &[E],
         _eval: &E, // Opening does not need eval, except for sanity check
         transcript: &mut impl Transcript<E>,
@@ -547,7 +547,7 @@ where
     fn batch_open(
         pp: &Self::ProverParam,
         polys: &[DenseMultilinearExtension<E>],
-        comms: &[Self::CommitmentWithData],
+        comms: &[Self::CommitmentWithWitness],
         points: &[Vec<E>],
         evals: &[Evaluation<E>],
         transcript: &mut impl Transcript<E>,
@@ -769,7 +769,7 @@ where
     fn simple_batch_open(
         pp: &Self::ProverParam,
         polys: &[ArcMultilinearExtension<E>],
-        comm: &Self::CommitmentWithData,
+        comm: &Self::CommitmentWithWitness,
         point: &[E],
         evals: &[E],
         transcript: &mut impl Transcript<E>,
diff --git a/mpcs/src/basefold/commit_phase.rs b/mpcs/src/basefold/commit_phase.rs
index 7d6e0b949..55a6acea5 100644
--- a/mpcs/src/basefold/commit_phase.rs
+++ b/mpcs/src/basefold/commit_phase.rs
@@ -27,13 +27,13 @@ use rayon::prelude::{
     ParallelSlice,
 };
 
-use super::structure::BasefoldCommitmentWithData;
+use super::structure::BasefoldCommitmentWithWitness;
 
 // outputs (trees, sumcheck_oracles, oracles, bh_evals, eq, eval)
 pub fn commit_phase<E: ExtensionField, Spec: BasefoldSpec<E>>(
     pp: &<Spec::EncodingScheme as EncodingScheme<E>>::ProverParameters,
     point: &[E],
-    comm: &BasefoldCommitmentWithData<E>,
+    comm: &BasefoldCommitmentWithWitness<E>,
     transcript: &mut impl Transcript<E>,
     num_vars: usize,
     num_rounds: usize,
@@ -179,7 +179,7 @@ where
 pub fn batch_commit_phase<E: ExtensionField, Spec: BasefoldSpec<E>>(
     pp: &<Spec::EncodingScheme as EncodingScheme<E>>::ProverParameters,
     point: &[E],
-    comms: &[BasefoldCommitmentWithData<E>],
+    comms: &[BasefoldCommitmentWithWitness<E>],
     transcript: &mut impl Transcript<E>,
     num_vars: usize,
     num_rounds: usize,
@@ -350,7 +350,7 @@ pub fn simple_batch_commit_phase<E: ExtensionField, Spec: BasefoldSpec<E>>(
     pp: &<Spec::EncodingScheme as EncodingScheme<E>>::ProverParameters,
     point: &[E],
     batch_coeffs: &[E],
-    comm: &BasefoldCommitmentWithData<E>,
+    comm: &BasefoldCommitmentWithWitness<E>,
     transcript: &mut impl Transcript<E>,
     num_vars: usize,
     num_rounds: usize,
diff --git a/mpcs/src/basefold/query_phase.rs b/mpcs/src/basefold/query_phase.rs
index f0db40890..9ec15d36b 100644
--- a/mpcs/src/basefold/query_phase.rs
+++ b/mpcs/src/basefold/query_phase.rs
@@ -25,12 +25,12 @@ use rayon::{
 
 use super::{
     encoding::EncodingScheme,
-    structure::{BasefoldCommitment, BasefoldCommitmentWithData, BasefoldSpec},
+    structure::{BasefoldCommitment, BasefoldCommitmentWithWitness, BasefoldSpec},
 };
 
 pub fn prover_query_phase<E: ExtensionField>(
     transcript: &mut impl Transcript<E>,
-    comm: &BasefoldCommitmentWithData<E>,
+    comm: &BasefoldCommitmentWithWitness<E>,
     trees: &[MerkleTree<E>],
     num_verifier_queries: usize,
 ) -> QueriesResult<E>
@@ -67,7 +67,7 @@ where
 pub fn batch_prover_query_phase<E: ExtensionField>(
     transcript: &mut impl Transcript<E>,
     codeword_size: usize,
-    comms: &[BasefoldCommitmentWithData<E>],
+    comms: &[BasefoldCommitmentWithWitness<E>],
     trees: &[MerkleTree<E>],
     num_verifier_queries: usize,
 ) -> BatchedQueriesResult<E>
@@ -103,7 +103,7 @@ where
 
