diff --git a/src/field/crandall_field.rs b/src/field/crandall_field.rs
index 051b0fd0..36cea17b 100644
--- a/src/field/crandall_field.rs
+++ b/src/field/crandall_field.rs
@@ -4,7 +4,6 @@ use std::hash::{Hash, Hasher};
 use std::iter::{Product, Sum};
 use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
-use itertools::Itertools;
 use num::bigint::BigUint;
 use num::Integer;
 use rand::Rng;
diff --git a/src/field/extension_field/quadratic.rs b/src/field/extension_field/quadratic.rs
index 5324ad2a..fbbb535f 100644
--- a/src/field/extension_field/quadratic.rs
+++ b/src/field/extension_field/quadratic.rs
@@ -3,7 +3,6 @@ use std::hash::Hash;
 use std::iter::{Product, Sum};
 use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
-use itertools::Itertools;
 use num::bigint::BigUint;
 use rand::Rng;
 use serde::{Deserialize, Serialize};
diff --git a/src/field/extension_field/quartic.rs b/src/field/extension_field/quartic.rs
index f38f103a..ef356232 100644
--- a/src/field/extension_field/quartic.rs
+++ b/src/field/extension_field/quartic.rs
@@ -3,7 +3,6 @@ use std::hash::Hash;
 use std::iter::{Product, Sum};
 use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};
 
-use itertools::Itertools;
 use num::bigint::BigUint;
 use num::traits::Pow;
 use rand::Rng;
diff --git a/src/field/field_testing.rs b/src/field/field_testing.rs
index ffab1d9d..d0930fb9 100644
--- a/src/field/field_testing.rs
+++ b/src/field/field_testing.rs
@@ -151,7 +151,7 @@ macro_rules! test_field_arithmetic {
     ($field:ty) => {
         mod field_arithmetic {
             use num::{bigint::BigUint, One, Zero};
-            use rand::{thread_rng, Rng};
+            use rand::Rng;
 
             use crate::field::field::Field;
 
diff --git a/src/gadgets/arithmetic.rs b/src/gadgets/arithmetic.rs
index bf16a820..48104117 100644
--- a/src/gadgets/arithmetic.rs
+++ b/src/gadgets/arithmetic.rs
@@ -181,11 +181,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         let gate = ExponentiationGate::new(self.config.clone());
         let gate_index = self.add_gate(gate.clone(), vec![]);
 
-        let two = self.constant(F::TWO);
-        let exponent = reduce_with_powers_recursive(self, &exp_bits_vec[..], two);
-
         self.route(base, Target::wire(gate_index, gate.wire_base()));
-        self.route(exponent, Target::wire(gate_index, gate.wire_power()));
         exp_bits_vec.iter().enumerate().for_each(|(i, bit)| {
             self.route(*bit, Target::wire(gate_index, gate.wire_power_bit(i)));
         });
diff --git a/src/gadgets/insert.rs b/src/gadgets/insert.rs
index 8406b83a..e9345ef7 100644
--- a/src/gadgets/insert.rs
+++ b/src/gadgets/insert.rs
@@ -1,5 +1,3 @@
-use std::marker::PhantomData;
-
 use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::Extendable;
diff --git a/src/gates/base_sum.rs b/src/gates/base_sum.rs
index 3ec017f3..723cc84f 100644
--- a/src/gates/base_sum.rs
+++ b/src/gates/base_sum.rs
@@ -4,7 +4,7 @@ use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::Extendable;
 use crate::field::field::Field;
-use crate::gates::gate::{Gate, GateRef};
+use crate::gates::gate::Gate;
 use crate::generator::{GeneratedValues, SimpleGenerator, WitnessGenerator};
 use crate::plonk_common::{reduce_with_powers, reduce_with_powers_ext_recursive};
 use crate::target::Target;
@@ -186,7 +186,6 @@ impl<F: Field, const B: usize> SimpleGenerator<F> for BaseSplitGenerator<B> {
 mod tests {
     use crate::field::crandall_field::CrandallField;
     use crate::gates::base_sum::BaseSumGate;
-    use crate::gates::gate::GateRef;
     use crate::gates::gate_testing::test_low_degree;
 
