diff --git a/src/fri/mod.rs b/src/fri/mod.rs
index 716260a3..5e8936fa 100644
--- a/src/fri/mod.rs
+++ b/src/fri/mod.rs
@@ -17,3 +17,9 @@ pub struct FriConfig {
     /// Number of query rounds to perform.
     pub num_query_rounds: usize,
 }
+
+impl FriConfig {
+    pub(crate) fn total_arities(&self) -> usize {
+        self.reduction_arity_bits.iter().sum()
+    }
+}
diff --git a/src/fri/proof.rs b/src/fri/proof.rs
index 1761744d..62e836af 100644
--- a/src/fri/proof.rs
+++ b/src/fri/proof.rs
@@ -83,6 +83,8 @@ pub struct FriQueryRoundTarget<const D: usize> {
 #[derive(Serialize, Deserialize, Clone, Debug, Eq, PartialEq)]
 #[serde(bound = "")]
 pub struct CompressedFriQueryRounds<F: Extendable<D>, const D: usize> {
+    /// Query indices.
+    pub indices: Vec<usize>,
     /// Map from initial indices `i` to the `FriInitialProof` for the `i`th leaf.
     pub initial_trees_proofs: HashMap<usize, FriInitialTreeProof<F>>,
     /// For each FRI query step, a map from indices `i` to the `FriQueryStep` for the `i`th leaf.
@@ -182,6 +184,7 @@ impl<F: RichField + Extendable<D>, const D: usize> FriProof<F, D> {
             .collect::<Vec<_>>();
 
         let mut compressed_query_proofs = CompressedFriQueryRounds {
+            indices: indices.to_vec(),
             initial_trees_proofs: HashMap::new(),
             steps: vec![HashMap::new(); num_reductions],
         };
diff --git a/src/fri/recursive_verifier.rs b/src/fri/recursive_verifier.rs
index 2c0d009f..78d5f007 100644
--- a/src/fri/recursive_verifier.rs
+++ b/src/fri/recursive_verifier.rs
@@ -81,15 +81,14 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         common_data: &CommonCircuitData<F, D>,
     ) {
         let config = &common_data.config;
-        let total_arities = config.fri_config.reduction_arity_bits.iter().sum::<usize>();
         debug_assert_eq!(
-            common_data.degree_bits,
-            log2_strict(proof.final_poly.len()) + total_arities,
+            common_data.final_poly_len(),
+            proof.final_poly.len(),
             "Final polynomial has wrong degree."
         );
 
         // Size of the LDE domain.
-        let n = proof.final_poly.len() << (total_arities + config.rate_bits);
+        let n = common_data.lde_size();
 
         challenger.observe_opening_set(os);
 
diff --git a/src/fri/verifier.rs b/src/fri/verifier.rs
index 6662830a..f9c1d998 100644
--- a/src/fri/verifier.rs
+++ b/src/fri/verifier.rs
@@ -64,14 +64,13 @@ pub(crate) fn verify_fri_proof<F: RichField + Extendable<D>, const D: usize>(
     common_data: &CommonCircuitData<F, D>,
 ) -> Result<()> {
     let config = &common_data.config;
-    let total_arities = config.fri_config.reduction_arity_bits.iter().sum::<usize>();
     ensure!(
-        common_data.degree_bits == log2_strict(proof.final_poly.len()) + total_arities,
+        common_data.final_poly_len() == proof.final_poly.len(),
         "Final polynomial has wrong degree."
     );
 
     // Size of the LDE domain.
-    let n = proof.final_poly.len() << (total_arities + config.rate_bits);
+    let n = common_data.lde_size();
 
     // Check PoW.
     fri_verify_proof_of_work(challenges.fri_pow_response, &config.fri_config)?;
diff --git a/src/plonk/circuit_builder.rs b/src/plonk/circuit_builder.rs
index 035a6d90..18f84681 100644
--- a/src/plonk/circuit_builder.rs
+++ b/src/plonk/circuit_builder.rs
@@ -579,12 +579,7 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         info!("Degree after blinding & padding: {}", degree);
         let degree_bits = log2_strict(degree);
         assert!(
-            self.config
-                .fri_config
-                .reduction_arity_bits
-                .iter()
-                .sum::<usize>()
-                <= degree_bits,
+            self.config.fri_config.total_arities() <= degree_bits,
             "FRI total reduction arity is too large."
