diff --git a/plonky2/examples/bench_recursion.rs b/plonky2/examples/bench_recursion.rs
index ce94fca7..d0c273d1 100644
--- a/plonky2/examples/bench_recursion.rs
+++ b/plonky2/examples/bench_recursion.rs
@@ -248,18 +248,18 @@ where
 fn test_serialization<F: RichField + Extendable<D>, C: GenericConfig<D, F = F>, const D: usize>(
     proof: &ProofWithPublicInputs<F, C, D>,
     vd: &VerifierOnlyCircuitData<C, D>,
-    cd: &CommonCircuitData<F, D>,
+    common_data: &CommonCircuitData<F, D>,
 ) -> Result<()> {
     let proof_bytes = proof.to_bytes();
     info!("Proof length: {} bytes", proof_bytes.len());
-    let proof_from_bytes = ProofWithPublicInputs::from_bytes(proof_bytes, cd)?;
+    let proof_from_bytes = ProofWithPublicInputs::from_bytes(proof_bytes, common_data)?;
     assert_eq!(proof, &proof_from_bytes);
 
     let now = std::time::Instant::now();
-    let compressed_proof = proof.clone().compress(&vd.circuit_digest, cd)?;
+    let compressed_proof = proof.clone().compress(&vd.circuit_digest, common_data)?;
     let decompressed_compressed_proof = compressed_proof
         .clone()
-        .decompress(&vd.circuit_digest, cd)?;
+        .decompress(&vd.circuit_digest, common_data)?;
     info!("{:.4}s to compress proof", now.elapsed().as_secs_f64());
     assert_eq!(proof, &decompressed_compressed_proof);
 
@@ -269,11 +269,11 @@ fn test_serialization<F: RichField + Extendable<D>, C: GenericConfig<D, F = F>,
         compressed_proof_bytes.len()
     );
     let compressed_proof_from_bytes =
-        CompressedProofWithPublicInputs::from_bytes(compressed_proof_bytes, cd)?;
+        CompressedProofWithPublicInputs::from_bytes(compressed_proof_bytes, common_data)?;
     assert_eq!(compressed_proof, compressed_proof_from_bytes);
 
     let gate_serializer = DefaultGateSerializer;
-    let common_data_bytes = cd
+    let common_data_bytes = common_data
         .to_bytes(&gate_serializer)
         .map_err(|_| anyhow::Error::msg("CommonCircuitData serialization failed."))?;
     info!(
@@ -283,7 +283,7 @@ fn test_serialization<F: RichField + Extendable<D>, C: GenericConfig<D, F = F>,
     let common_data_from_bytes =
         CommonCircuitData::<F, D>::from_bytes(common_data_bytes, &gate_serializer)
             .map_err(|_| anyhow::Error::msg("CommonCircuitData deserialization failed."))?;
-    assert_eq!(cd, &common_data_from_bytes);
+    assert_eq!(common_data, &common_data_from_bytes);
 
     Ok(())
 }
@@ -312,35 +312,35 @@ pub fn benchmark_function(
     };
     // Start with a dummy proof of specified size
     let inner = dummy_proof_function(config, log2_inner_size)?;
-    let (_, _, cd) = &inner;
+    let (_, _, common_data) = &inner;
     info!(
         "Initial {} degree {} = 2^{}",
         name,
-        cd.degree(),
-        cd.degree_bits()
+        common_data.degree(),
+        common_data.degree_bits()
     );
 
     // Recursively verify the proof
     let middle = recursive_proof::<F, C, C, D>(&inner, config, None)?;
-    let (_, _, cd) = &middle;
+    let (_, _, common_data) = &middle;
     info!(
         "Single recursion {} degree {} = 2^{}",
         name,
-        cd.degree(),
-        cd.degree_bits()
+        common_data.degree(),
+        common_data.degree_bits()
     );
 
     // Add a second layer of recursion to shrink the proof size further
     let outer = recursive_proof::<F, C, C, D>(&middle, config, None)?;
-    let (proof, vd, cd) = &outer;
+    let (proof, vd, common_data) = &outer;
     info!(
         "Double recursion {} degree {} = 2^{}",
         name,
-        cd.degree(),
-        cd.degree_bits()
+        common_data.degree(),
+        common_data.degree_bits()
     );
 
-    test_serialization(proof, vd, cd)?;
+    test_serialization(proof, vd, common_data)?;
 