     #[test]
diff --git a/src/gates/constant.rs b/src/gates/constant.rs
index 1d8f50dd..11592c90 100644
--- a/src/gates/constant.rs
+++ b/src/gates/constant.rs
@@ -2,7 +2,7 @@ use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::Extendable;
 use crate::field::field::Field;
-use crate::gates::gate::{Gate, GateRef};
+use crate::gates::gate::Gate;
 use crate::generator::{GeneratedValues, SimpleGenerator, WitnessGenerator};
 use crate::target::Target;
 use crate::vars::{EvaluationTargets, EvaluationVars, EvaluationVarsBase};
diff --git a/src/gates/exponentiation.rs b/src/gates/exponentiation.rs
index a7cd8b1c..d114874a 100644
--- a/src/gates/exponentiation.rs
+++ b/src/gates/exponentiation.rs
@@ -39,23 +39,19 @@ impl<F: Extendable<D>, const D: usize> ExponentiationGate<F, D> {
         0
     }
 
-    pub fn wire_power(&self) -> usize {
-        1
-    }
-
     /// The `i`th bit of the exponent, in little-endian order.
     pub fn wire_power_bit(&self, i: usize) -> usize {
         debug_assert!(i < self.num_power_bits);
-        2 + i
+        1 + i
     }
 
     pub fn wire_output(&self) -> usize {
-        2 + self.num_power_bits
+        1 + self.num_power_bits
     }
 
     pub fn wire_intermediate_value(&self, i: usize) -> usize {
         debug_assert!(i < self.num_power_bits);
-        3 + self.num_power_bits + i
+        2 + self.num_power_bits + i
     }
 }
 
@@ -66,7 +62,6 @@ impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
 
     fn eval_unfiltered(&self, vars: EvaluationVars<F, D>) -> Vec<F::Extension> {
         let base = vars.local_wires[self.wire_base()];
-        let power = vars.local_wires[self.wire_power()];
 
         let power_bits: Vec<_> = (0..self.num_power_bits)
             .map(|i| vars.local_wires[self.wire_power_bit(i)])
@@ -79,9 +74,6 @@ impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
 
         let mut constraints = Vec::new();
 
-        let computed_power = reduce_with_powers(&power_bits, F::Extension::TWO);
-        constraints.push(power - computed_power);
-
         for i in 0..self.num_power_bits {
             let prev_intermediate_value = if i == 0 {
                 F::Extension::ONE
@@ -105,7 +97,6 @@ impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
 
     fn eval_unfiltered_base(&self, vars: EvaluationVarsBase<F>) -> Vec<F> {
         let base = vars.local_wires[self.wire_base()];
-        let power = vars.local_wires[self.wire_power()];
 
         let power_bits: Vec<_> = (0..self.num_power_bits)
             .map(|i| vars.local_wires[self.wire_power_bit(i)])
@@ -118,9 +109,6 @@ impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
 
         let mut constraints = Vec::new();
 
-        let computed_power = reduce_with_powers(&power_bits, F::TWO);
-        constraints.push(power - computed_power);
-
         for i in 0..self.num_power_bits {
             let prev_intermediate_value = if i == 0 {
                 F::ONE
@@ -148,7 +136,6 @@ impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
         vars: EvaluationTargets<D>,
     ) -> Vec<ExtensionTarget<D>> {
         let base = vars.local_wires[self.wire_base()];
-        let power = vars.local_wires[self.wire_power()];
 
         let power_bits: Vec<_> = (0..self.num_power_bits)
             .map(|i| vars.local_wires[self.wire_power_bit(i)])
@@ -161,11 +148,6 @@ impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
 
         let mut constraints = Vec::new();
 