         );
 
diff --git a/src/plonk/circuit_data.rs b/src/plonk/circuit_data.rs
index 644ab370..f8bb0b6a 100644
--- a/src/plonk/circuit_data.rs
+++ b/src/plonk/circuit_data.rs
@@ -252,6 +252,10 @@ impl<F: RichField + Extendable<D>, const D: usize> CommonCircuitData<F, D> {
     pub fn partial_products_range(&self) -> RangeFrom<usize> {
         self.config.num_challenges..
     }
+
+    pub fn final_poly_len(&self) -> usize {
+        1 << (self.degree_bits - self.config.fri_config.total_arities())
+    }
 }
 
 /// The `Target` version of `VerifierCircuitData`, for use inside recursive circuits. Note that this
diff --git a/src/plonk/proof.rs b/src/plonk/proof.rs
index 9510687a..4021f3cb 100644
--- a/src/plonk/proof.rs
+++ b/src/plonk/proof.rs
@@ -11,6 +11,7 @@ use crate::hash::hashing::hash_n_to_hash;
 use crate::hash::merkle_tree::MerkleCap;
 use crate::iop::target::Target;
 use crate::plonk::circuit_data::CommonCircuitData;
+use crate::util::serialization::Buffer;
 
 #[derive(Serialize, Deserialize, Clone, Debug, Eq, PartialEq)]
 #[serde(bound = "")]
@@ -83,6 +84,21 @@ impl<F: RichField + Extendable<D>, const D: usize> ProofWithPublicInputs<F, D> {
     pub(crate) fn get_public_inputs_hash(&self) -> HashOut<F> {
         hash_n_to_hash(self.public_inputs.clone(), true)
     }
+
+    pub fn to_bytes(&self) -> anyhow::Result<Vec<u8>> {
+        let mut buffer = Buffer::new(Vec::new());
+        buffer.write_proof_with_public_inputs(self)?;
+        Ok(buffer.bytes())
+    }
+
+    pub fn from_bytes(
+        bytes: Vec<u8>,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> anyhow::Result<Self> {
+        let mut buffer = Buffer::new(bytes);
+        let proof = buffer.read_proof_with_public_inputs(common_data)?;
+        Ok(proof)
+    }
 }
 
 #[derive(Serialize, Deserialize, Clone, Debug, Eq, PartialEq)]
@@ -148,6 +164,21 @@ impl<F: RichField + Extendable<D>, const D: usize> CompressedProofWithPublicInpu
     pub(crate) fn get_public_inputs_hash(&self) -> HashOut<F> {
         hash_n_to_hash(self.public_inputs.clone(), true)
     }
+
+    pub fn to_bytes(&self) -> anyhow::Result<Vec<u8>> {
+        let mut buffer = Buffer::new(Vec::new());
+        buffer.write_compressed_proof_with_public_inputs(self)?;
+        Ok(buffer.bytes())
+    }
+
+    pub fn from_bytes(
+        bytes: Vec<u8>,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> anyhow::Result<Self> {
+        let mut buffer = Buffer::new(bytes);
+        let proof = buffer.read_compressed_proof_with_public_inputs(common_data)?;
+        Ok(proof)
+    }
 }
 
 pub(crate) struct ProofChallenges<F: RichField + Extendable<D>, const D: usize> {
diff --git a/src/plonk/recursive_verifier.rs b/src/plonk/recursive_verifier.rs
index 15dd7830..33bcbb67 100644
--- a/src/plonk/recursive_verifier.rs
+++ b/src/plonk/recursive_verifier.rs
@@ -136,7 +136,10 @@ mod tests {
     use crate::gadgets::polynomial::PolynomialCoeffsExtTarget;
     use crate::hash::merkle_proofs::MerkleProofTarget;
     use crate::iop::witness::{PartialWitness, Witness};
-    use crate::plonk::proof::{OpeningSetTarget, Proof, ProofTarget, ProofWithPublicInputs};
+    use crate::plonk::proof::{
+        CompressedProofWithPublicInputs, OpeningSetTarget, Proof, ProofTarget,
+        ProofWithPublicInputs,
+    };
     use crate::plonk::verifier::verify;
     use crate::util::log2_strict;
 
@@ -480,6 +483,10 @@ mod tests {
         builder.print_gate_counts(0);
