diff --git a/src/core/air/accumulation.rs b/src/core/air/accumulation.rs
index c96d37a96..338dda855 100644
--- a/src/core/air/accumulation.rs
+++ b/src/core/air/accumulation.rs
@@ -2,12 +2,12 @@
 //! Given N polynomials, sort them by size: u_0(P), ... u_{N-1}(P).
 //! Given a random alpha, the combined polynomial is defined as
 //!   f(p) = sum_i alpha^{N-1-i} u_i (P).
-use crate::core::backend::cpu::CPUCircleEvaluation;
 use crate::core::backend::{Backend, CPUBackend};
+use crate::core::fields::m31::BaseField;
 use crate::core::fields::qm31::SecureField;
 use crate::core::fields::secure_column::SecureColumn;
-use crate::core::fields::FieldExpOps;
-use crate::core::poly::circle::{CanonicCoset, CirclePoly, SecureCirclePoly};
+use crate::core::fields::{FieldExpOps, FieldOps};
+use crate::core::poly::circle::{CanonicCoset, CircleEvaluation, CirclePoly, SecureCirclePoly};
 use crate::core::poly::BitReversedOrder;
 
 /// Accumulates evaluations of u_i(P0) at a single point.
@@ -119,12 +119,17 @@ impl<B: Backend> DomainEvaluationAccumulator<B> {
     }
 }
 
-impl DomainEvaluationAccumulator<CPUBackend> {
+pub trait AccumulationOps: FieldOps<BaseField> + Sized {
+    /// Accumulates other into column:
+    ///   column = column * alpha + other.
+    fn accumulate(column: &mut SecureColumn<Self>, alpha: SecureField, other: &SecureColumn<Self>);
+}
+
+impl<B: Backend> DomainEvaluationAccumulator<B> {
     /// Computes f(P) as coefficients.
-    pub fn finalize(self) -> SecureCirclePoly {
-        let mut res_coeffs = SecureColumn::<CPUBackend>::zeros(1 << self.log_size());
+    pub fn finalize(self) -> SecureCirclePoly<B> {
+        let mut res_coeffs = SecureColumn::<B>::zeros(1 << self.log_size());
         let res_log_size = self.log_size();
-        let res_size = 1 << res_log_size;
 
         for ((log_size, values), n_cols) in self
             .sub_accumulations
@@ -133,9 +138,9 @@ impl DomainEvaluationAccumulator<CPUBackend> {
             .zip(self.n_cols_per_size.iter())
             .skip(1)
         {
-            let coeffs = SecureColumn::<CPUBackend> {
+            let coeffs = SecureColumn::<B> {
                 columns: values.columns.map(|c| {
-                    CPUCircleEvaluation::<_, BitReversedOrder>::new(
+                    CircleEvaluation::<B, BaseField, BitReversedOrder>::new(
                         CanonicCoset::new(log_size as u32).circle_domain(),
                         c,
                     )
@@ -146,10 +151,7 @@ impl DomainEvaluationAccumulator<CPUBackend> {
             };
             // Add column coefficients into result coefficients, element-wise, in-place.
             let multiplier = self.random_coeff.pow(*n_cols as u128);
-            for i in 0..res_size {
-                let res_coeff = res_coeffs.at(i) * multiplier + coeffs.at(i);
-                res_coeffs.set(i, res_coeff);
-            }
+            B::accumulate(&mut res_coeffs, multiplier, &coeffs);
         }
 
         SecureCirclePoly(res_coeffs.columns.map(CirclePoly::new))
@@ -163,6 +165,9 @@ pub struct ColumnAccumulator<'a, B: Backend> {
 }
 impl<'a> ColumnAccumulator<'a, CPUBackend> {
     pub fn accumulate(&mut self, index: usize, evaluation: SecureField) {
+        // TODO(spapini): Multiplying QM31 by QM31 is not the best way to do this.
+        // It's probably better to cache all the coefficient powers and multiply QM31 by M31,
+        // and only add in QM31.
         let val = self.col.at(index) * self.random_coeff_pow + evaluation;
         self.col.set(index, val);
     }
diff --git a/src/core/air/air_ext.rs b/src/core/air/air_ext.rs
index 6fe86ac35..16396dac1 100644
--- a/src/core/air/air_ext.rs
+++ b/src/core/air/air_ext.rs
@@ -4,14 +4,14 @@ use itertools::Itertools;
 
 use super::accumulation::{DomainEvaluationAccumulator, PointEvaluationAccumulator};
 use super::{Air, ComponentTrace};
-use crate::core::backend::CPUBackend;
+use crate::core::backend::Backend;
 use crate::core::circle::CirclePoint;
 use crate::core::fields::qm31::SecureField;
 use crate::core::poly::circle::{CanonicCoset, CirclePoly, SecureCirclePoly};
 use crate::core::prover::LOG_BLOWUP_FACTOR;
 use crate::core::ComponentVec;
 