     Ok(())
 }
diff --git a/plonky2/examples/square_root.rs b/plonky2/examples/square_root.rs
index 48883e3f..40c2e1fc 100644
--- a/plonky2/examples/square_root.rs
+++ b/plonky2/examples/square_root.rs
@@ -52,12 +52,12 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         out_buffer.set_target(self.x, x);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.x)?;
         dst.write_target(self.x_squared)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let x = src.read_target()?;
         let x_squared = src.read_target()?;
         Ok(Self {
diff --git a/plonky2/src/gadgets/arithmetic.rs b/plonky2/src/gadgets/arithmetic.rs
index d245aef5..858a4eaf 100644
--- a/plonky2/src/gadgets/arithmetic.rs
+++ b/plonky2/src/gadgets/arithmetic.rs
@@ -399,14 +399,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Equ
         out_buffer.set_target(self.inv, inv);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.x)?;
         dst.write_target(self.y)?;
         dst.write_target_bool(self.equal)?;
         dst.write_target(self.inv)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let x = src.read_target()?;
         let y = src.read_target()?;
         let equal = src.read_target_bool()?;
diff --git a/plonky2/src/gadgets/arithmetic_extension.rs b/plonky2/src/gadgets/arithmetic_extension.rs
index 11e3cc83..0fe8083a 100644
--- a/plonky2/src/gadgets/arithmetic_extension.rs
+++ b/plonky2/src/gadgets/arithmetic_extension.rs
@@ -522,13 +522,13 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         out_buffer.set_extension_target(self.quotient, quotient)
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target_ext(self.numerator)?;
         dst.write_target_ext(self.denominator)?;
         dst.write_target_ext(self.quotient)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let numerator = src.read_target_ext()?;
         let denominator = src.read_target_ext()?;
         let quotient = src.read_target_ext()?;
diff --git a/plonky2/src/gadgets/range_check.rs b/plonky2/src/gadgets/range_check.rs
index c7bfc6a5..bdb35f9e 100644
--- a/plonky2/src/gadgets/range_check.rs
+++ b/plonky2/src/gadgets/range_check.rs
@@ -79,14 +79,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Low
         out_buffer.set_target(self.high, F::from_canonical_u64(high));
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.integer)?;
         dst.write_usize(self.n_log)?;
         dst.write_target(self.low)?;
         dst.write_target(self.high)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let integer = src.read_target()?;
         let n_log = src.read_usize()?;
         let low = src.read_target()?;
diff --git a/plonky2/src/gadgets/split_base.rs b/plonky2/src/gadgets/split_base.rs
index c300723e..0a39b8f0 100644
--- a/plonky2/src/gadgets/split_base.rs
+++ b/plonky2/src/gadgets/split_base.rs
@@ -110,12 +110,12 @@ impl<F: RichField + Extendable<D>, const B: usize, const D: usize> SimpleGenerat
         out_buffer.set_target(Target::wire(self.row, BaseSumGate::<B>::WIRE_SUM), sum);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_target_bool_vec(&self.limbs)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let limbs = src.read_target_bool_vec()?;
         Ok(Self { row, limbs })
diff --git a/plonky2/src/gadgets/split_join.rs b/plonky2/src/gadgets/split_join.rs
index b8daeec8..fb83c3a6 100644
--- a/plonky2/src/gadgets/split_join.rs
+++ b/plonky2/src/gadgets/split_join.rs
@@ -87,12 +87,12 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Spl
         );
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.integer)?;
         dst.write_target_vec(&self.bits)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let integer = src.read_target()?;
         let bits = src.read_target_vec()?;
         Ok(Self { integer, bits })
@@ -142,13 +142,13 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Wir
         );
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.integer)?;
         dst.write_usize_vec(&self.gates)?;
         dst.write_usize(self.num_limbs)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let integer = src.read_target()?;
         let gates = src.read_usize_vec()?;
         let num_limbs = src.read_usize()?;
diff --git a/plonky2/src/gates/arithmetic_base.rs b/plonky2/src/gates/arithmetic_base.rs
index dae84fb1..cf111a71 100644
--- a/plonky2/src/gates/arithmetic_base.rs
+++ b/plonky2/src/gates/arithmetic_base.rs
@@ -60,11 +60,11 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ArithmeticGate
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_ops)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_ops = src.read_usize()?;
         Ok(Self { num_ops })
     }
@@ -221,14 +221,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         out_buffer.set_target(output_target, computed_output)
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_field(self.const_0)?;
         dst.write_field(self.const_1)?;
         dst.write_usize(self.i)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let const_0 = src.read_field()?;
         let const_1 = src.read_field()?;
diff --git a/plonky2/src/gates/arithmetic_extension.rs b/plonky2/src/gates/arithmetic_extension.rs
index bd21b22c..82ca8a56 100644
--- a/plonky2/src/gates/arithmetic_extension.rs
+++ b/plonky2/src/gates/arithmetic_extension.rs
@@ -56,11 +56,11 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ArithmeticExte
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_ops)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_ops = src.read_usize()?;
         Ok(Self { num_ops })
     }
@@ -214,14 +214,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         out_buffer.set_extension_target(output_target, computed_output)
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_field(self.const_0)?;
         dst.write_field(self.const_1)?;
         dst.write_usize(self.i)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let const_0 = src.read_field()?;
         let const_1 = src.read_field()?;
diff --git a/plonky2/src/gates/base_sum.rs b/plonky2/src/gates/base_sum.rs
index 5d999e92..181252a2 100644
--- a/plonky2/src/gates/base_sum.rs
+++ b/plonky2/src/gates/base_sum.rs
@@ -55,11 +55,11 @@ impl<F: RichField + Extendable<D>, const D: usize, const B: usize> Gate<F, D> fo
         format!("{self:?} + Base: {B}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_limbs)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_limbs = src.read_usize()?;
         Ok(Self { num_limbs })
     }
@@ -211,12 +211,12 @@ impl<F: RichField + Extendable<D>, const B: usize, const D: usize> SimpleGenerat
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_usize(self.num_limbs)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let num_limbs = src.read_usize()?;
         Ok(Self { row, num_limbs })
diff --git a/plonky2/src/gates/constant.rs b/plonky2/src/gates/constant.rs
index 4a41ab9f..067855ca 100644
--- a/plonky2/src/gates/constant.rs
+++ b/plonky2/src/gates/constant.rs
@@ -43,11 +43,11 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ConstantGate {
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_consts)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_consts = src.read_usize()?;
         Ok(Self { num_consts })
     }
diff --git a/plonky2/src/gates/coset_interpolation.rs b/plonky2/src/gates/coset_interpolation.rs
index ea0b00e3..c701b8cf 100644
--- a/plonky2/src/gates/coset_interpolation.rs
+++ b/plonky2/src/gates/coset_interpolation.rs
@@ -169,14 +169,14 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for CosetInterpola
         format!("{self:?}<D={D}>")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.subgroup_bits)?;
         dst.write_usize(self.degree)?;
         dst.write_usize(self.barycentric_weights.len())?;
         dst.write_field_vec(&self.barycentric_weights)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let subgroup_bits = src.read_usize()?;
         let degree = src.read_usize()?;
         let length = src.read_usize()?;
@@ -497,14 +497,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         out_buffer.set_ext_wires(evaluation_value_wires, computed_eval);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
-        self.gate.serialize(dst, _cd)
+        self.gate.serialize(dst, _common_data)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
-        let gate = CosetInterpolationGate::deserialize(src, _cd)?;
+        let gate = CosetInterpolationGate::deserialize(src, _common_data)?;
         Ok(Self::new(row, gate))
     }
 }
diff --git a/plonky2/src/gates/exponentiation.rs b/plonky2/src/gates/exponentiation.rs
index 38053e65..a8629a84 100644
--- a/plonky2/src/gates/exponentiation.rs
+++ b/plonky2/src/gates/exponentiation.rs
@@ -76,11 +76,11 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for Exponentiation
         format!("{self:?}<D={D}>")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_power_bits)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_power_bits = src.read_usize()?;
         Ok(Self::new(num_power_bits))
     }
@@ -295,14 +295,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         out_buffer.set_wire(output_wire, intermediate_values[num_power_bits - 1]);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
-        self.gate.serialize(dst, _cd)
+        self.gate.serialize(dst, _common_data)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
-        let gate = ExponentiationGate::deserialize(src, _cd)?;
+        let gate = ExponentiationGate::deserialize(src, _common_data)?;
         Ok(Self { row, gate })
     }
 }
diff --git a/plonky2/src/gates/gate.rs b/plonky2/src/gates/gate.rs
index 6783c38f..2c52076b 100644
--- a/plonky2/src/gates/gate.rs
+++ b/plonky2/src/gates/gate.rs
@@ -29,9 +29,9 @@ use crate::util::serialization::{Buffer, IoResult};
 pub trait Gate<F: RichField + Extendable<D>, const D: usize>: 'static + Send + Sync {
     fn id(&self) -> String;
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()>;
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()>;
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self>
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self>
     where
         Self: Sized;
 