         let data = builder.build();
         let recursive_proof = data.prove(pw)?;
+        let proof_bytes = recursive_proof.to_bytes()?;
+        info!("Proof length: {} bytes", proof_bytes.len());
+        let proof_from_bytes = ProofWithPublicInputs::from_bytes(proof_bytes, &data.common)?;
+        assert_eq!(recursive_proof, proof_from_bytes);
         let now = std::time::Instant::now();
         let compressed_recursive_proof = recursive_proof.clone().compress(&data.common)?;
         let decompressed_compressed_proof = compressed_recursive_proof
@@ -487,13 +494,14 @@ mod tests {
             .decompress(&data.common)?;
         assert_eq!(recursive_proof, decompressed_compressed_proof);
         info!("{:.4} to compress proof", now.elapsed().as_secs_f64());
-        let proof_bytes = serde_cbor::to_vec(&recursive_proof).unwrap();
-        info!("Proof length: {} bytes", proof_bytes.len());
-        let compressed_proof_bytes = serde_cbor::to_vec(&compressed_recursive_proof).unwrap();
+        let compressed_proof_bytes = compressed_recursive_proof.to_bytes()?;
         info!(
             "Compressed proof length: {} bytes",
             compressed_proof_bytes.len()
         );
+        let compressed_proof_from_bytes =
+            CompressedProofWithPublicInputs::from_bytes(compressed_proof_bytes, &data.common)?;
+        assert_eq!(compressed_recursive_proof, compressed_proof_from_bytes);
         verify(recursive_proof, &data.verifier_only, &data.common)
     }
 }
diff --git a/src/util/mod.rs b/src/util/mod.rs
index daa6716b..d8e05a18 100644
--- a/src/util/mod.rs
+++ b/src/util/mod.rs
@@ -6,6 +6,7 @@ pub(crate) mod context_tree;
 pub(crate) mod marking;
 pub(crate) mod partial_products;
 pub mod reducing;
+pub mod serialization;
 pub(crate) mod timing;
 
 pub(crate) fn bits_u64(n: u64) -> usize {
diff --git a/src/util/serialization.rs b/src/util/serialization.rs
new file mode 100644
index 00000000..21b12c11
--- /dev/null
+++ b/src/util/serialization.rs
@@ -0,0 +1,537 @@
+use std::collections::HashMap;
+use std::convert::TryInto;
+use std::io::Cursor;
+use std::io::{Read, Result, Write};
+use std::iter::FromIterator;
+
+use crate::field::extension_field::{Extendable, FieldExtension};
+use crate::field::field_types::{PrimeField, RichField};
+use crate::fri::proof::{
+    CompressedFriProof, CompressedFriQueryRounds, FriInitialTreeProof, FriProof, FriQueryRound,
+    FriQueryStep,
+};
+use crate::hash::hash_types::HashOut;
+use crate::hash::merkle_proofs::MerkleProof;
+use crate::hash::merkle_tree::MerkleCap;
+use crate::plonk::circuit_data::CommonCircuitData;
+use crate::plonk::proof::{
+    CompressedProof, CompressedProofWithPublicInputs, OpeningSet, Proof, ProofWithPublicInputs,
+};
+use crate::polynomial::polynomial::PolynomialCoeffs;
+
+#[derive(Debug)]
+pub struct Buffer(Cursor<Vec<u8>>);
+
+impl Buffer {
+    pub fn new(buffer: Vec<u8>) -> Self {
+        Self(Cursor::new(buffer))
+    }
+
+    pub fn len(&self) -> usize {
+        self.0.get_ref().len()
+    }
+
+    pub fn bytes(self) -> Vec<u8> {
+        self.0.into_inner()
+    }
+
+    fn write_u8(&mut self, x: u8) -> Result<()> {
+        self.0.write_all(&[x])
+    }
+    fn read_u8(&mut self) -> Result<u8> {
+        let mut buf = [0; std::mem::size_of::<u8>()];
+        self.0.read_exact(&mut buf)?;
+        Ok(buf[0])
+    }
+
+    fn write_u32(&mut self, x: u32) -> Result<()> {
+        self.0.write_all(&x.to_le_bytes())
+    }