-pub trait AirExt: Air<CPUBackend> {
+pub trait AirExt<B: Backend>: Air<B> {
     fn composition_log_degree_bound(&self) -> u32 {
         self.components()
             .iter()
@@ -30,8 +30,8 @@ pub trait AirExt: Air<CPUBackend> {
     fn compute_composition_polynomial(
         &self,
         random_coeff: SecureField,
-        component_traces: &[ComponentTrace<'_, CPUBackend>],
-    ) -> SecureCirclePoly {
+        component_traces: &[ComponentTrace<'_, B>],
+    ) -> SecureCirclePoly<B> {
         let mut accumulator =
             DomainEvaluationAccumulator::new(random_coeff, self.composition_log_degree_bound());
         zip(self.components(), component_traces).for_each(|(component, trace)| {
@@ -43,7 +43,7 @@ pub trait AirExt: Air<CPUBackend> {
     fn mask_points_and_values(
         &self,
         point: CirclePoint<SecureField>,
-        component_traces: &[ComponentTrace<'_, CPUBackend>],
+        component_traces: &[ComponentTrace<'_, B>],
     ) -> (
         ComponentVec<Vec<CirclePoint<SecureField>>>,
         ComponentVec<Vec<SecureField>>,
@@ -101,8 +101,8 @@ pub trait AirExt: Air<CPUBackend> {
 
     fn component_traces<'a>(
         &'a self,
-        polynomials: &'a [CirclePoly<CPUBackend>],
-    ) -> Vec<ComponentTrace<'_, CPUBackend>> {
+        polynomials: &'a [CirclePoly<B>],
+    ) -> Vec<ComponentTrace<'_, B>> {
         let poly_iter = &mut polynomials.iter();
         self.components()
             .iter()
@@ -115,4 +115,4 @@ pub trait AirExt: Air<CPUBackend> {
     }
 }
 
-impl<A: Air<CPUBackend>> AirExt for A {}
+impl<B: Backend, A: Air<B>> AirExt<B> for A {}
diff --git a/src/core/backend/cpu/accumulation.rs b/src/core/backend/cpu/accumulation.rs
new file mode 100644
index 000000000..cbcf8cbde
--- /dev/null
+++ b/src/core/backend/cpu/accumulation.rs
@@ -0,0 +1,13 @@
+use super::CPUBackend;
+use crate::core::air::accumulation::AccumulationOps;
+use crate::core::fields::qm31::SecureField;
+use crate::core::fields::secure_column::SecureColumn;
+
+impl AccumulationOps for CPUBackend {
+    fn accumulate(column: &mut SecureColumn<Self>, alpha: SecureField, other: &SecureColumn<Self>) {
+        for i in 0..column.len() {
+            let res_coeff = column.at(i) * alpha + other.at(i);
+            column.set(i, res_coeff);
+        }
+    }
+}
diff --git a/src/core/backend/cpu/mod.rs b/src/core/backend/cpu/mod.rs
index 589bde681..fab217d1a 100644
--- a/src/core/backend/cpu/mod.rs
+++ b/src/core/backend/cpu/mod.rs
@@ -1,3 +1,4 @@
+mod accumulation;
 mod circle;
 mod fri;
 pub mod quotients;
diff --git a/src/core/backend/mod.rs b/src/core/backend/mod.rs
index 7f290386d..5319ace9e 100644
--- a/src/core/backend/mod.rs
+++ b/src/core/backend/mod.rs
@@ -2,6 +2,7 @@ use std::fmt::Debug;
 
 pub use cpu::CPUBackend;
 
+use super::air::accumulation::AccumulationOps;
 use super::commitment_scheme::quotients::QuotientOps;
 use super::fields::m31::BaseField;
 use super::fields::qm31::SecureField;
@@ -13,7 +14,15 @@ pub mod avx512;
 pub mod cpu;
 
 pub trait Backend:
-    Copy + Clone + Debug + FieldOps<BaseField> + FieldOps<SecureField> + PolyOps + QuotientOps + FriOps
+    Copy
+    + Clone
+    + Debug
+    + FieldOps<BaseField>
+    + FieldOps<SecureField>
+    + PolyOps
+    + QuotientOps
+    + FriOps
+    + AccumulationOps
 {
 }
 
diff --git a/src/core/commitment_scheme/mod.rs b/src/core/commitment_scheme/mod.rs
index 093dcef0f..8d3efa64a 100644
--- a/src/core/commitment_scheme/mod.rs
+++ b/src/core/commitment_scheme/mod.rs
@@ -8,7 +8,7 @@
 
 mod prover;
 pub mod quotients;
-pub mod utils;
+mod utils;
 mod verifier;
 
 pub use self::prover::{CommitmentSchemeProof, CommitmentSchemeProver};
diff --git a/src/core/poly/circle/secure_poly.rs b/src/core/poly/circle/secure_poly.rs
index a7ee0946e..24aabc917 100644
--- a/src/core/poly/circle/secure_poly.rs
+++ b/src/core/poly/circle/secure_poly.rs
@@ -1,7 +1,7 @@
 use std::ops::Deref;
 