 pub fn simple_batch_prover_query_phase<E: ExtensionField>(
     transcript: &mut impl Transcript<E>,
-    comm: &BasefoldCommitmentWithData<E>,
+    comm: &BasefoldCommitmentWithWitness<E>,
     trees: &[MerkleTree<E>],
     num_verifier_queries: usize,
 ) -> SimpleBatchQueriesResult<E>
@@ -411,7 +411,7 @@ where
 }
 
 fn batch_basefold_get_query<E: ExtensionField>(
-    comms: &[BasefoldCommitmentWithData<E>],
+    comms: &[BasefoldCommitmentWithWitness<E>],
     trees: &[MerkleTree<E>],
     codeword_size: usize,
     x_index: usize,
@@ -828,7 +828,7 @@ where
     pub fn from_single_query_result(
         single_query_result: SingleQueryResult<E>,
         oracle_trees: &[MerkleTree<E>],
-        commitment: &BasefoldCommitmentWithData<E>,
+        commitment: &BasefoldCommitmentWithWitness<E>,
     ) -> Self {
         assert!(commitment.codeword_tree.height() > 0);
         Self {
@@ -927,7 +927,7 @@ where
     pub fn from_query_result(
         query_result: QueriesResult<E>,
         oracle_trees: &[MerkleTree<E>],
-        commitment: &BasefoldCommitmentWithData<E>,
+        commitment: &BasefoldCommitmentWithWitness<E>,
     ) -> Self {
         Self {
             inner: query_result
@@ -1002,7 +1002,7 @@ where
     pub fn from_batched_single_query_result(
         batched_single_query_result: BatchedSingleQueryResult<E>,
         oracle_trees: &[MerkleTree<E>],
-        commitments: &[BasefoldCommitmentWithData<E>],
+        commitments: &[BasefoldCommitmentWithWitness<E>],
     ) -> Self {
         Self {
             oracle_query: OracleListQueryResultWithMerklePath::from_query_and_trees(
@@ -1147,7 +1147,7 @@ where
     pub fn from_batched_query_result(
         batched_query_result: BatchedQueriesResult<E>,
         oracle_trees: &[MerkleTree<E>],
-        commitments: &[BasefoldCommitmentWithData<E>],
+        commitments: &[BasefoldCommitmentWithWitness<E>],
     ) -> Self {
         Self {
             inner: batched_query_result
@@ -1307,7 +1307,7 @@ where
     pub fn from_single_query_result(
         single_query_result: SimpleBatchSingleQueryResult<E>,
         oracle_trees: &[MerkleTree<E>],
-        commitment: &BasefoldCommitmentWithData<E>,
+        commitment: &BasefoldCommitmentWithWitness<E>,
     ) -> Self {
         Self {
             oracle_query: OracleListQueryResultWithMerklePath::from_query_and_trees(
@@ -1403,7 +1403,7 @@ where
     pub fn from_query_result(
         query_result: SimpleBatchQueriesResult<E>,
         oracle_trees: &[MerkleTree<E>],
-        commitment: &BasefoldCommitmentWithData<E>,
+        commitment: &BasefoldCommitmentWithWitness<E>,
     ) -> Self {
         Self {
             inner: query_result
diff --git a/mpcs/src/basefold/structure.rs b/mpcs/src/basefold/structure.rs
index 0cb0748a0..129409efd 100644
--- a/mpcs/src/basefold/structure.rs
+++ b/mpcs/src/basefold/structure.rs
@@ -58,9 +58,8 @@ pub struct BasefoldVerifierParams<E: ExtensionField, Spec: BasefoldSpec<E>> {
 
 /// A polynomial commitment together with all the data (e.g., the codeword, and Merkle tree)
 /// used to generate this commitment and for assistant in opening
-#[derive(Clone, Debug, Default, Serialize, Deserialize)]
-#[serde(bound(serialize = "E: Serialize", deserialize = "E: DeserializeOwned"))]
-pub struct BasefoldCommitmentWithData<E: ExtensionField>
+#[derive(Clone, Debug, Default)]
+pub struct BasefoldCommitmentWithWitness<E: ExtensionField>
 where
     E::BaseField: Serialize + DeserializeOwned,
 {
@@ -71,7 +70,7 @@ where
     pub(crate) num_polys: usize,
 }
 