+    fn read_u32(&mut self) -> Result<u32> {
+        let mut buf = [0; std::mem::size_of::<u32>()];
+        self.0.read_exact(&mut buf)?;
+        Ok(u32::from_le_bytes(buf))
+    }
+
+    fn write_field<F: PrimeField>(&mut self, x: F) -> Result<()> {
+        self.0.write_all(&x.to_canonical_u64().to_le_bytes())
+    }
+    fn read_field<F: PrimeField>(&mut self) -> Result<F> {
+        let mut buf = [0; std::mem::size_of::<u64>()];
+        self.0.read_exact(&mut buf)?;
+        Ok(F::from_canonical_u64(u64::from_le_bytes(
+            buf.try_into().unwrap(),
+        )))
+    }
+
+    fn write_field_ext<F: Extendable<D>, const D: usize>(&mut self, x: F::Extension) -> Result<()> {
+        for &a in &x.to_basefield_array() {
+            self.write_field(a)?;
+        }
+        Ok(())
+    }
+    fn read_field_ext<F: Extendable<D>, const D: usize>(&mut self) -> Result<F::Extension> {
+        let mut arr = [F::ZERO; D];
+        for a in arr.iter_mut() {
+            *a = self.read_field()?;
+        }
+        Ok(<F::Extension as FieldExtension<D>>::from_basefield_array(
+            arr,
+        ))
+    }
+
+    fn write_hash<F: PrimeField>(&mut self, h: HashOut<F>) -> Result<()> {
+        for &a in &h.elements {
+            self.write_field(a)?;
+        }
+        Ok(())
+    }
+    fn read_hash<F: PrimeField>(&mut self) -> Result<HashOut<F>> {
+        let mut elements = [F::ZERO; 4];
+        for a in elements.iter_mut() {
+            *a = self.read_field()?;
+        }
+        Ok(HashOut { elements })
+    }
+
+    fn write_merkle_cap<F: PrimeField>(&mut self, cap: &MerkleCap<F>) -> Result<()> {
+        for &a in &cap.0 {
+            self.write_hash(a)?;
+        }
+        Ok(())
+    }
+    fn read_merkle_cap<F: PrimeField>(&mut self, cap_height: usize) -> Result<MerkleCap<F>> {
+        let cap_length = 1 << cap_height;
+        Ok(MerkleCap(
+            (0..cap_length)
+                .map(|_| self.read_hash())
+                .collect::<Result<Vec<_>>>()?,
+        ))
+    }
+
+    fn write_field_vec<F: PrimeField>(&mut self, v: &[F]) -> Result<()> {
+        for &a in v {
+            self.write_field(a)?;
+        }
+        Ok(())
+    }
+    fn read_field_vec<F: PrimeField>(&mut self, length: usize) -> Result<Vec<F>> {
+        (0..length)
+            .map(|_| self.read_field())
+            .collect::<Result<Vec<_>>>()
+    }
+
+    fn write_field_ext_vec<F: Extendable<D>, const D: usize>(
+        &mut self,
+        v: &[F::Extension],
+    ) -> Result<()> {
+        for &a in v {
+            self.write_field_ext::<F, D>(a)?;
+        }
+        Ok(())
+    }
+    fn read_field_ext_vec<F: Extendable<D>, const D: usize>(
+        &mut self,
+        length: usize,
+    ) -> Result<Vec<F::Extension>> {
+        (0..length)
+            .map(|_| self.read_field_ext::<F, D>())
+            .collect::<Result<Vec<_>>>()
+    }
+
+    fn write_opening_set<F: Extendable<D>, const D: usize>(
+        &mut self,
+        os: &OpeningSet<F, D>,
+    ) -> Result<()> {
+        self.write_field_ext_vec::<F, D>(&os.constants)?;
+        self.write_field_ext_vec::<F, D>(&os.plonk_sigmas)?;
+        self.write_field_ext_vec::<F, D>(&os.wires)?;
+        self.write_field_ext_vec::<F, D>(&os.plonk_zs)?;
+        self.write_field_ext_vec::<F, D>(&os.plonk_zs_right)?;
+        self.write_field_ext_vec::<F, D>(&os.partial_products)?;
+        self.write_field_ext_vec::<F, D>(&os.quotient_polys)
