diff --git a/src/gadgets/select.rs b/src/gadgets/select.rs
index bbd36d76..9df395d8 100644
--- a/src/gadgets/select.rs
+++ b/src/gadgets/select.rs
@@ -10,24 +10,24 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
     /// Note: This does not range-check `b`.
     pub fn select_ext(
         &mut self,
-        b: Target,
+        b: ExtensionTarget<D>,
         x: ExtensionTarget<D>,
         y: ExtensionTarget<D>,
     ) -> ExtensionTarget<D> {
-        let b_ext = self.convert_to_ext(b);
         let gate = self.num_gates();
         // Holds `by - y`.
         let first_out =
             ExtensionTarget::from_range(gate, ArithmeticExtensionGate::<D>::wires_first_output());
-        self.double_arithmetic_extension(F::ONE, F::NEG_ONE, b_ext, y, y, b_ext, x, first_out)
+        self.double_arithmetic_extension(F::ONE, F::NEG_ONE, b, y, y, b, x, first_out)
             .1
     }
 
     /// See `select_ext`.
     pub fn select(&mut self, b: Target, x: Target, y: Target) -> Target {
+        let b_ext = self.convert_to_ext(b);
         let x_ext = self.convert_to_ext(x);
         let y_ext = self.convert_to_ext(y);
-        self.select_ext(b, x_ext, y_ext).to_target_array()[0]
+        self.select_ext(b_ext, x_ext, y_ext).to_target_array()[0]
     }
 }
 
@@ -54,13 +54,13 @@ mod tests {
         let (x, y) = (FF::rand(), FF::rand());
         let xt = builder.add_virtual_extension_target();
         let yt = builder.add_virtual_extension_target();
-        let truet = builder.add_virtual_target();
-        let falset = builder.add_virtual_target();
+        let truet = builder.add_virtual_extension_target();
+        let falset = builder.add_virtual_extension_target();
 
         pw.set_extension_target(xt, x);
         pw.set_extension_target(yt, y);
-        pw.set_target(truet, F::ONE);
-        pw.set_target(falset, F::ZERO);
+        pw.set_extension_target(truet, FF::ONE);
+        pw.set_extension_target(falset, FF::ZERO);
 
