diff --git a/src/gadgets/arithmetic_u32.rs b/src/gadgets/arithmetic_u32.rs
index ce35d4f0..0d88d52c 100644
--- a/src/gadgets/arithmetic_u32.rs
+++ b/src/gadgets/arithmetic_u32.rs
@@ -5,7 +5,7 @@ use crate::gates::subtraction_u32::U32SubtractionGate;
 use crate::iop::target::Target;
 use crate::plonk::circuit_builder::CircuitBuilder;
 
-#[derive(Clone, Copy)]
+#[derive(Clone, Copy, Debug)]
 pub struct U32Target(pub Target);
 
 impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
@@ -28,6 +28,14 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         U32Target(self.one())
     }
 
+    pub fn connect_u32(&mut self, x: U32Target, y: U32Target) {
+        self.connect(x.0, y.0)
+    }
+
+    pub fn assert_zero_u32(&self, x: U32Target) {
+        self.assert_zero(x.0)
+    }
+
     // Returns x * y + z.
     pub fn mul_add_u32(
         &mut self,
diff --git a/src/gadgets/biguint.rs b/src/gadgets/biguint.rs
index 4adc2382..b30b4680 100644
--- a/src/gadgets/biguint.rs
+++ b/src/gadgets/biguint.rs
@@ -1,17 +1,81 @@
 use std::marker::PhantomData;
+use std::ops::Neg;
 
 use num::BigUint;
 
 use crate::field::field_types::RichField;
 use crate::field::{extension_field::Extendable, field_types::Field};
 use crate::gadgets::arithmetic_u32::U32Target;
+use crate::iop::generator::{GeneratedValues, SimpleGenerator};
+use crate::iop::target::{BoolTarget, Target};
+use crate::iop::witness::PartitionWitness;
 use crate::plonk::circuit_builder::CircuitBuilder;
 
+#[derive(Clone, Debug)]
 pub struct BigUintTarget {
-    limbs: Vec<U32Target>,
+    pub limbs: Vec<U32Target>,
+}
+
+impl BigUintTarget {
+    pub fn num_limbs(&self) -> usize {
+        self.limbs.len()
+    }
+
+    pub fn get_limb(&self, i: usize) -> U32Target {
+        self.limbs[i]
+    }
 }
 
 impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
+    fn connect_biguint(&self, lhs: BigUintTarget, rhs: BigUintTarget) {
+        let min_limbs = lhs.num_limbs().min(rhs.num_limbs());
+        for i in 0..min_limbs {
+            self.connect_u32(lhs.get_limb(i), rhs.get_limb(i));
+        }
+
+        for i in min_limbs..lhs.num_limbs() {
+            self.assert_zero_u32(lhs.get_limb(i));
+        }
+        for i in min_limbs..rhs.num_limbs() {
+            self.assert_zero_u32(rhs.get_limb(i));
+        }
+    }
+
+    fn pad_biguints(&mut self, a: BigUintTarget, b: BigUintTarget) -> (BigUintTarget, BigUintTarget) {
+        if a.num_limbs() > b.num_limbs() {
+            let mut padded_b_limbs = b.limbs.clone();
+            padded_b_limbs.extend(self.add_virtual_u32_targets(a.num_limbs() - b.num_limbs()));
+            let padded_b = BigUintTarget {
+                limbs: padded_b_limbs,
+            };
+            (a, padded_b)
+        } else {
+            let mut padded_a_limbs = a.limbs.clone();
+            padded_a_limbs.extend(self.add_virtual_u32_targets(b.num_limbs() - a.num_limbs()));
+            let padded_a = BigUintTarget {
+                limbs: padded_a_limbs,
+            };
+            (padded_a, b)
+        }
+    }
+
+    fn cmp_biguint(&mut self, a: BigUintTarget, b: BigUintTarget) -> BoolTarget {
+        let (padded_a, padded_b) = self.pad_biguints(a, b);
+
+        let a_vec = a.limbs.iter().map(|&x| x.0).collect();
+        let b_vec = b.limbs.iter().map(|&x| x.0).collect();
+
+        self.list_le(a_vec, b_vec, 32)
+    }
+
+    fn add_virtual_biguint_target(&mut self, num_limbs: usize) -> BigUintTarget {
+        let limbs = (0..num_limbs).map(|_| self.add_virtual_u32_target()).collect();
+
+        BigUintTarget {
+            limbs,
+        }
+    }
+
     // Add two `BigUintTarget`s.
     pub fn add_biguint(&mut self, a: BigUintTarget, b: BigUintTarget) -> BigUintTarget {
         let num_limbs = a.limbs.len();
@@ -79,4 +143,49 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
             limbs: combined_limbs,
         }
     }
+
+    pub fn div_rem_biguint(&mut self, a: BigUintTarget, b: BigUintTarget) -> (BigUintTarget, BigUintTarget) {
+        let num_limbs = a.limbs.len();
+        let div = self.add_virtual_biguint_target(num_limbs);
+        let rem = self.add_virtual_biguint_target(num_limbs);
+
+        self.add_simple_generator(BigUintDivRemGenerator::<F, D> {
+            a: a.clone(),
+            b: b.clone(),
+            div: div.clone(),
+            rem: rem.clone(),
+            _phantom: PhantomData,
+        });
+
+        let div_b = self.mul_biguint(div, b);
+        let div_b_plus_rem = self.add_biguint(div_b, rem);
+        self.connect_biguint(x, div_b_plus_rem);
+
+        let 
+
+        self.assert_one()
+
+        (div, rem)
+    }
 }
+
+#[derive(Debug)]
+struct BigUintDivRemGenerator<F: RichField + Extendable<D>, const D: usize> {
+    a: BigUintTarget,
+    b: BigUintTarget,
+    div: BigUintTarget,
+    rem: BigUintTarget,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F>
+    for BigUintDivRemGenerator<F, D>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.a.limbs.iter().map(|&l| l.0).chain(self.b.limbs.iter().map(|&l| l.0)).collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        
+    }
+}
\ No newline at end of file
diff --git a/src/gadgets/multiple_comparison.rs b/src/gadgets/multiple_comparison.rs
index 11225ca5..3c2ac0f5 100644
--- a/src/gadgets/multiple_comparison.rs
+++ b/src/gadgets/multiple_comparison.rs
@@ -11,7 +11,7 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         assert_eq!(
             a.len(),
             b.len(),
-            "Permutation must have same number of inputs and outputs"
+            "Comparison must be between same number of inputs and outputs"
         );
         let n = a.len();
 
