diff --git a/src/gadgets/mod.rs b/src/gadgets/mod.rs
index 4b3371ef..aa18fbeb 100644
--- a/src/gadgets/mod.rs
+++ b/src/gadgets/mod.rs
@@ -8,5 +8,6 @@ pub mod polynomial;
 pub mod random_access;
 pub mod range_check;
 pub mod select;
+pub mod sorting;
 pub mod split_base;
 pub(crate) mod split_join;
diff --git a/src/gadgets/permutation.rs b/src/gadgets/permutation.rs
index 126846ec..0f320dfd 100644
--- a/src/gadgets/permutation.rs
+++ b/src/gadgets/permutation.rs
@@ -384,7 +384,7 @@ impl<F: Field> SimpleGenerator<F> for PermutationGenerator<F> {
 #[cfg(test)]
 mod tests {
     use anyhow::Result;
-    use rand::{seq::SliceRandom, thread_rng};
+    use rand::{seq::SliceRandom, thread_rng, Rng};
 
     use super::*;
     use crate::field::crandall_field::CrandallField;
@@ -418,6 +418,35 @@ mod tests {
         verify(proof, &data.verifier_only, &data.common)
     }
 
+    fn test_permutation_duplicates(size: usize) -> Result<()> {
+        type F = CrandallField;
+        const D: usize = 4;
+
+        let config = CircuitConfig::large_zk_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let mut rng = thread_rng();
+        let lst: Vec<F> = (0..size * 2)
+            .map(|_| F::from_canonical_usize(rng.gen_range(0..2usize)))
+            .collect();
+        let a: Vec<Vec<Target>> = lst[..]
+            .chunks(2)
+            .map(|pair| vec![builder.constant(pair[0]), builder.constant(pair[1])])
+            .collect();
+
+        let mut b = a.clone();
+        b.shuffle(&mut thread_rng());
+
+        builder.assert_permutation(a, b);
+
+        let data = builder.build();
+        let proof = data.prove(pw).unwrap();
+
+        verify(proof, &data.verifier_only, &data.common)
+    }
+
     fn test_permutation_bad(size: usize) -> Result<()> {
         type F = CrandallField;
         const D: usize = 4;
@@ -446,6 +475,15 @@ mod tests {
         Ok(())
     }
 