diff --git a/plonky2/src/gates/lookup.rs b/plonky2/src/gates/lookup.rs
index 5c751989..03cddbf7 100644
--- a/plonky2/src/gates/lookup.rs
+++ b/plonky2/src/gates/lookup.rs
@@ -59,9 +59,9 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for LookupGate {
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_slots)?;
-        for (i, lut) in cd.luts.iter().enumerate() {
+        for (i, lut) in common_data.luts.iter().enumerate() {
             if lut == &self.lut {
                 return dst.write_usize(i);
             }
@@ -70,13 +70,13 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for LookupGate {
         panic!("The associated lookup table couldn't be found.")
     }
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_slots = src.read_usize()?;
         let lut_index = src.read_usize()?;
 
         Ok(Self {
             num_slots,
-            lut: cd.luts[lut_index].clone(),
+            lut: common_data.luts[lut_index].clone(),
         })
     }
 
@@ -191,10 +191,10 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Loo
         };
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_usize(self.slot_nb)?;
-        for (i, lut) in cd.luts.iter().enumerate() {
+        for (i, lut) in common_data.luts.iter().enumerate() {
             if lut == &self.lut {
                 return dst.write_usize(i);
             }
@@ -203,14 +203,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Loo
         panic!("The associated lookup table couldn't be found.")
     }
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let slot_nb = src.read_usize()?;
         let lut_index = src.read_usize()?;
 