-        let two = builder.constant(F::TWO);
-        let computed_power = reduce_with_powers_ext_recursive(builder, &power_bits, two);
-        let power_diff = builder.sub_extension(power, computed_power);
-        constraints.push(power_diff);
-
         let one = builder.constant_extension(F::Extension::ONE);
         for i in 0..self.num_power_bits {
             let prev_intermediate_value = if i == 0 {
@@ -216,7 +198,7 @@ impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
     }
 
     fn num_constraints(&self) -> usize {
-        self.num_power_bits + 2
+        self.num_power_bits + 1
     }
 }
 
@@ -232,7 +214,6 @@ impl<F: Extendable<D>, const D: usize> SimpleGenerator<F> for ExponentiationGene
 
         let mut deps = Vec::with_capacity(self.gate.num_power_bits + 2);
         deps.push(local_target(self.gate.wire_base()));
-        deps.push(local_target(self.gate.wire_power()));
         for i in 0..self.gate.num_power_bits {
             deps.push(local_target(self.gate.wire_power_bit(i)));
         }
@@ -281,7 +262,7 @@ impl<F: Extendable<D>, const D: usize> SimpleGenerator<F> for ExponentiationGene
 mod tests {
     use std::marker::PhantomData;
 
-    use rand::{thread_rng, Rng};
+    use rand::Rng;
 
     use crate::circuit_data::CircuitConfig;
     use crate::field::crandall_field::CrandallField;
@@ -305,12 +286,11 @@ mod tests {
         };
 
         assert_eq!(gate.wire_base(), 0);
-        assert_eq!(gate.wire_power(), 1);
-        assert_eq!(gate.wire_power_bit(0), 2);
-        assert_eq!(gate.wire_power_bit(4), 6);
-        assert_eq!(gate.wire_output(), 7);
-        assert_eq!(gate.wire_intermediate_value(0), 8);
-        assert_eq!(gate.wire_intermediate_value(4), 12);
+        assert_eq!(gate.wire_power_bit(0), 1);
+        assert_eq!(gate.wire_power_bit(4), 5);
+        assert_eq!(gate.wire_output(), 6);
+        assert_eq!(gate.wire_intermediate_value(0), 7);
+        assert_eq!(gate.wire_intermediate_value(4), 11);
     }
 
     #[test]
@@ -350,7 +330,6 @@ mod tests {
 
             let num_power_bits = power_bits.len();
 
-            let power_F = F::from_canonical_u64(power);
             let power_bits_F: Vec<_> = power_bits
                 .iter()
                 .map(|b| F::from_canonical_u64(*b))
@@ -358,7 +337,6 @@ mod tests {
 
             let mut v = Vec::new();
             v.push(base);
-            v.push(power_F);
             v.extend(power_bits_F.clone());
 