+    #[test]
+    fn test_permutations_duplicates() -> Result<()> {
+        for n in 2..9 {
+            test_permutation_duplicates(n)?;
+        }
+
+        Ok(())
+    }
+
     #[test]
     fn test_permutations_good() -> Result<()> {
         for n in 2..9 {
diff --git a/src/gadgets/sorting.rs b/src/gadgets/sorting.rs
new file mode 100644
index 00000000..f202620c
--- /dev/null
+++ b/src/gadgets/sorting.rs
@@ -0,0 +1,263 @@
+use std::marker::PhantomData;
+
+use itertools::izip;
+
+use crate::field::extension_field::Extendable;
+use crate::field::field_types::{Field, RichField};
+use crate::gates::comparison::ComparisonGate;
+use crate::iop::generator::{GeneratedValues, SimpleGenerator};
+use crate::iop::target::{BoolTarget, Target};
+use crate::iop::witness::{PartitionWitness, Witness};
+use crate::plonk::circuit_builder::CircuitBuilder;
+use crate::util::ceil_div_usize;
+
+pub struct MemoryOp<F: Field> {
+    is_write: bool,
+    address: F,
+    timestamp: F,
+    value: F,
+}
+
+#[derive(Clone, Debug)]
+pub struct MemoryOpTarget {
+    is_write: BoolTarget,
+    address: Target,
+    timestamp: Target,
+    value: Target,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
+    pub fn assert_permutation_memory_ops(&mut self, a: &[MemoryOpTarget], b: &[MemoryOpTarget]) {
+        let a_chunks: Vec<Vec<Target>> = a
+            .iter()
+            .map(|op| vec![op.address, op.timestamp, op.is_write.target, op.value])
+            .collect();
+        let b_chunks: Vec<Vec<Target>> = b
+            .iter()
+            .map(|op| vec![op.address, op.timestamp, op.is_write.target, op.value])
+            .collect();
+
+        self.assert_permutation(a_chunks, b_chunks);
+    }
+
+    /// Add a ComparisonGate to assert that `lhs` is less than `rhs`, where their values are at most `bits` bits.
+    pub fn assert_le(&mut self, lhs: Target, rhs: Target, bits: usize, num_chunks: usize) {
+        let gate = ComparisonGate::new(bits, num_chunks);
+        let gate_index = self.add_gate(gate.clone(), vec![]);
+
+        self.connect(Target::wire(gate_index, gate.wire_first_input()), lhs);
+        self.connect(Target::wire(gate_index, gate.wire_second_input()), rhs);
+    }
+
+    /// Sort memory operations by address value, then by timestamp value.
+    /// This is done by combining address and timestamp into one field element (using their given bit lengths).
+    pub fn sort_memory_ops(
+        &mut self,
+        ops: &[MemoryOpTarget],
+        address_bits: usize,
+        timestamp_bits: usize,
+    ) -> Vec<MemoryOpTarget> {
+        let n = ops.len();
+
+        let combined_bits = address_bits + timestamp_bits;
+        let chunk_bits = 3;
+        let num_chunks = ceil_div_usize(combined_bits, chunk_bits);
+
+        // This is safe because `assert_permutation` will force these targets (in the output list) to match the boolean values from the input list.
+        let is_write_targets: Vec<_> = self
+            .add_virtual_targets(n)
+            .iter()
+            .map(|&t| BoolTarget::new_unsafe(t))
+            .collect();
+
+        let address_targets = self.add_virtual_targets(n);
+        let timestamp_targets = self.add_virtual_targets(n);
+        let value_targets = self.add_virtual_targets(n);
+
+        let output_targets: Vec<_> = izip!(
+            is_write_targets,
+            address_targets,
+            timestamp_targets,
+            value_targets
+        )
+        .map(|(i, a, t, v)| MemoryOpTarget {
+            is_write: i,
+            address: a,
+            timestamp: t,
+            value: v,
+        })
+        .collect();
+
+        let two_n = self.constant(F::from_canonical_usize(1 << timestamp_bits));
+        let address_timestamp_combined: Vec<_> = output_targets
+            .iter()
+            .map(|op| self.mul_add(op.address, two_n, op.timestamp))
+            .collect();
+
+        for i in 1..n {
+            self.assert_le(
+                address_timestamp_combined[i - 1],
+                address_timestamp_combined[i],
+                combined_bits,
+                num_chunks,
+            );
+        }
+
+        self.assert_permutation_memory_ops(ops, &output_targets);
+
+        self.add_simple_generator(MemoryOpSortGenerator::<F, D> {
+            input_ops: ops.to_vec(),
+            output_ops: output_targets.clone(),
+            _phantom: PhantomData,
+        });
+
+        output_targets
+    }
+}
+
+#[derive(Debug)]
+struct MemoryOpSortGenerator<F: RichField + Extendable<D>, const D: usize> {
+    input_ops: Vec<MemoryOpTarget>,
+    output_ops: Vec<MemoryOpTarget>,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F>
+    for MemoryOpSortGenerator<F, D>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.input_ops
+            .iter()
+            .map(|op| vec![op.is_write.target, op.address, op.timestamp, op.value])
+            .flatten()