         Ok(Self {
             row,
-            lut: cd.luts[lut_index].clone(),
+            lut: common_data.luts[lut_index].clone(),
             slot_nb,
         })
     }
diff --git a/plonky2/src/gates/lookup_table.rs b/plonky2/src/gates/lookup_table.rs
index 99109f04..39ef1953 100644
--- a/plonky2/src/gates/lookup_table.rs
+++ b/plonky2/src/gates/lookup_table.rs
@@ -72,10 +72,10 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for LookupTableGat
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_slots)?;
         dst.write_usize(self.last_lut_row)?;
-        for (i, lut) in cd.luts.iter().enumerate() {
+        for (i, lut) in common_data.luts.iter().enumerate() {
             if lut == &self.lut {
                 return dst.write_usize(i);
             }
@@ -84,14 +84,14 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for LookupTableGat
         panic!("The associated lookup table couldn't be found.")
     }
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_slots = src.read_usize()?;
         let last_lut_row = src.read_usize()?;
         let lut_index = src.read_usize()?;
 
         Ok(Self {
             num_slots,
-            lut: cd.luts[lut_index].clone(),
+            lut: common_data.luts[lut_index].clone(),
             last_lut_row,
         })
     }
@@ -203,12 +203,12 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Loo
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_usize(self.slot_nb)?;
         dst.write_usize(self.num_slots)?;
         dst.write_usize(self.last_lut_row)?;
-        for (i, lut) in cd.luts.iter().enumerate() {
+        for (i, lut) in common_data.luts.iter().enumerate() {
             if lut == &self.lut {
                 return dst.write_usize(i);
             }
@@ -217,7 +217,7 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Loo
         panic!("The associated lookup table couldn't be found.")
     }
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let slot_nb = src.read_usize()?;
         let num_slots = src.read_usize()?;
@@ -226,7 +226,7 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Loo
 
         Ok(Self {
             row,
-            lut: cd.luts[lut_index].clone(),
+            lut: common_data.luts[lut_index].clone(),
             slot_nb,
             num_slots,
             last_lut_row,
diff --git a/plonky2/src/gates/multiplication_extension.rs b/plonky2/src/gates/multiplication_extension.rs
index ed330e85..6799bd37 100644
--- a/plonky2/src/gates/multiplication_extension.rs
+++ b/plonky2/src/gates/multiplication_extension.rs
@@ -53,11 +53,11 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for MulExtensionGa
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_ops)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let num_ops = src.read_usize()?;
         Ok(Self { num_ops })
     }
@@ -191,13 +191,13 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         out_buffer.set_extension_target(output_target, computed_output)
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_field(self.const_0)?;
         dst.write_usize(self.i)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let const_0 = src.read_field()?;
         let i = src.read_usize()?;
diff --git a/plonky2/src/gates/noop.rs b/plonky2/src/gates/noop.rs
index f1baae52..8752f380 100644
--- a/plonky2/src/gates/noop.rs
+++ b/plonky2/src/gates/noop.rs
@@ -19,11 +19,15 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for NoopGate {
         "NoopGate".into()
     }
 
-    fn serialize(&self, _dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(
+        &self,
+        _dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<F, D>,
+    ) -> IoResult<()> {
         Ok(())
     }
 
-    fn deserialize(_src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(_src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         Ok(Self)
     }
 
diff --git a/plonky2/src/gates/poseidon.rs b/plonky2/src/gates/poseidon.rs
index 0f68ea99..f6d06572 100644
--- a/plonky2/src/gates/poseidon.rs
+++ b/plonky2/src/gates/poseidon.rs
@@ -100,11 +100,15 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for PoseidonGate<F
         format!("{self:?}<WIDTH={SPONGE_WIDTH}>")
     }
 
-    fn serialize(&self, _dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(
+        &self,
+        _dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<F, D>,
+    ) -> IoResult<()> {
         Ok(())
     }
 
-    fn deserialize(_src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(_src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         Ok(PoseidonGate::new())
     }
 
@@ -513,11 +517,11 @@ impl<F: RichField + Extendable<D> + Poseidon, const D: usize> SimpleGenerator<F,
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         Ok(Self {
             row,
diff --git a/plonky2/src/gates/poseidon_mds.rs b/plonky2/src/gates/poseidon_mds.rs
index 59bbf48b..8e2f4a76 100644
--- a/plonky2/src/gates/poseidon_mds.rs
+++ b/plonky2/src/gates/poseidon_mds.rs
@@ -119,11 +119,15 @@ impl<F: RichField + Extendable<D> + Poseidon, const D: usize> Gate<F, D> for Pos
         format!("{self:?}<WIDTH={SPONGE_WIDTH}>")
     }
 
-    fn serialize(&self, _dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(
+        &self,
+        _dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<F, D>,
+    ) -> IoResult<()> {
         Ok(())
     }
 
-    fn deserialize(_src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(_src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         Ok(PoseidonMdsGate::new())
     }
 
@@ -251,11 +255,11 @@ impl<F: RichField + Extendable<D> + Poseidon, const D: usize> SimpleGenerator<F,
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         Ok(Self { row })
     }
diff --git a/plonky2/src/gates/public_input.rs b/plonky2/src/gates/public_input.rs
index 1d081819..f770e2e6 100644
--- a/plonky2/src/gates/public_input.rs
+++ b/plonky2/src/gates/public_input.rs
@@ -32,11 +32,15 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for PublicInputGat
         "PublicInputGate".into()
     }
 