             let mut intermediate_values = Vec::new();
diff --git a/src/gates/gate_testing.rs b/src/gates/gate_testing.rs
index 4a6c80df..e0b72477 100644
--- a/src/gates/gate_testing.rs
+++ b/src/gates/gate_testing.rs
@@ -1,6 +1,6 @@
 use crate::field::extension_field::Extendable;
 use crate::field::field::Field;
-use crate::gates::gate::{Gate, GateRef};
+use crate::gates::gate::Gate;
 use crate::polynomial::polynomial::{PolynomialCoeffs, PolynomialValues};
 use crate::proof::Hash;
 use crate::util::{log2_ceil, transpose};
diff --git a/src/gates/gmimc.rs b/src/gates/gmimc.rs
index 2c8ca297..ec2aab6c 100644
--- a/src/gates/gmimc.rs
+++ b/src/gates/gmimc.rs
@@ -334,7 +334,7 @@ mod tests {
     use crate::field::crandall_field::CrandallField;
     use crate::field::extension_field::quartic::QuarticCrandallField;
     use crate::field::field::Field;
-    use crate::gates::gate::{Gate, GateRef};
+    use crate::gates::gate::Gate;
     use crate::gates::gate_testing::test_low_degree;
     use crate::gates::gmimc::{GMiMCGate, W};
     use crate::generator::generate_partial_witness;
diff --git a/src/gates/interpolation.rs b/src/gates/interpolation.rs
index 47fa6d4a..36e90c53 100644
--- a/src/gates/interpolation.rs
+++ b/src/gates/interpolation.rs
@@ -8,7 +8,7 @@ use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::{Extendable, FieldExtension};
 use crate::field::interpolation::interpolant;
 use crate::gadgets::polynomial::PolynomialCoeffsExtAlgebraTarget;
-use crate::gates::gate::{Gate, GateRef};
+use crate::gates::gate::Gate;
 use crate::generator::{GeneratedValues, SimpleGenerator, WitnessGenerator};
 use crate::polynomial::polynomial::PolynomialCoeffs;
 use crate::target::Target;
@@ -288,7 +288,6 @@ mod tests {
 
     use crate::field::crandall_field::CrandallField;
     use crate::field::extension_field::quartic::QuarticCrandallField;
-    use crate::field::extension_field::FieldExtension;
     use crate::field::field::Field;
     use crate::gates::gate::Gate;
     use crate::gates::gate_testing::test_low_degree;
@@ -319,7 +318,6 @@ mod tests {
 
     #[test]
     fn low_degree() {
-        type F = CrandallField;
         test_low_degree::<CrandallField, _, 4>(InterpolationGate::new(4));
     }
 
diff --git a/src/gates/noop.rs b/src/gates/noop.rs
index 0787c9fa..3245f7b7 100644
--- a/src/gates/noop.rs
+++ b/src/gates/noop.rs
@@ -1,7 +1,7 @@
 use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::Extendable;
-use crate::gates::gate::{Gate, GateRef};
+use crate::gates::gate::Gate;
 use crate::generator::WitnessGenerator;
 use crate::vars::{EvaluationTargets, EvaluationVars, EvaluationVarsBase};
 
diff --git a/src/gates/public_input.rs b/src/gates/public_input.rs
index e8299b26..ae9d2377 100644
--- a/src/gates/public_input.rs
+++ b/src/gates/public_input.rs
@@ -3,7 +3,7 @@ use std::ops::Range;
 use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::Extendable;
-use crate::gates::gate::{Gate, GateRef};
+use crate::gates::gate::Gate;
 use crate::generator::WitnessGenerator;
 use crate::vars::{EvaluationTargets, EvaluationVars, EvaluationVarsBase};
 
diff --git a/src/gates/reducing.rs b/src/gates/reducing.rs
index 0129a156..cea7267a 100644
--- a/src/gates/reducing.rs
+++ b/src/gates/reducing.rs
@@ -219,7 +219,6 @@ mod tests {
 
     #[test]
     fn low_degree() {
-        type F = CrandallField;
         test_low_degree::<CrandallField, _, 4>(ReducingGate::new(22));
     }
 }
diff --git a/src/prover.rs b/src/prover.rs
index 7d209667..865489a0 100644
--- a/src/prover.rs
+++ b/src/prover.rs
@@ -129,7 +129,7 @@ pub(crate) fn prove<F: Extendable<D>, const D: usize>(
             &gammas,
             &alphas,
         ),
-        "to compute vanishing polys"
+        "to compute quotient polys"
     );
 
     // Compute the quotient polynomials, aka `t` in the Plonk paper.
@@ -146,7 +146,7 @@ pub(crate) fn prove<F: Extendable<D>, const D: usize>(
                 quotient_poly.chunks(degree)
             })
             .collect(),
-        "to compute quotient polys"
+        "to split up quotient polys"
     );
 
     let quotient_polys_commitment = timed!(