+            .collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let n = self.input_ops.len();
+        debug_assert!(self.output_ops.len() == n);
+
+        let mut ops: Vec<_> = self
+            .input_ops
+            .iter()
+            .map(|op| {
+                let is_write = witness.get_bool_target(op.is_write);
+                let address = witness.get_target(op.address);
+                let timestamp = witness.get_target(op.timestamp);
+                let value = witness.get_target(op.value);
+                MemoryOp {
+                    is_write,
+                    address,
+                    timestamp,
+                    value,
+                }
+            })
+            .collect();
+
+        ops.sort_unstable_by_key(|op| {
+            (
+                op.address.to_canonical_u64(),
+                op.timestamp.to_canonical_u64(),
+            )
+        });
+
+        for (op, out_op) in ops.iter().zip(&self.output_ops) {
+            out_buffer.set_target(out_op.is_write.target, F::from_bool(op.is_write));
+            out_buffer.set_target(out_op.address, op.address);
+            out_buffer.set_target(out_op.timestamp, op.timestamp);
+            out_buffer.set_target(out_op.value, op.value);
+        }
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use rand::{thread_rng, Rng};
+
+    use super::*;
+    use crate::field::crandall_field::CrandallField;
+    use crate::field::field_types::{Field, PrimeField};
+    use crate::iop::witness::PartialWitness;
+    use crate::plonk::circuit_data::CircuitConfig;
+    use crate::plonk::verifier::verify;
+
+    fn test_sorting(size: usize, address_bits: usize, timestamp_bits: usize) -> Result<()> {
+        type F = CrandallField;
+        const D: usize = 4;
+
+        let config = CircuitConfig::large_zk_config();
+
+        let mut pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let mut rng = thread_rng();
+        let is_write_vals: Vec<_> = (0..size).map(|_| rng.gen_range(0..2) != 0).collect();
+        let address_vals: Vec<_> = (0..size)
+            .map(|_| F::from_canonical_u64(rng.gen_range(0..1 << address_bits as u64)))
+            .collect();
+        let timestamp_vals: Vec<_> = (0..size)
+            .map(|_| F::from_canonical_u64(rng.gen_range(0..1 << timestamp_bits as u64)))
+            .collect();
+        let value_vals: Vec<_> = (0..size).map(|_| F::rand()).collect();
+
+        let input_ops: Vec<MemoryOpTarget> = izip!(
+            is_write_vals.clone(),
+            address_vals.clone(),
+            timestamp_vals.clone(),
+            value_vals.clone()
+        )
+        .map(|(is_write, address, timestamp, value)| MemoryOpTarget {
+            is_write: builder.constant_bool(is_write),
+            address: builder.constant(address),
+            timestamp: builder.constant(timestamp),
+            value: builder.constant(value),
+        })
+        .collect();
+
+        let combined_vals_u64: Vec<_> = timestamp_vals
+            .iter()
+            .zip(&address_vals)
+            .map(|(&t, &a)| (a.to_canonical_u64() << timestamp_bits as u64) + t.to_canonical_u64())
+            .collect();
+        let mut input_ops_and_keys: Vec<_> =
+            izip!(is_write_vals, address_vals, timestamp_vals, value_vals)
+                .zip(combined_vals_u64)
+                .collect::<Vec<_>>();
+        input_ops_and_keys.sort_by_key(|(_, val)| val.clone());
+        let input_ops_sorted: Vec<_> = input_ops_and_keys.iter().map(|(x, _)| x).collect();
+
+        let output_ops =
+            builder.sort_memory_ops(input_ops.as_slice(), address_bits, timestamp_bits);
+
+        for i in 0..size {
+            pw.set_bool_target(output_ops[i].is_write, input_ops_sorted[i].0);
+            pw.set_target(output_ops[i].address, input_ops_sorted[i].1);
+            pw.set_target(output_ops[i].timestamp, input_ops_sorted[i].2);
+            pw.set_target(output_ops[i].value, input_ops_sorted[i].3);
+        }
+
+        let data = builder.build();
+        let proof = data.prove(pw).unwrap();
+
+        verify(proof, &data.verifier_only, &data.common)
+    }
+
+    #[test]
+    fn test_sorting_small() -> Result<()> {
+        let size = 5;
+        let address_bits = 20;
+        let timestamp_bits = 20;
+
+        test_sorting(size, address_bits, timestamp_bits)
+    }
+
+    #[test]
+    fn test_sorting_large() -> Result<()> {
+        let size = 20;
+        let address_bits = 20;
+        let timestamp_bits = 20;
+
+        test_sorting(size, address_bits, timestamp_bits)
+    }
+}
diff --git a/src/gates/comparison.rs b/src/gates/comparison.rs
index 773b8bf3..44f2923a 100644
--- a/src/gates/comparison.rs
+++ b/src/gates/comparison.rs
@@ -13,9 +13,9 @@ use crate::plonk::plonk_common::{reduce_with_powers, reduce_with_powers_ext_recu
 use crate::plonk::vars::{EvaluationTargets, EvaluationVars, EvaluationVarsBase};
 use crate::util::ceil_div_usize;
 