+    }
+    fn read_opening_set<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<OpeningSet<F, D>> {
+        let config = &common_data.config;
+        let constants = self.read_field_ext_vec::<F, D>(common_data.num_constants)?;
+        let plonk_sigmas = self.read_field_ext_vec::<F, D>(config.num_routed_wires)?;
+        let wires = self.read_field_ext_vec::<F, D>(config.num_wires)?;
+        let plonk_zs = self.read_field_ext_vec::<F, D>(config.num_challenges)?;
+        let plonk_zs_right = self.read_field_ext_vec::<F, D>(config.num_challenges)?;
+        let partial_products = self.read_field_ext_vec::<F, D>(
+            common_data.num_partial_products.0 * config.num_challenges,
+        )?;
+        let quotient_polys = self.read_field_ext_vec::<F, D>(
+            common_data.quotient_degree_factor * config.num_challenges,
+        )?;
+        Ok(OpeningSet {
+            constants,
+            plonk_sigmas,
+            wires,
+            plonk_zs,
+            plonk_zs_right,
+            partial_products,
+            quotient_polys,
+        })
+    }
+
+    fn write_merkle_proof<F: PrimeField>(&mut self, p: &MerkleProof<F>) -> Result<()> {
+        let length = p.siblings.len();
+        self.write_u8(
+            length
+                .try_into()
+                .expect("Merkle proof length must fit in u8."),
+        )?;
+        for &h in &p.siblings {
+            self.write_hash(h)?;
+        }
+        Ok(())
+    }
+    fn read_merkle_proof<F: PrimeField>(&mut self) -> Result<MerkleProof<F>> {
+        let length = self.read_u8()?;
+        Ok(MerkleProof {
+            siblings: (0..length)
+                .map(|_| self.read_hash())
+                .collect::<Result<Vec<_>>>()?,
+        })
+    }
+
+    fn write_fri_initial_proof<F: PrimeField>(
+        &mut self,
+        fitp: &FriInitialTreeProof<F>,
+    ) -> Result<()> {
+        for (v, p) in &fitp.evals_proofs {
+            self.write_field_vec(v)?;
+            self.write_merkle_proof(p)?;
+        }
+        Ok(())
+    }
+    fn read_fri_initial_proof<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<FriInitialTreeProof<F>> {
+        let config = &common_data.config;
+        let mut evals_proofs = Vec::with_capacity(4);
+
+        let constants_sigmas_v =
+            self.read_field_vec(common_data.num_constants + config.num_routed_wires)?;
+        let constants_sigmas_p = self.read_merkle_proof()?;
+        evals_proofs.push((constants_sigmas_v, constants_sigmas_p));
+
+        let wires_v = self.read_field_vec(config.num_wires)?;
+        let wires_p = self.read_merkle_proof()?;
+        evals_proofs.push((wires_v, wires_p));
+
+        let zs_partial_v =
+            self.read_field_vec(config.num_challenges * (1 + common_data.num_partial_products.0))?;
+        let zs_partial_p = self.read_merkle_proof()?;
+        evals_proofs.push((zs_partial_v, zs_partial_p));
+
+        let quotient_v =
+            self.read_field_vec(config.num_challenges * common_data.quotient_degree_factor)?;
+        let quotient_p = self.read_merkle_proof()?;
+        evals_proofs.push((quotient_v, quotient_p));
+
+        Ok(FriInitialTreeProof { evals_proofs })
+    }
+
+    fn write_fri_query_step<F: Extendable<D>, const D: usize>(
+        &mut self,
+        fqs: &FriQueryStep<F, D>,
+    ) -> Result<()> {
+        self.write_field_ext_vec::<F, D>(&fqs.evals)?;
+        self.write_merkle_proof(&fqs.merkle_proof)