-    fn serialize(&self, _dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(
+        &self,
+        _dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<F, D>,
+    ) -> IoResult<()> {
         Ok(())
     }
 
-    fn deserialize(_src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(_src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         Ok(Self)
     }
 
diff --git a/plonky2/src/gates/random_access.rs b/plonky2/src/gates/random_access.rs
index 38ba5a47..9110a59b 100644
--- a/plonky2/src/gates/random_access.rs
+++ b/plonky2/src/gates/random_access.rs
@@ -122,14 +122,14 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for RandomAccessGa
         format!("{self:?}<D={D}>")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.bits)?;
         dst.write_usize(self.num_copies)?;
         dst.write_usize(self.num_extra_constants)?;
         Ok(())
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let bits = src.read_usize()?;
         let num_copies = src.read_usize()?;
         let num_extra_constants = src.read_usize()?;
@@ -394,16 +394,16 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D>
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_usize(self.copy)?;
-        self.gate.serialize(dst, _cd)
+        self.gate.serialize(dst, _common_data)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let copy = src.read_usize()?;
-        let gate = RandomAccessGate::<F, D>::deserialize(src, _cd)?;
+        let gate = RandomAccessGate::<F, D>::deserialize(src, _common_data)?;
         Ok(Self { row, gate, copy })
     }
 }
diff --git a/plonky2/src/gates/reducing.rs b/plonky2/src/gates/reducing.rs
index e0587f02..b313efe6 100644
--- a/plonky2/src/gates/reducing.rs
+++ b/plonky2/src/gates/reducing.rs
@@ -61,12 +61,12 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ReducingGate<D
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_coeffs)?;
         Ok(())
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self>
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self>
     where
         Self: Sized,
     {
@@ -226,14 +226,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Red
         out_buffer.set_extension_target(output, acc);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
-        <ReducingGate<D> as Gate<F, D>>::serialize(&self.gate, dst, _cd)
+        <ReducingGate<D> as Gate<F, D>>::serialize(&self.gate, dst, _common_data)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
-        let gate = <ReducingGate<D> as Gate<F, D>>::deserialize(src, _cd)?;
+        let gate = <ReducingGate<D> as Gate<F, D>>::deserialize(src, _common_data)?;
         Ok(Self { row, gate })
     }
 }
diff --git a/plonky2/src/gates/reducing_extension.rs b/plonky2/src/gates/reducing_extension.rs
index 2c4e5aa6..5492c506 100644
--- a/plonky2/src/gates/reducing_extension.rs
+++ b/plonky2/src/gates/reducing_extension.rs
@@ -64,12 +64,12 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ReducingExtens
         format!("{self:?}")
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.num_coeffs)?;
         Ok(())
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self>
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self>
     where
         Self: Sized,
     {
@@ -220,14 +220,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Red
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
-        <ReducingExtensionGate<D> as Gate<F, D>>::serialize(&self.gate, dst, _cd)
+        <ReducingExtensionGate<D> as Gate<F, D>>::serialize(&self.gate, dst, _common_data)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
-        let gate = <ReducingExtensionGate<D> as Gate<F, D>>::deserialize(src, _cd)?;
+        let gate = <ReducingExtensionGate<D> as Gate<F, D>>::deserialize(src, _common_data)?;
         Ok(Self { row, gate })
     }
 }
diff --git a/plonky2/src/iop/generator.rs b/plonky2/src/iop/generator.rs
index a0cd8e76..478c5ff5 100644
--- a/plonky2/src/iop/generator.rs
+++ b/plonky2/src/iop/generator.rs
@@ -112,9 +112,9 @@ pub trait WitnessGenerator<F: RichField + Extendable<D>, const D: usize>:
     /// run next time a target in its watch list is populated.
     fn run(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) -> bool;
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()>;
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()>;
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self>
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self>
     where
         Self: Sized;
 }
@@ -213,9 +213,9 @@ pub trait SimpleGenerator<F: RichField + Extendable<D>, const D: usize>:
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()>;
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()>;
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self>
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self>
     where
         Self: Sized;
 }
@@ -250,13 +250,13 @@ impl<F: RichField + Extendable<D>, SG: SimpleGenerator<F, D>, const D: usize> Wi
         }
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, cd: &CommonCircuitData<F, D>) -> IoResult<()> {
-        self.inner.serialize(dst, cd)
+    fn serialize(&self, dst: &mut Vec<u8>, common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
+        self.inner.serialize(dst, common_data)
     }
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         Ok(Self {
-            inner: SG::deserialize(src, cd)?,
+            inner: SG::deserialize(src, common_data)?,
             _phantom: PhantomData,
         })
     }
@@ -283,12 +283,12 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Cop
         out_buffer.set_target(self.dst, value);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.src)?;
         dst.write_target(self.dst)
     }
 