-impl<E: ExtensionField> BasefoldCommitmentWithData<E>
+impl<E: ExtensionField> BasefoldCommitmentWithWitness<E>
 where
     E::BaseField: Serialize + DeserializeOwned,
 {
@@ -133,20 +132,20 @@ where
     }
 }
 
-impl<E: ExtensionField> From<BasefoldCommitmentWithData<E>> for Digest<E::BaseField>
+impl<E: ExtensionField> From<BasefoldCommitmentWithWitness<E>> for Digest<E::BaseField>
 where
     E::BaseField: Serialize + DeserializeOwned,
 {
-    fn from(val: BasefoldCommitmentWithData<E>) -> Self {
+    fn from(val: BasefoldCommitmentWithWitness<E>) -> Self {
         val.get_root_as()
     }
 }
 
-impl<E: ExtensionField> From<&BasefoldCommitmentWithData<E>> for BasefoldCommitment<E>
+impl<E: ExtensionField> From<&BasefoldCommitmentWithWitness<E>> for BasefoldCommitment<E>
 where
     E::BaseField: Serialize + DeserializeOwned,
 {
-    fn from(val: &BasefoldCommitmentWithData<E>) -> Self {
+    fn from(val: &BasefoldCommitmentWithWitness<E>) -> Self {
         val.to_commitment()
     }
 }
@@ -194,7 +193,7 @@ where
     }
 }
 
-impl<E: ExtensionField> PartialEq for BasefoldCommitmentWithData<E>
+impl<E: ExtensionField> PartialEq for BasefoldCommitmentWithWitness<E>
 where
     E::BaseField: Serialize + DeserializeOwned,
 {
@@ -204,7 +203,7 @@ where
     }
 }
 
-impl<E: ExtensionField> Eq for BasefoldCommitmentWithData<E> where
+impl<E: ExtensionField> Eq for BasefoldCommitmentWithWitness<E> where
     E::BaseField: Serialize + DeserializeOwned
 {
 }
@@ -266,7 +265,7 @@ where
     }
 }
 
-impl<E: ExtensionField> AsRef<[Digest<E::BaseField>]> for BasefoldCommitmentWithData<E>
+impl<E: ExtensionField> AsRef<[Digest<E::BaseField>]> for BasefoldCommitmentWithWitness<E>
 where
     E::BaseField: Serialize + DeserializeOwned,
 {
diff --git a/mpcs/src/lib.rs b/mpcs/src/lib.rs
index e9c67b866..dcc6b6725 100644
--- a/mpcs/src/lib.rs
+++ b/mpcs/src/lib.rs
@@ -12,7 +12,8 @@ pub mod util;
 
 pub type Commitment<E, Pcs> = <Pcs as PolynomialCommitmentScheme<E>>::Commitment;
 pub type CommitmentChunk<E, Pcs> = <Pcs as PolynomialCommitmentScheme<E>>::CommitmentChunk;
-pub type CommitmentWithData<E, Pcs> = <Pcs as PolynomialCommitmentScheme<E>>::CommitmentWithData;
+pub type CommitmentWithWitness<E, Pcs> =
+    <Pcs as PolynomialCommitmentScheme<E>>::CommitmentWithWitness;
 
 pub type Param<E, Pcs> = <Pcs as PolynomialCommitmentScheme<E>>::Param;
 pub type ProverParam<E, Pcs> = <Pcs as PolynomialCommitmentScheme<E>>::ProverParam;
@@ -34,7 +35,7 @@ pub fn pcs_trim<E: ExtensionField, Pcs: PolynomialCommitmentScheme<E>>(
 pub fn pcs_commit<E: ExtensionField, Pcs: PolynomialCommitmentScheme<E>>(
     pp: &Pcs::ProverParam,
     poly: &DenseMultilinearExtension<E>,
-) -> Result<Pcs::CommitmentWithData, Error> {
+) -> Result<Pcs::CommitmentWithWitness, Error> {
     Pcs::commit(pp, poly)
 }
 