         let should_be_x = builder.select_ext(truet, xt, yt);
         let should_be_y = builder.select_ext(falset, xt, yt);
diff --git a/src/gates/exponentiation.rs b/src/gates/exponentiation.rs
new file mode 100644
index 00000000..2dd3b009
--- /dev/null
+++ b/src/gates/exponentiation.rs
@@ -0,0 +1,382 @@
+use std::marker::PhantomData;
+
+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;
+use crate::generator::{GeneratedValues, SimpleGenerator, WitnessGenerator};
+use crate::plonk_common::{reduce_with_powers, reduce_with_powers_recursive};
+use crate::target::Target;
+use crate::vars::{EvaluationTargets, EvaluationVars, EvaluationVarsBase};
+use crate::wire::Wire;
+use crate::witness::PartialWitness;
+
+/// A gate for raising a value to a power.
+#[derive(Clone, Debug)]
+pub(crate) struct ExponentiationGate<F: Extendable<D>, const D: usize> {
+    pub num_power_bits: usize,
+    pub _phantom: PhantomData<F>,
+}
+
+impl<F: Extendable<D>, const D: usize> ExponentiationGate<F, D> {
+    pub fn new(num_power_bits: usize) -> Self {
+        Self {
+            num_power_bits,
+            _phantom: PhantomData,
+        }
+    }
+
+    pub fn max_power_bits(num_wires: usize, num_routed_wires: usize) -> usize {
+        let max_for_routed_wires = num_routed_wires - 3;
+        let max_for_wires = (num_wires - 2) / 2;
+        max_for_routed_wires.min(max_for_wires)
+    }
+
+    pub fn wire_base(&self) -> usize {
+        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
+    }
+
+    pub fn wire_output(&self) -> usize {
+        2 + 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
+    }
+}
+
+impl<F: Extendable<D>, const D: usize> Gate<F, D> for ExponentiationGate<F, D> {
+    fn id(&self) -> String {
+        format!("{:?}<D={}>", self, 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)])
+            .collect();
+        let intermediate_values: Vec<_> = (0..self.num_power_bits)
+            .map(|i| vars.local_wires[self.wire_intermediate_value(i)])
+            .collect();
+
+        let output = vars.local_wires[self.wire_output()];
+
+        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
+            } else {
+                intermediate_values[i - 1].square()
+            };
+
+            // power_bits is in LE order, but we accumulate in BE order.
+            let cur_bit = power_bits[self.num_power_bits - i - 1];
+
+            let not_cur_bit = F::Extension::ONE - cur_bit;
+            let computed_intermediate_value =
+                prev_intermediate_value * (cur_bit * base + not_cur_bit);
+            constraints.push(computed_intermediate_value - intermediate_values[i]);
+        }
+
+        constraints.push(output - intermediate_values[self.num_power_bits - 1]);
+
+        constraints
+    }
+
+    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)])
+            .collect();
+        let intermediate_values: Vec<_> = (0..self.num_power_bits)
+            .map(|i| vars.local_wires[self.wire_intermediate_value(i)])
+            .collect();
+
+        let output = vars.local_wires[self.wire_output()];
+
+        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
+            } else {
+                intermediate_values[i - 1].square()
+            };
+
+            // power_bits is in LE order, but we accumulate in BE order.
+            let cur_bit = power_bits[self.num_power_bits - i - 1];
+
+            let not_cur_bit = F::ONE - cur_bit;
+            let computed_intermediate_value =
+                prev_intermediate_value * (cur_bit * base + not_cur_bit);
+            constraints.push(computed_intermediate_value - intermediate_values[i]);
+        }
+
+        constraints.push(output - intermediate_values[self.num_power_bits - 1]);
+
+        constraints
+    }
+
+    fn eval_unfiltered_recursively(
+        &self,
+        builder: &mut CircuitBuilder<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)])
+            .collect();
+        let intermediate_values: Vec<_> = (0..self.num_power_bits)
+            .map(|i| vars.local_wires[self.wire_intermediate_value(i)])
+            .collect();
+
+        let output = vars.local_wires[self.wire_output()];
+
+        let mut constraints = Vec::new();
+
+        let two = builder.constant(F::TWO);
+        let computed_power = reduce_with_powers_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 {
+                one
+            } else {
+                builder.square_extension(intermediate_values[i - 1])
+            };
+
+            // power_bits is in LE order, but we accumulate in BE order.
+            let cur_bit = power_bits[self.num_power_bits - i - 1];
+            let mul_by = builder.select_ext(cur_bit, base, one);
+            let computed_intermediate_value =
+                builder.mul_extension(prev_intermediate_value, mul_by);
+            let intermediate_value_diff =
+                builder.sub_extension(computed_intermediate_value, intermediate_values[i]);
+            constraints.push(intermediate_value_diff);
+        }
+
+        let output_diff =
+            builder.sub_extension(output, intermediate_values[self.num_power_bits - 1]);
+        constraints.push(output_diff);
+
+        constraints
+    }
+
+    fn generators(
+        &self,
+        gate_index: usize,
+        _local_constants: &[F],
+    ) -> Vec<Box<dyn WitnessGenerator<F>>> {
+        let gen = ExponentiationGenerator::<F, D> {
+            gate_index,
+            gate: self.clone(),
+        };
+        vec![Box::new(gen)]
+    }
+
+    fn num_wires(&self) -> usize {
+        self.wire_intermediate_value(self.num_power_bits - 1) + 1
+    }
+
+    fn num_constants(&self) -> usize {
+        0
+    }
+
+    fn degree(&self) -> usize {
+        4
+    }
+
+    fn num_constraints(&self) -> usize {
+        self.num_power_bits + 2
+    }
+}
+
+#[derive(Debug)]
+struct ExponentiationGenerator<F: Extendable<D>, const D: usize> {
+    gate_index: usize,
