diff --git a/src/gadgets/arithmetic.rs b/src/gadgets/arithmetic.rs
index 0761c0ab..df7fb895 100644
--- a/src/gadgets/arithmetic.rs
+++ b/src/gadgets/arithmetic.rs
@@ -3,7 +3,9 @@ use std::borrow::Borrow;
 use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::Extendable;
 use crate::gates::exponentiation::ExponentiationGate;
+use crate::plonk_common::reduce_with_powers_recursive;
 use crate::target::Target;
+use crate::util::log2_ceil;
 
 impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
     /// Computes `-x`.
@@ -168,7 +170,6 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         base
     }
 
-    // TODO: Optimize this, maybe with a new gate.
     // TODO: Test
     /// Exponentiate `base` to the power of `exponent`, given by its little-endian bits.
     pub fn exp_from_bits(
@@ -176,61 +177,36 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         base: Target,
         exponent_bits: impl Iterator<Item = impl Borrow<Target>>,
     ) -> Target {
-        let mut current = base;
-        let one = self.one();
-        let mut product = one;
+        let exp_bits_vec: Vec<Target> = exponent_bits.map(|b| *b.borrow()).collect::<Vec<_>>();
+        let gate = ExponentiationGate::new(exp_bits_vec.len());
+        let gate_index = self.add_gate(gate.clone(), vec![]);
 
-        for bit in exponent_bits {
-            let multiplicand = self.select(*bit.borrow(), current, one);
-            product = self.mul(product, multiplicand);
-            current = self.mul(current, current);
-        }
+        let two = self.constant(F::TWO);
+        let exponent = reduce_with_powers_recursive(self, &exp_bits_vec[..], two);
 
-        product
+        self.route(exponent, 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)));
+        });
+
+        Target::wire(gate_index, gate.wire_output())
     }
 
-    // TODO: Optimize this, maybe with a new gate.
     // TODO: Test
     /// Exponentiate `base` to the power of `exponent`, where `exponent < 2^num_bits`.
     pub fn exp(&mut self, base: Target, exponent: Target, num_bits: usize) -> Target {
         let exponent_bits = self.split_le(exponent, num_bits);
 
-        let gate = ExponentiationGate::new(exponent_bits.len());
-        let gate_index = self.add_gate(gate.clone(), vec![]);
-
-        self.route(exponent, Target::wire(gate_index, gate.wire_power()));
-        exponent_bits.iter().enumerate().for_each(|(i, &bit)| {
-            self.route(bit, Target::wire(gate_index, gate.wire_power_bit(i)));
-        });
-
-        Target::wire(gate_index, gate.wire_output())
+        self.exp_from_bits(base, exponent_bits.iter())
     }
 
     /// Exponentiate `base` to the power of a known `exponent`.
     // TODO: Test
     pub fn exp_u64(&mut self, base: Target, exponent: u64) -> Target {
-        let mut exp_bits = Vec::new();
-        let mut cur_exp = exponent;
-        while cur_exp > 0 {
-            exp_bits.push(cur_exp % 2);
-            cur_exp /= 2;
-        }
-
         let exp_target = self.constant(F::from_canonical_u64(exponent));
-        let exp_bits_targets: Vec<_> = exp_bits
-            .iter()
-            .map(|b| self.constant(F::from_canonical_u64(*b)))
-            .collect();
-
-        let gate = ExponentiationGate::new(exp_bits.len());
-        let gate_index = self.add_gate(gate.clone(), vec![]);
-
-        self.route(exp_target, Target::wire(gate_index, gate.wire_power()));
-        exp_bits_targets.iter().enumerate().for_each(|(i, &bit)| {
-            self.route(bit, Target::wire(gate_index, gate.wire_power_bit(i)));
-        });
-
-        Target::wire(gate_index, gate.wire_output())
+        let num_bits = log2_ceil(exponent as usize + 1);
+        self.exp(base, exp_target, num_bits)
     }
 
     /// Computes `x / y`. Results in an unsatisfiable instance if `y = 0`.
diff --git a/src/gates/base_sum.rs b/src/gates/base_sum.rs
index cd1694e4..3ec017f3 100644
--- a/src/gates/base_sum.rs
+++ b/src/gates/base_sum.rs
@@ -6,7 +6,7 @@ use crate::field::extension_field::Extendable;
 use crate::field::field::Field;
 use crate::gates::gate::{Gate, GateRef};
 use crate::generator::{GeneratedValues, SimpleGenerator, WitnessGenerator};
-use crate::plonk_common::{reduce_with_powers, reduce_with_powers_recursive};
+use crate::plonk_common::{reduce_with_powers, reduce_with_powers_ext_recursive};
 use crate::target::Target;
 use crate::vars::{EvaluationTargets, EvaluationVars, EvaluationVarsBase};
 use crate::witness::PartialWitness;
@@ -80,9 +80,9 @@ impl<F: Extendable<D>, const D: usize, const B: usize> Gate<F, D> for BaseSumGat
         let sum = vars.local_wires[Self::WIRE_SUM];
         let reversed_sum = vars.local_wires[Self::WIRE_REVERSED_SUM];
         let mut limbs = vars.local_wires[self.limbs()].to_vec();
-        let computed_sum = reduce_with_powers_recursive(builder, &limbs, base);
+        let computed_sum = reduce_with_powers_ext_recursive(builder, &limbs, base);
         limbs.reverse();
-        let computed_reversed_sum = reduce_with_powers_recursive(builder, &limbs, base);
+        let computed_reversed_sum = reduce_with_powers_ext_recursive(builder, &limbs, base);
         let mut constraints = vec![
             builder.sub_extension(computed_sum, sum),
             builder.sub_extension(computed_reversed_sum, reversed_sum),
diff --git a/src/gates/exponentiation.rs b/src/gates/exponentiation.rs
index baaabc2b..900559e6 100644
--- a/src/gates/exponentiation.rs
+++ b/src/gates/exponentiation.rs
@@ -6,7 +6,7 @@ use crate::field::extension_field::Extendable;
 use crate::field::field::Field;
 use crate::gates::gate::Gate;
 use crate::generator::{GeneratedValues, SimpleGenerator, WitnessGenerator};
-use crate::plonk_common::{reduce_with_powers, reduce_with_powers_recursive};
+use crate::plonk_common::{reduce_with_powers, reduce_with_powers_ext_recursive};
 use crate::target::Target;
 use crate::vars::{EvaluationTargets, EvaluationVars, EvaluationVarsBase};
 use crate::wire::Wire;
@@ -160,7 +160,7 @@ 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_recursive(builder, &power_bits, 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);
 
diff --git a/src/plonk_common.rs b/src/plonk_common.rs
index 62552c0a..595fed91 100644
--- a/src/plonk_common.rs
+++ b/src/plonk_common.rs
@@ -158,6 +158,18 @@ pub(crate) fn reduce_with_powers<F: Field>(terms: &[F], alpha: F) -> F {
 }
 
 pub(crate) fn reduce_with_powers_recursive<F: Extendable<D>, const D: usize>(
+    builder: &mut CircuitBuilder<F, D>,
+    terms: &[Target],
+    alpha: Target,
+) -> Target {
+    let mut sum = builder.zero();
+    for &term in terms.iter().rev() {
+        sum = builder.mul_add(sum, alpha, term);
+    }
+    sum
+}
+
+pub(crate) fn reduce_with_powers_ext_recursive<F: Extendable<D>, const D: usize>(
     builder: &mut CircuitBuilder<F, D>,
     terms: &[ExtensionTarget<D>],
     alpha: Target,