@@ -42,14 +43,14 @@ pub fn pcs_commit_and_write<E: ExtensionField, Pcs: PolynomialCommitmentScheme<E
     pp: &Pcs::ProverParam,
     poly: &DenseMultilinearExtension<E>,
     transcript: &mut impl Transcript<E>,
-) -> Result<Pcs::CommitmentWithData, Error> {
+) -> Result<Pcs::CommitmentWithWitness, Error> {
     Pcs::commit_and_write(pp, poly, transcript)
 }
 
 pub fn pcs_batch_commit<E: ExtensionField, Pcs: PolynomialCommitmentScheme<E>>(
     pp: &Pcs::ProverParam,
     polys: &[DenseMultilinearExtension<E>],
-) -> Result<Pcs::CommitmentWithData, Error> {
+) -> Result<Pcs::CommitmentWithWitness, Error> {
     Pcs::batch_commit(pp, polys)
 }
 
@@ -57,14 +58,14 @@ pub fn pcs_batch_commit_and_write<E: ExtensionField, Pcs: PolynomialCommitmentSc
     pp: &Pcs::ProverParam,
     polys: &[DenseMultilinearExtension<E>],
     transcript: &mut impl Transcript<E>,
-) -> Result<Pcs::CommitmentWithData, Error> {
+) -> Result<Pcs::CommitmentWithWitness, Error> {
     Pcs::batch_commit_and_write(pp, polys, transcript)
 }
 
 pub fn pcs_open<E: ExtensionField, Pcs: PolynomialCommitmentScheme<E>>(
     pp: &Pcs::ProverParam,
     poly: &DenseMultilinearExtension<E>,
-    comm: &Pcs::CommitmentWithData,
+    comm: &Pcs::CommitmentWithWitness,
     point: &[E],
     eval: &E,
     transcript: &mut impl Transcript<E>,
@@ -75,7 +76,7 @@ pub fn pcs_open<E: ExtensionField, Pcs: PolynomialCommitmentScheme<E>>(
 pub fn pcs_batch_open<E: ExtensionField, Pcs: PolynomialCommitmentScheme<E>>(
     pp: &Pcs::ProverParam,
     polys: &[DenseMultilinearExtension<E>],
-    comms: &[Pcs::CommitmentWithData],
+    comms: &[Pcs::CommitmentWithWitness],
     points: &[Vec<E>],
     evals: &[Evaluation<E>],
     transcript: &mut impl Transcript<E>,
@@ -112,7 +113,7 @@ pub trait PolynomialCommitmentScheme<E: ExtensionField>: Clone + Debug {
     type Param: Clone + Debug + Serialize + DeserializeOwned;
     type ProverParam: Clone + Debug + Serialize + DeserializeOwned;
     type VerifierParam: Clone + Debug + Serialize + DeserializeOwned;
-    type CommitmentWithData: Clone + Debug + Default + Serialize + DeserializeOwned;
+    type CommitmentWithWitness: Clone + Debug;
     type Commitment: Clone + Debug + Default + Serialize + DeserializeOwned;
     type CommitmentChunk: Clone + Debug + Default;
     type Proof: Clone + Debug + Serialize + DeserializeOwned;
@@ -127,13 +128,13 @@ pub trait PolynomialCommitmentScheme<E: ExtensionField>: Clone + Debug {
     fn commit(
         pp: &Self::ProverParam,
         poly: &DenseMultilinearExtension<E>,
-    ) -> Result<Self::CommitmentWithData, Error>;
+    ) -> Result<Self::CommitmentWithWitness, Error>;
 
     fn commit_and_write(
         pp: &Self::ProverParam,
         poly: &DenseMultilinearExtension<E>,
         transcript: &mut impl Transcript<E>,
-    ) -> Result<Self::CommitmentWithData, Error> {
+    ) -> Result<Self::CommitmentWithWitness, Error> {
         let comm = Self::commit(pp, poly)?;
         Self::write_commitment(&Self::get_pure_commitment(&comm), transcript)?;
         Ok(comm)
@@ -144,18 +145,18 @@ pub trait PolynomialCommitmentScheme<E: ExtensionField>: Clone + Debug {
         transcript: &mut impl Transcript<E>,
     ) -> Result<(), Error>;
 