+    }
+    fn read_fri_query_step<F: Extendable<D>, const D: usize>(
+        &mut self,
+        arity: usize,
+    ) -> Result<FriQueryStep<F, D>> {
+        let evals = self.read_field_ext_vec::<F, D>(arity)?;
+        let merkle_proof = self.read_merkle_proof()?;
+        Ok(FriQueryStep {
+            evals,
+            merkle_proof,
+        })
+    }
+
+    fn write_fri_query_rounds<F: Extendable<D>, const D: usize>(
+        &mut self,
+        fqrs: &[FriQueryRound<F, D>],
+    ) -> Result<()> {
+        for fqr in fqrs {
+            self.write_fri_initial_proof(&fqr.initial_trees_proof)?;
+            for fqs in &fqr.steps {
+                self.write_fri_query_step(fqs)?;
+            }
+        }
+        Ok(())
+    }
+    fn read_fri_query_rounds<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<Vec<FriQueryRound<F, D>>> {
+        let config = &common_data.config;
+        let mut fqrs = Vec::with_capacity(config.fri_config.num_query_rounds);
+        for _ in 0..config.fri_config.num_query_rounds {
+            let initial_trees_proof = self.read_fri_initial_proof(common_data)?;
+            let steps = config
+                .fri_config
+                .reduction_arity_bits
+                .iter()
+                .map(|&ar| self.read_fri_query_step(1 << ar))
+                .collect::<Result<_>>()?;
+            fqrs.push(FriQueryRound {
+                initial_trees_proof,
+                steps,
+            })
+        }
+        Ok(fqrs)
+    }
+
+    fn write_fri_proof<F: Extendable<D>, const D: usize>(
+        &mut self,
+        fp: &FriProof<F, D>,
+    ) -> Result<()> {
+        for cap in &fp.commit_phase_merkle_caps {
+            self.write_merkle_cap(cap)?;
+        }
+        self.write_fri_query_rounds(&fp.query_round_proofs)?;
+        self.write_field_ext_vec::<F, D>(&fp.final_poly.coeffs)?;
+        self.write_field(fp.pow_witness)
+    }
+    fn read_fri_proof<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<FriProof<F, D>> {
+        let config = &common_data.config;
+        let commit_phase_merkle_caps = (0..config.fri_config.reduction_arity_bits.len())
+            .map(|_| self.read_merkle_cap(config.cap_height))
+            .collect::<Result<Vec<_>>>()?;
+        let query_round_proofs = self.read_fri_query_rounds(common_data)?;
+        let final_poly =
+            PolynomialCoeffs::new(self.read_field_ext_vec::<F, D>(common_data.final_poly_len())?);
+        let pow_witness = self.read_field()?;
+        Ok(FriProof {
+            commit_phase_merkle_caps,
+            query_round_proofs,
+            final_poly,
+            pow_witness,
+        })
+    }
+
+    pub fn write_proof<F: Extendable<D>, const D: usize>(
+        &mut self,
+        proof: &Proof<F, D>,
+    ) -> Result<()> {
+        self.write_merkle_cap(&proof.wires_cap)?;
+        self.write_merkle_cap(&proof.plonk_zs_partial_products_cap)?;
+        self.write_merkle_cap(&proof.quotient_polys_cap)?;
+        self.write_opening_set(&proof.openings)?;
+        self.write_fri_proof(&proof.opening_proof)
+    }
+    pub fn read_proof<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<Proof<F, D>> {
+        let config = &common_data.config;
+        let wires_cap = self.read_merkle_cap(config.cap_height)?;
+        let plonk_zs_partial_products_cap = self.read_merkle_cap(config.cap_height)?;
+        let quotient_polys_cap = self.read_merkle_cap(config.cap_height)?;