-    fn deserialize(source: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(source: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let src = source.read_target()?;
         let dst = source.read_target()?;
         Ok(Self { src, dst })
@@ -315,11 +315,11 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Ran
         out_buffer.set_target(self.target, random_value);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.target)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let target = src.read_target()?;
         Ok(Self { target })
     }
@@ -353,12 +353,12 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Non
         out_buffer.set_target(self.dummy, dummy_value);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target(self.to_test)?;
         dst.write_target(self.dummy)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let to_test = src.read_target()?;
         let dummy = src.read_target()?;
         Ok(Self { to_test, dummy })
@@ -393,14 +393,14 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F, D> for Con
         out_buffer.set_target(Target::wire(self.row, self.wire_index), self.constant);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_usize(self.row)?;
         dst.write_usize(self.constant_index)?;
         dst.write_usize(self.wire_index)?;
         dst.write_field(self.constant)
     }
 
-    fn deserialize(src: &mut Buffer, _cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let row = src.read_usize()?;
         let constant_index = src.read_usize()?;
         let wire_index = src.read_usize()?;
diff --git a/plonky2/src/recursion/dummy_circuit.rs b/plonky2/src/recursion/dummy_circuit.rs
index 897abb75..620c979f 100644
--- a/plonky2/src/recursion/dummy_circuit.rs
+++ b/plonky2/src/recursion/dummy_circuit.rs
@@ -234,16 +234,16 @@ where
         out_buffer.set_verifier_data_target(&self.verifier_data_target, &self.verifier_data);
     }
 
-    fn serialize(&self, dst: &mut Vec<u8>, _cd: &CommonCircuitData<F, D>) -> IoResult<()> {
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
         dst.write_target_proof_with_public_inputs(&self.proof_with_pis_target)?;
         dst.write_proof_with_public_inputs(&self.proof_with_pis)?;
         dst.write_target_verifier_circuit(&self.verifier_data_target)?;
         dst.write_verifier_only_circuit_data(&self.verifier_data)
     }
 
-    fn deserialize(src: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<Self> {
+    fn deserialize(src: &mut Buffer, common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
         let proof_with_pis_target = src.read_target_proof_with_public_inputs()?;
-        let proof_with_pis = src.read_proof_with_public_inputs(cd)?;
+        let proof_with_pis = src.read_proof_with_public_inputs(common_data)?;
         let verifier_data_target = src.read_target_verifier_circuit()?;
         let verifier_data = src.read_verifier_only_circuit_data()?;
         Ok(Self {
diff --git a/plonky2/src/recursion/recursive_verifier.rs b/plonky2/src/recursion/recursive_verifier.rs
index 613766e4..ada2b002 100644
--- a/plonky2/src/recursion/recursive_verifier.rs
+++ b/plonky2/src/recursion/recursive_verifier.rs
@@ -218,10 +218,10 @@ mod tests {
         type F = <C as GenericConfig<D>>::F;
         let config = CircuitConfig::standard_recursion_zk_config();
 
-        let (proof, vd, cd) = dummy_proof::<F, C, D>(&config, 4_000)?;
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &config, None, true, true)?;
-        test_serialization(&proof, &vd, &cd)?;
+        let (proof, vd, common_data) = dummy_proof::<F, C, D>(&config, 4_000)?;
+        let (proof, vd, common_data) =
+            recursive_proof::<F, C, C, D>(proof, vd, common_data, &config, None, true, true)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
         Ok(())
     }
@@ -234,10 +234,10 @@ mod tests {
         type F = <C as GenericConfig<D>>::F;
         let config = CircuitConfig::standard_recursion_zk_config();
 
-        let (proof, vd, cd) = dummy_lookup_proof::<F, C, D>(&config, 10)?;
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &config, None, true, true)?;
-        test_serialization(&proof, &vd, &cd)?;
+        let (proof, vd, common_data) = dummy_lookup_proof::<F, C, D>(&config, 10)?;
+        let (proof, vd, common_data) =
+            recursive_proof::<F, C, C, D>(proof, vd, common_data, &config, None, true, true)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
         Ok(())
     }
@@ -250,10 +250,10 @@ mod tests {
         type F = <C as GenericConfig<D>>::F;
         let config = CircuitConfig::standard_recursion_config();
 
-        let (proof, vd, cd) = dummy_two_luts_proof::<F, C, D>(&config)?;
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &config, None, true, true)?;
-        test_serialization(&proof, &vd, &cd)?;
+        let (proof, vd, common_data) = dummy_two_luts_proof::<F, C, D>(&config)?;
+        let (proof, vd, common_data) =
+            recursive_proof::<F, C, C, D>(proof, vd, common_data, &config, None, true, true)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
         Ok(())
     }
@@ -266,10 +266,10 @@ mod tests {
         type F = <C as GenericConfig<D>>::F;
         let config = CircuitConfig::standard_recursion_config();
 
-        let (proof, vd, cd) = dummy_too_many_rows_proof::<F, C, D>(&config)?;
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &config, None, true, true)?;
-        test_serialization(&proof, &vd, &cd)?;
+        let (proof, vd, common_data) = dummy_too_many_rows_proof::<F, C, D>(&config)?;
+        let (proof, vd, common_data) =
+            recursive_proof::<F, C, C, D>(proof, vd, common_data, &config, None, true, true)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
         Ok(())
     }
@@ -284,20 +284,20 @@ mod tests {
         let config = CircuitConfig::standard_recursion_config();
 