diff --git a/src/gadgets/nonnative.rs b/src/gadgets/nonnative.rs
index 7faa6d4e..e41f8fb5 100644
--- a/src/gadgets/nonnative.rs
+++ b/src/gadgets/nonnative.rs
@@ -1,21 +1,12 @@
-use std::collections::BTreeMap;
 use std::marker::PhantomData;
 
-use num::bigint::BigUint;
 
 use crate::field::field_types::RichField;
 use crate::field::{extension_field::Extendable, field_types::Field};
 use crate::gadgets::arithmetic_u32::U32Target;
-use crate::gates::arithmetic_u32::U32ArithmeticGate;
-use crate::gates::switch::SwitchGate;
-use crate::iop::generator::{GeneratedValues, SimpleGenerator};
-use crate::iop::target::Target;
-use crate::iop::witness::{PartitionWitness, Witness};
 use crate::plonk::circuit_builder::CircuitBuilder;
-use crate::util::bimap::bimap_from_lists;
 
 pub struct ForeignFieldTarget<FF: Field> {
-    /// These F elements are assumed to contain 32-bit values.
     limbs: Vec<U32Target>,
     _phantom: PhantomData<FF>,
 }
@@ -30,6 +21,16 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
             .collect()
     }
 
+    fn power_of_2_mod_order<FF: Field>(&mut self, i: usize) -> Vec<U32Target> {
+
+    }
+
+    pub fn powers_of_2_mod_order<FF: Field>(&mut self, max: usize) -> Vec<Vec<U32Target>> {
+        for i in 0..max {
+
+        }
+    }
+
     // Add two `ForeignFieldTarget`s.
     pub fn add_nonnative<FF: Field>(
         &mut self,
@@ -63,7 +64,7 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         let mut modulus_limbs = self.order_u32_limbs::<FF>();
         modulus_limbs.append(self.zero_u32());
 
-        let needs_reduce = self.list_le(modulus, limbs);
+        let needs_reduce = self.list_le(modulus_limbs, limbs);
 
         let mut to_subtract = vec![];
         for i in 0..num_limbs {
@@ -98,7 +99,7 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         let mut borrow = self.zero_u32();
         for i in 0..num_limbs {
             let (result, new_borrow) = self.sub_u32(a.limbs[i], b.limbs[i], borrow);
-            reduced_limbs[i] = result;
+            result_limbs[i] = result;
             borrow = new_borrow;
         }
         // Borrow should be zero here.
@@ -147,6 +148,7 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
     }
 
     pub fn reduce_mul_result<FF: Field>(&mut self, limbs: Vec<U32Target>) -> Vec<U32Target> {
+
         todo!()
     }
 }
diff --git a/src/iop/witness.rs b/src/iop/witness.rs
index 858bacd9..12186fb1 100644
--- a/src/iop/witness.rs
+++ b/src/iop/witness.rs
@@ -1,9 +1,13 @@
 use std::collections::HashMap;
 use std::convert::TryInto;
 
+use num::BigUint;
+
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::{Extendable, FieldExtension};
-use crate::field::field_types::Field;
+use crate::field::field_types::{Field, PrimeField};
+use crate::gadgets::arithmetic_u32::U32Target;
+use crate::gadgets::biguint::BigUintTarget;
 use crate::hash::hash_types::HashOutTarget;
 use crate::hash::hash_types::{HashOut, MerkleCapTarget};
 use crate::hash::merkle_tree::MerkleCap;
@@ -53,12 +57,24 @@ pub trait Witness<F: Field> {
         panic!("not a bool")
     }
 
+    fn get_u32_target(&self, target: U32Target) -> F {
+        let result = self.get_target(target.0);
+        debug_assert!(result.to_canonical_u64() < 1 << 32u64);
+        result
+    }
+
     fn get_hash_target(&self, ht: HashOutTarget) -> HashOut<F> {
         HashOut {
             elements: self.get_targets(&ht.elements).try_into().unwrap(),
         }
     }
 
+    fn get_biguint_target(&self, target: BigUintTarget) -> BigUint {
+        let mut result = BigUint::zero();
+        for (i, &limb) in target
+        result
+    }
+
     fn get_wire(&self, wire: Wire) -> F {
         self.get_target(Target::Wire(wire))
     }
diff --git a/src/plonk/circuit_builder.rs b/src/plonk/circuit_builder.rs
index f21c68cb..804e7be2 100644
--- a/src/plonk/circuit_builder.rs
+++ b/src/plonk/circuit_builder.rs
@@ -246,6 +246,11 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         self.connect(x, zero);
     }
 
+    pub fn assert_one(&mut self, x: Target) {
+        let one = self.one();
+        self.connect(x, one);
+    }
+
     pub fn add_generators(&mut self, generators: Vec<Box<dyn WitnessGenerator<F>>>) {
         self.generators.extend(generators);
     }