+        let openings = self.read_opening_set(common_data)?;
+        let opening_proof = self.read_fri_proof(common_data)?;
+
+        Ok(Proof {
+            wires_cap,
+            plonk_zs_partial_products_cap,
+            quotient_polys_cap,
+            openings,
+            opening_proof,
+        })
+    }
+
+    pub fn write_proof_with_public_inputs<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        proof_with_pis: &ProofWithPublicInputs<F, D>,
+    ) -> Result<()> {
+        let ProofWithPublicInputs {
+            proof,
+            public_inputs,
+        } = proof_with_pis;
+        self.write_proof(proof)?;
+        self.write_field_vec(public_inputs)
+    }
+    pub fn read_proof_with_public_inputs<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<ProofWithPublicInputs<F, D>> {
+        let proof = self.read_proof(common_data)?;
+        let public_inputs = self.read_field_vec(
+            (self.len() - self.0.position() as usize) / std::mem::size_of::<u64>(),
+        )?;
+
+        Ok(ProofWithPublicInputs {
+            proof,
+            public_inputs,
+        })
+    }
+
+    fn write_compressed_fri_query_rounds<F: Extendable<D>, const D: usize>(
+        &mut self,
+        cfqrs: &CompressedFriQueryRounds<F, D>,
+    ) -> Result<()> {
+        for &i in &cfqrs.indices {
+            self.write_u32(i as u32)?;
+        }
+
+        let mut initial_trees_proofs = cfqrs.initial_trees_proofs.iter().collect::<Vec<_>>();
+        initial_trees_proofs.sort_by_key(|&x| x.0);
+        for (_, itp) in initial_trees_proofs {
+            self.write_fri_initial_proof(itp)?;
+        }
+        for h in &cfqrs.steps {
+            let mut fri_query_steps = h.iter().collect::<Vec<_>>();
+            fri_query_steps.sort_by_key(|&x| x.0);
+            for (_, fqs) in fri_query_steps {
+                self.write_fri_query_step(fqs)?;
+            }
+        }
+        Ok(())
+    }
+    fn read_compressed_fri_query_rounds<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<CompressedFriQueryRounds<F, D>> {
+        let config = &common_data.config;
+        let original_indices = (0..config.fri_config.num_query_rounds)
+            .map(|_| self.read_u32().map(|i| i as usize))
+            .collect::<Result<Vec<_>>>()?;
+        let mut indices = original_indices.clone();
+        indices.sort_unstable();
+        indices.dedup();
+        let mut pairs = Vec::new();
+        for &i in &indices {
+            pairs.push((i, self.read_fri_initial_proof(common_data)?));
+        }
+        let initial_trees_proofs = HashMap::from_iter(pairs);
+
+        let mut steps = Vec::with_capacity(config.fri_config.reduction_arity_bits.len());
+        for &a in &config.fri_config.reduction_arity_bits {
+            indices.iter_mut().for_each(|x| {
+                *x >>= a;
+            });
+            indices.dedup();
+            let query_steps = (0..indices.len())
+                .map(|_| self.read_fri_query_step(1 << a))
+                .collect::<Result<Vec<_>>>()?;
+            steps.push(
+                indices
+                    .iter()
+                    .copied()
+                    .zip(query_steps)
+                    .collect::<HashMap<_, _>>(),
+            );
+        }
+
+        Ok(CompressedFriQueryRounds {
+            indices: original_indices,
+            initial_trees_proofs,
+            steps,
+        })
+    }
+
+    fn write_compressed_fri_proof<F: Extendable<D>, const D: usize>(