         // Start with a degree 2^14 proof
-        let (proof, vd, cd) = dummy_proof::<F, C, D>(&config, 16_000)?;
-        assert_eq!(cd.degree_bits(), 14);
+        let (proof, vd, common_data) = dummy_proof::<F, C, D>(&config, 16_000)?;
+        assert_eq!(common_data.degree_bits(), 14);
 
         // Shrink it to 2^13.
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &config, Some(13), false, false)?;
-        assert_eq!(cd.degree_bits(), 13);
+        let (proof, vd, common_data) =
+            recursive_proof::<F, C, C, D>(proof, vd, common_data, &config, Some(13), false, false)?;
+        assert_eq!(common_data.degree_bits(), 13);
 
         // Shrink it to 2^12.
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &config, None, true, true)?;
-        assert_eq!(cd.degree_bits(), 12);
+        let (proof, vd, common_data) =
+            recursive_proof::<F, C, C, D>(proof, vd, common_data, &config, None, true, true)?;
+        assert_eq!(common_data.degree_bits(), 12);
 
-        test_serialization(&proof, &vd, &cd)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
         Ok(())
     }
@@ -316,13 +316,20 @@ mod tests {
         let standard_config = CircuitConfig::standard_recursion_config();
 
         // An initial dummy proof.
-        let (proof, vd, cd) = dummy_proof::<F, C, D>(&standard_config, 4_000)?;
-        assert_eq!(cd.degree_bits(), 12);
+        let (proof, vd, common_data) = dummy_proof::<F, C, D>(&standard_config, 4_000)?;
+        assert_eq!(common_data.degree_bits(), 12);
 
         // A standard recursive proof.
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &standard_config, None, false, false)?;
-        assert_eq!(cd.degree_bits(), 12);
+        let (proof, vd, common_data) = recursive_proof::<F, C, C, D>(
+            proof,
+            vd,
+            common_data,
+            &standard_config,
+            None,
+            false,
+            false,
+        )?;
+        assert_eq!(common_data.degree_bits(), 12);
 
         // A high-rate recursive proof, designed to be verifiable with fewer routed wires.
         let high_rate_config = CircuitConfig {
@@ -334,9 +341,16 @@ mod tests {
             },
             ..standard_config
         };
-        let (proof, vd, cd) =
-            recursive_proof::<F, C, C, D>(proof, vd, cd, &high_rate_config, None, true, true)?;
-        assert_eq!(cd.degree_bits(), 12);
+        let (proof, vd, common_data) = recursive_proof::<F, C, C, D>(
+            proof,
+            vd,
+            common_data,
+            &high_rate_config,
+            None,
+            true,
+            true,
+        )?;
+        assert_eq!(common_data.degree_bits(), 12);
 
         // A final proof, optimized for size.
         let final_config = CircuitConfig {
@@ -350,11 +364,18 @@ mod tests {
             },
             ..high_rate_config
         };
-        let (proof, vd, cd) =
-            recursive_proof::<F, KC, C, D>(proof, vd, cd, &final_config, None, true, true)?;
-        assert_eq!(cd.degree_bits(), 12, "final proof too large");
+        let (proof, vd, common_data) = recursive_proof::<F, KC, C, D>(
+            proof,
+            vd,
+            common_data,
+            &final_config,
+            None,
+            true,
+            true,
+        )?;
+        assert_eq!(common_data.degree_bits(), 12, "final proof too large");
 
-        test_serialization(&proof, &vd, &cd)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
         Ok(())
     }
@@ -368,15 +389,15 @@ mod tests {
         type F = <PC as GenericConfig<D>>::F;
 
         let config = CircuitConfig::standard_recursion_config();
-        let (proof, vd, cd) = dummy_proof::<F, PC, D>(&config, 4_000)?;
+        let (proof, vd, common_data) = dummy_proof::<F, PC, D>(&config, 4_000)?;
 
-        let (proof, vd, cd) =
-            recursive_proof::<F, PC, PC, D>(proof, vd, cd, &config, None, false, false)?;
-        test_serialization(&proof, &vd, &cd)?;
+        let (proof, vd, common_data) =
+            recursive_proof::<F, PC, PC, D>(proof, vd, common_data, &config, None, false, false)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
-        let (proof, vd, cd) =
-            recursive_proof::<F, KC, PC, D>(proof, vd, cd, &config, None, false, false)?;
-        test_serialization(&proof, &vd, &cd)?;
+        let (proof, vd, common_data) =
+            recursive_proof::<F, KC, PC, D>(proof, vd, common_data, &config, None, false, false)?;
+        test_serialization(&proof, &vd, &common_data)?;
 