-    fn get_pure_commitment(comm: &Self::CommitmentWithData) -> Self::Commitment;
+    fn get_pure_commitment(comm: &Self::CommitmentWithWitness) -> Self::Commitment;
 
     fn batch_commit(
         pp: &Self::ProverParam,
         polys: &[DenseMultilinearExtension<E>],
-    ) -> Result<Self::CommitmentWithData, Error>;
+    ) -> Result<Self::CommitmentWithWitness, Error>;
 
     fn batch_commit_and_write(
         pp: &Self::ProverParam,
         polys: &[DenseMultilinearExtension<E>],
         transcript: &mut impl Transcript<E>,
-    ) -> Result<Self::CommitmentWithData, Error> {
+    ) -> Result<Self::CommitmentWithWitness, Error> {
         let comm = Self::batch_commit(pp, polys)?;
         Self::write_commitment(&Self::get_pure_commitment(&comm), transcript)?;
         Ok(comm)
@@ -164,7 +165,7 @@ pub trait PolynomialCommitmentScheme<E: ExtensionField>: Clone + Debug {
     fn open(
         pp: &Self::ProverParam,
         poly: &DenseMultilinearExtension<E>,
-        comm: &Self::CommitmentWithData,
+        comm: &Self::CommitmentWithWitness,
         point: &[E],
         eval: &E,
         transcript: &mut impl Transcript<E>,
@@ -173,7 +174,7 @@ pub trait PolynomialCommitmentScheme<E: ExtensionField>: Clone + Debug {
     fn batch_open(
         pp: &Self::ProverParam,
         polys: &[DenseMultilinearExtension<E>],
-        comms: &[Self::CommitmentWithData],
+        comms: &[Self::CommitmentWithWitness],
         points: &[Vec<E>],
         evals: &[Evaluation<E>],
         transcript: &mut impl Transcript<E>,
@@ -186,7 +187,7 @@ pub trait PolynomialCommitmentScheme<E: ExtensionField>: Clone + Debug {
     fn simple_batch_open(
         pp: &Self::ProverParam,
         polys: &[ArcMultilinearExtension<E>],
-        comm: &Self::CommitmentWithData,
+        comm: &Self::CommitmentWithWitness,
         point: &[E],
         evals: &[E],
         transcript: &mut impl Transcript<E>,
@@ -228,7 +229,7 @@ where
     fn ni_open(
         pp: &Self::ProverParam,
         poly: &DenseMultilinearExtension<E>,
-        comm: &Self::CommitmentWithData,
+        comm: &Self::CommitmentWithWitness,
         point: &[E],
         eval: &E,
     ) -> Result<Self::Proof, Error> {
@@ -239,7 +240,7 @@ where
     fn ni_batch_open(
         pp: &Self::ProverParam,
         polys: &[DenseMultilinearExtension<E>],
-        comms: &[Self::CommitmentWithData],
+        comms: &[Self::CommitmentWithWitness],
         points: &[Vec<E>],
         evals: &[Evaluation<E>],
     ) -> Result<Self::Proof, Error> {
@@ -315,7 +316,7 @@ pub enum Error {
 mod basefold;
 pub use basefold::{
     Basecode, BasecodeDefaultSpec, Basefold, BasefoldBasecodeParams, BasefoldCommitment,
-    BasefoldCommitmentWithData, BasefoldDefault, BasefoldParams, BasefoldRSParams, BasefoldSpec,
+    BasefoldCommitmentWithWitness, BasefoldDefault, BasefoldParams, BasefoldRSParams, BasefoldSpec,
     EncodingScheme, RSCode, RSCodeDefaultSpec, coset_fft, fft, fft_root_table, one_level_eval_hc,
     one_level_interp_hc,
 };
@@ -436,7 +437,7 @@ pub mod test_util {
         pp: &Pcs::ProverParam,
         polys: &[DenseMultilinearExtension<E>],
         transcript: &mut impl Transcript<E>,
-    ) -> Vec<Pcs::CommitmentWithData> {
+    ) -> Vec<Pcs::CommitmentWithWitness> {
         polys
             .iter()
             .map(|poly| Pcs::commit_and_write(pp, poly, transcript).unwrap())