-use super::CircleDomain;
-use crate::core::backend::cpu::{CPUCircleEvaluation, CPUCirclePoly};
+use super::{CircleDomain, CirclePoly, PolyOps};
+use crate::core::backend::cpu::CPUCircleEvaluation;
 use crate::core::circle::CirclePoint;
 use crate::core::fields::m31::BaseField;
 use crate::core::fields::qm31::SecureField;
@@ -9,9 +9,9 @@ use crate::core::fields::secure_column::{SecureColumn, SECURE_EXTENSION_DEGREE};
 use crate::core::fields::FieldOps;
 use crate::core::poly::BitReversedOrder;
 
-pub struct SecureCirclePoly(pub [CPUCirclePoly; SECURE_EXTENSION_DEGREE]);
+pub struct SecureCirclePoly<B: FieldOps<BaseField>>(pub [CirclePoly<B>; SECURE_EXTENSION_DEGREE]);
 
-impl SecureCirclePoly {
+impl<B: PolyOps> SecureCirclePoly<B> {
     pub fn eval_at_point(&self, point: CirclePoint<SecureField>) -> SecureField {
         Self::eval_from_partial_evals(self.eval_columns_at_point(point))
     }
@@ -43,8 +43,8 @@ impl SecureCirclePoly {
     }
 }
 
-impl Deref for SecureCirclePoly {
-    type Target = [CPUCirclePoly; SECURE_EXTENSION_DEGREE];
+impl<B: FieldOps<BaseField>> Deref for SecureCirclePoly<B> {
+    type Target = [CirclePoly<B>; SECURE_EXTENSION_DEGREE];
 
     fn deref(&self) -> &Self::Target {
         &self.0
diff --git a/src/core/prover/mod.rs b/src/core/prover/mod.rs
index d8ace6156..711ed7bad 100644
--- a/src/core/prover/mod.rs
+++ b/src/core/prover/mod.rs
@@ -1,6 +1,7 @@
 use itertools::Itertools;
 use thiserror::Error;
 
+use super::backend::Backend;
 use super::commitment_scheme::{CommitmentSchemeProof, TreeVec};
 use super::fri::FriVerificationError;
 use super::poly::circle::{SecureCirclePoly, MAX_CIRCLE_DOMAIN_LOG_SIZE};
@@ -9,7 +10,6 @@ use super::ColumnVec;
 use crate::commitment_scheme::blake2_hash::Blake2sHasher;
 use crate::commitment_scheme::hasher::Hasher;
 use crate::core::air::{Air, AirExt};
-use crate::core::backend::cpu::CPUCircleEvaluation;
 use crate::core::backend::CPUBackend;
 use crate::core::channel::{Blake2sChannel, Channel as ChannelTrait};
 use crate::core::circle::CirclePoint;
@@ -42,10 +42,10 @@ pub struct AdditionalProofData {
     pub oods_quotients: Vec<CircleEvaluation<CPUBackend, SecureField, BitReversedOrder>>,
 }
 
-pub fn prove(
-    air: &impl Air<CPUBackend>,
+pub fn prove<B: Backend>(
+    air: &impl Air<B>,
     channel: &mut Channel,
-    trace: ColumnVec<CPUCircleEvaluation<BaseField, BitReversedOrder>>,
+    trace: ColumnVec<CircleEvaluation<B, BaseField, BitReversedOrder>>,
 ) -> Result<StarkProof, ProvingError> {
     // Check that traces are not too big.
     for (i, trace) in trace.iter().enumerate() {
@@ -154,11 +154,11 @@ pub fn verify(
     commitment_scheme.verify_values(open_points, proof.commitment_scheme_proof, channel)
 }
 
-fn opened_values_to_mask(
-    air: &impl Air<CPUBackend>,
+fn opened_values_to_mask<B: Backend>(
+    air: &impl Air<B>,
     mut opened_values: TreeVec<ColumnVec<Vec<SecureField>>>,
 ) -> Result<(ComponentVec<Vec<SecureField>>, SecureField), ()> {
-    let composition_oods_values = SecureCirclePoly::eval_from_partial_evals(
+    let composition_oods_values = SecureCirclePoly::<B>::eval_from_partial_evals(
         opened_values
             .pop()
             .unwrap()