         Ok(())
     }
@@ -662,18 +683,18 @@ mod tests {
     >(
         proof: &ProofWithPublicInputs<F, C, D>,
         vd: &VerifierOnlyCircuitData<C, D>,
-        cd: &CommonCircuitData<F, D>,
+        common_data: &CommonCircuitData<F, D>,
     ) -> Result<()> {
         let proof_bytes = proof.to_bytes();
         info!("Proof length: {} bytes", proof_bytes.len());
-        let proof_from_bytes = ProofWithPublicInputs::from_bytes(proof_bytes, cd)?;
+        let proof_from_bytes = ProofWithPublicInputs::from_bytes(proof_bytes, common_data)?;
         assert_eq!(proof, &proof_from_bytes);
 
         let now = std::time::Instant::now();
-        let compressed_proof = proof.clone().compress(&vd.circuit_digest, cd)?;
+        let compressed_proof = proof.clone().compress(&vd.circuit_digest, common_data)?;
         let decompressed_compressed_proof = compressed_proof
             .clone()
-            .decompress(&vd.circuit_digest, cd)?;
+            .decompress(&vd.circuit_digest, common_data)?;
         info!("{:.4}s to compress proof", now.elapsed().as_secs_f64());
         assert_eq!(proof, &decompressed_compressed_proof);
 
@@ -683,7 +704,7 @@ mod tests {
             compressed_proof_bytes.len()
         );
         let compressed_proof_from_bytes =
-            CompressedProofWithPublicInputs::from_bytes(compressed_proof_bytes, cd)?;
+            CompressedProofWithPublicInputs::from_bytes(compressed_proof_bytes, common_data)?;
         assert_eq!(compressed_proof, compressed_proof_from_bytes);
 
         Ok(())
diff --git a/plonky2/src/util/serialization/gate_serialization.rs b/plonky2/src/util/serialization/gate_serialization.rs
index 2ed43990..008e29c0 100644
--- a/plonky2/src/util/serialization/gate_serialization.rs
+++ b/plonky2/src/util/serialization/gate_serialization.rs
@@ -6,12 +6,16 @@ use crate::plonk::circuit_data::CommonCircuitData;
 use crate::util::serialization::{Buffer, IoResult};
 
 pub trait GateSerializer<F: RichField + Extendable<D>, const D: usize> {
-    fn read_gate(&self, buf: &mut Buffer, cd: &CommonCircuitData<F, D>) -> IoResult<GateRef<F, D>>;
+    fn read_gate(
+        &self,
+        buf: &mut Buffer,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> IoResult<GateRef<F, D>>;
     fn write_gate(
         &self,
         buf: &mut Vec<u8>,
         gate: &GateRef<F, D>,
-        cd: &CommonCircuitData<F, D>,
+        common_data: &CommonCircuitData<F, D>,
     ) -> IoResult<()>;
 }
 
diff --git a/plonky2/src/util/serialization/generator_serialization.rs b/plonky2/src/util/serialization/generator_serialization.rs
index a7377de8..6e003400 100644
--- a/plonky2/src/util/serialization/generator_serialization.rs
+++ b/plonky2/src/util/serialization/generator_serialization.rs
@@ -11,14 +11,14 @@ pub trait WitnessGeneratorSerializer<F: RichField + Extendable<D>, const D: usiz
     fn read_generator(
         &self,
         buf: &mut Buffer,
-        cd: &CommonCircuitData<F, D>,
+        common_data: &CommonCircuitData<F, D>,
     ) -> IoResult<WitnessGeneratorRef<F, D>>;
 
     fn write_generator(
         &self,
         buf: &mut Vec<u8>,
         generator: &WitnessGeneratorRef<F, D>,
-        cd: &CommonCircuitData<F, D>,
+        common_data: &CommonCircuitData<F, D>,
     ) -> IoResult<()>;
 }
 
diff --git a/plonky2/src/util/serialization/mod.rs b/plonky2/src/util/serialization/mod.rs
index 79c11c40..2751ec69 100644
--- a/plonky2/src/util/serialization/mod.rs
+++ b/plonky2/src/util/serialization/mod.rs
@@ -769,7 +769,7 @@ pub trait Read {
 
         // We construct the common data without gates first,
         // to pass it as argument when reading the gates.
-        let mut cd = CommonCircuitData {
+        let mut common_data = CommonCircuitData {
             config,
             fri_params,
             gates: vec![],
@@ -786,13 +786,13 @@ pub trait Read {
         };
 
         for _ in 0..gates_len {
-            let gate = self.read_gate::<F, D>(gate_serializer, &cd)?;
+            let gate = self.read_gate::<F, D>(gate_serializer, &common_data)?;
             gates.push(gate);
         }
 
-        cd.gates = gates;
+        common_data.gates = gates;
 
-        Ok(cd)
+        Ok(common_data)
     }
 
     fn read_circuit_data<
@@ -821,12 +821,12 @@ pub trait Read {
     >(
         &mut self,
         generator_serializer: &dyn WitnessGeneratorSerializer<F, D>,
-        cd: &CommonCircuitData<F, D>,
+        common_data: &CommonCircuitData<F, D>,
     ) -> IoResult<ProverOnlyCircuitData<F, C, D>> {
         let gen_len = self.read_usize()?;
         let mut generators = Vec::with_capacity(gen_len);
         for _ in 0..gen_len {
-            generators.push(self.read_generator(generator_serializer, cd)?);
+            generators.push(self.read_generator(generator_serializer, common_data)?);
         }
         let map_len = self.read_usize()?;
         let mut generator_indices_by_watches = BTreeMap::new();