+        &mut self,
+        fp: &CompressedFriProof<F, D>,
+    ) -> Result<()> {
+        for cap in &fp.commit_phase_merkle_caps {
+            self.write_merkle_cap(cap)?;
+        }
+        self.write_compressed_fri_query_rounds(&fp.query_round_proofs)?;
+        self.write_field_ext_vec::<F, D>(&fp.final_poly.coeffs)?;
+        self.write_field(fp.pow_witness)
+    }
+    fn read_compressed_fri_proof<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<CompressedFriProof<F, D>> {
+        let config = &common_data.config;
+        let commit_phase_merkle_caps = (0..config.fri_config.reduction_arity_bits.len())
+            .map(|_| self.read_merkle_cap(config.cap_height))
+            .collect::<Result<Vec<_>>>()?;
+        let query_round_proofs = self.read_compressed_fri_query_rounds(common_data)?;
+        let final_poly =
+            PolynomialCoeffs::new(self.read_field_ext_vec::<F, D>(common_data.final_poly_len())?);
+        let pow_witness = self.read_field()?;
+        Ok(CompressedFriProof {
+            commit_phase_merkle_caps,
+            query_round_proofs,
+            final_poly,
+            pow_witness,
+        })
+    }
+
+    pub fn write_compressed_proof<F: Extendable<D>, const D: usize>(
+        &mut self,
+        proof: &CompressedProof<F, D>,
+    ) -> Result<()> {
+        self.write_merkle_cap(&proof.wires_cap)?;
+        self.write_merkle_cap(&proof.plonk_zs_partial_products_cap)?;
+        self.write_merkle_cap(&proof.quotient_polys_cap)?;
+        self.write_opening_set(&proof.openings)?;
+        self.write_compressed_fri_proof(&proof.opening_proof)
+    }
+    pub fn read_compressed_proof<F: RichField + Extendable<D>, const D: usize>(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<CompressedProof<F, D>> {
+        let config = &common_data.config;
+        let wires_cap = self.read_merkle_cap(config.cap_height)?;
+        let plonk_zs_partial_products_cap = self.read_merkle_cap(config.cap_height)?;
+        let quotient_polys_cap = self.read_merkle_cap(config.cap_height)?;
+        let openings = self.read_opening_set(common_data)?;
+        let opening_proof = self.read_compressed_fri_proof(common_data)?;
+
+        Ok(CompressedProof {
+            wires_cap,
+            plonk_zs_partial_products_cap,
+            quotient_polys_cap,
+            openings,
+            opening_proof,
+        })
+    }
+
+    pub fn write_compressed_proof_with_public_inputs<
+        F: RichField + Extendable<D>,
+        const D: usize,
+    >(
+        &mut self,
+        proof_with_pis: &CompressedProofWithPublicInputs<F, D>,
+    ) -> Result<()> {
+        let CompressedProofWithPublicInputs {
+            proof,
+            public_inputs,
+        } = proof_with_pis;
+        self.write_compressed_proof(proof)?;
+        self.write_field_vec(public_inputs)
+    }
+    pub fn read_compressed_proof_with_public_inputs<
+        F: RichField + Extendable<D>,
+        const D: usize,
+    >(
+        &mut self,
+        common_data: &CommonCircuitData<F, D>,
+    ) -> Result<CompressedProofWithPublicInputs<F, D>> {
+        let proof = self.read_compressed_proof(common_data)?;
+        let public_inputs = self.read_field_vec(
+            (self.len() - self.0.position() as usize) / std::mem::size_of::<u64>(),
+        )?;
+
+        Ok(CompressedProofWithPublicInputs {
+            proof,
+            public_inputs,
+        })
+    }
+}