+    gate: ExponentiationGate<F, D>,
+}
+
+impl<F: Extendable<D>, const D: usize> SimpleGenerator<F> for ExponentiationGenerator<F, D> {
+    fn dependencies(&self) -> Vec<Target> {
+        let local_target = |input| Target::wire(self.gate_index, input);
+
+        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)));
+        }
+        deps
+    }
+
+    fn run_once(&self, witness: &PartialWitness<F>) -> GeneratedValues<F> {
+        let local_wire = |input| Wire {
+            gate: self.gate_index,
+            input,
+        };
+
+        let get_local_wire = |input| witness.get_wire(local_wire(input));
+
+        let num_power_bits = self.gate.num_power_bits;
+        let base = get_local_wire(self.gate.wire_base());
+
+        let power_bits = (0..num_power_bits)
+            .map(|i| get_local_wire(self.gate.wire_power_bit(i)))
+            .collect::<Vec<_>>();
+        let mut intermediate_values = Vec::new();
+
+        let mut current_intermediate_value = F::ONE;
+        for i in 0..num_power_bits {
+            if power_bits[num_power_bits - i - 1] == F::ONE {
+                current_intermediate_value *= base;
+            }
+            intermediate_values.push(current_intermediate_value);
+            current_intermediate_value *= current_intermediate_value;
+        }
+
+        let mut result = GeneratedValues::with_capacity(num_power_bits + 1);
+        for i in 0..num_power_bits {
+            let intermediate_value_wire = local_wire(self.gate.wire_intermediate_value(i));
+            result.set_wire(intermediate_value_wire, intermediate_values[i]);
+        }
+
+        let output_wire = local_wire(self.gate.wire_output());
+        result.set_wire(output_wire, intermediate_values[num_power_bits - 1]);
+
+        result
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use std::marker::PhantomData;
+
+    use rand::{thread_rng, Rng};
+
+    use crate::field::crandall_field::CrandallField;
+    use crate::field::extension_field::quartic::QuarticCrandallField;
+    use crate::field::field::Field;
+    use crate::gates::exponentiation::ExponentiationGate;
+    use crate::gates::gate::Gate;
+    use crate::gates::gate_testing::test_low_degree;
+    use crate::proof::Hash;
+    use crate::util::log2_ceil;
+    use crate::vars::EvaluationVars;
+
+    const MAX_POWER_BITS: usize = 17;
+
+    #[test]
+    fn wire_indices() {
+        let gate = ExponentiationGate::<CrandallField, 4> {
+            num_power_bits: 5,
+            _phantom: PhantomData,
+        };
+
+        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);
+    }
+
+    #[test]
+    fn low_degree() {
+        test_low_degree::<CrandallField, _, 4>(ExponentiationGate::new(5));
+    }
+
+    #[test]
+    fn test_gate_constraint() {
+        type F = CrandallField;
+        type FF = QuarticCrandallField;
+        const D: usize = 4;
+
+        /// Returns the local wires for an exponentiation gate given the base, power, and power bit
+        /// values.
+        fn get_wires(base: F, power: u64) -> Vec<FF> {
+            let mut power_bits = Vec::new();
+            let mut cur_power = power;
+            while cur_power > 0 {
+                power_bits.push(cur_power % 2);
+                cur_power /= 2;
+            }
+
+            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))
+                .collect();
+
+            let mut v = Vec::new();
+            v.push(base);
+            v.push(power_F);
+            v.extend(power_bits_F.clone());
+
+            let mut intermediate_values = Vec::new();
+            let mut current_intermediate_value = F::ONE;
+            for i in 0..num_power_bits {
+                if power_bits[num_power_bits - i - 1] == 1 {
+                    current_intermediate_value *= base;
+                }
+                intermediate_values.push(current_intermediate_value);
+                current_intermediate_value *= current_intermediate_value;
+            }
+            let output_value = intermediate_values[num_power_bits - 1];
+            v.push(output_value);
+            v.extend(intermediate_values);
+
+            v.iter().map(|&x| x.into()).collect::<Vec<_>>()
+        }
+
+        let mut rng = rand::thread_rng();
+
+        let base = F::TWO;
+        let power = rng.gen::<usize>() % (1 << MAX_POWER_BITS);
+        let num_power_bits = log2_ceil(power + 1);
+        let gate = ExponentiationGate::<F, D> {
+            num_power_bits,
+            _phantom: PhantomData,
+        };
+
+        let vars = EvaluationVars {
+            local_constants: &[],
+            local_wires: &get_wires(base, power as u64),
+            public_inputs_hash: &Hash::rand(),
+        };
+        assert!(
+            gate.eval_unfiltered(vars).iter().all(|x| x.is_zero()),
+            "Gate constraints are not satisfied."
+        );
+    }
+}
diff --git a/src/gates/insertion.rs b/src/gates/insertion.rs
index 17df4024..41e3e6e9 100644
--- a/src/gates/insertion.rs
+++ b/src/gates/insertion.rs
@@ -6,7 +6,7 @@ use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::{Extendable, FieldExtension};
 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};
@@ -358,7 +358,7 @@ mod tests {
         type FF = QuarticCrandallField;
         const D: usize = 4;
 
-        /// Returns the local wires for an insertion gate for given the original vector, element to
+        /// Returns the local wires for an insertion gate given the original vector, element to
         /// insert, and index.
         fn get_wires(orig_vec: Vec<FF>, insertion_index: usize, element_to_insert: FF) -> Vec<FF> {
             let vec_size = orig_vec.len();
diff --git a/src/gates/mod.rs b/src/gates/mod.rs
index 987a276d..20680141 100644
--- a/src/gates/mod.rs
+++ b/src/gates/mod.rs
@@ -4,6 +4,7 @@
 pub mod arithmetic;
 pub mod base_sum;
 pub mod constant;
+pub mod exponentiation;
 pub(crate) mod gate;
 pub mod gate_tree;
 pub mod gmimc;