-/// A gate for checking that one value is less than another.
+/// A gate for checking that one value is less than or equal to another.
 #[derive(Clone, Debug)]
-pub(crate) struct ComparisonGate<F: PrimeField + Extendable<D>, const D: usize> {
+pub struct ComparisonGate<F: PrimeField + Extendable<D>, const D: usize> {
     pub(crate) num_bits: usize,
     pub(crate) num_chunks: usize,
     _phantom: PhantomData<F>,
@@ -137,7 +137,7 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ComparisonGate
         constraints.push(most_significant_diff - most_significant_diff_so_far);
 
         // Range check `most_significant_diff` to be less than `chunk_size`.
-        let product = (1..chunk_size)
+        let product = (0..chunk_size)
             .map(|x| most_significant_diff - F::Extension::from_canonical_usize(x))
             .product();
         constraints.push(product);
@@ -205,7 +205,7 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ComparisonGate
         constraints.push(most_significant_diff - most_significant_diff_so_far);
 
         // Range check `most_significant_diff` to be less than `chunk_size`.
-        let product = (1..chunk_size)
+        let product = (0..chunk_size)
             .map(|x| most_significant_diff - F::from_canonical_usize(x))
             .product();
         constraints.push(product);
@@ -286,7 +286,7 @@ impl<F: RichField + Extendable<D>, const D: usize> Gate<F, D> for ComparisonGate
 
         // Range check `most_significant_diff` to be less than `chunk_size`.
         let mut product = builder.one_extension();
-        for x in 1..chunk_size {
+        for x in 0..chunk_size {
             let x_F = builder.constant_extension(F::Extension::from_canonical_usize(x));
             let diff = builder.sub_extension(most_significant_diff, x_F);
             product = builder.mul_extension(product, diff);
@@ -386,7 +386,7 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F>
 
         let mut most_significant_diff_so_far = F::ZERO;
         let mut intermediate_values = Vec::new();
-        for i in 1..self.gate.num_chunks {
+        for i in 0..self.gate.num_chunks {
             if first_input_chunks[i] != second_input_chunks[i] {
                 most_significant_diff_so_far = second_input_chunks[i] - first_input_chunks[i];
                 intermediate_values.push(F::ZERO);
@@ -552,7 +552,7 @@ mod tests {
         let first_input_u64 = rng.gen_range(0..max);
         let second_input_u64 = {
             let mut val = rng.gen_range(0..max);
-            while val <= first_input_u64 {
+            while val < first_input_u64 {
                 val = rng.gen_range(0..max);
             }
             val
@@ -561,20 +561,39 @@ mod tests {
         let first_input = F::from_canonical_u64(first_input_u64);
         let second_input = F::from_canonical_u64(second_input_u64);
 
-        let gate = ComparisonGate::<F, D> {
+        let less_than_gate = ComparisonGate::<F, D> {
             num_bits,
             num_chunks,
             _phantom: PhantomData,
         };
-
-        let vars = EvaluationVars {
+        let less_than_vars = EvaluationVars {
             local_constants: &[],
             local_wires: &get_wires(first_input, second_input),
             public_inputs_hash: &HashOut::rand(),
         };
-
         assert!(
-            gate.eval_unfiltered(vars).iter().all(|x| x.is_zero()),
+            less_than_gate
+                .eval_unfiltered(less_than_vars)
+                .iter()
+                .all(|x| x.is_zero()),
+            "Gate constraints are not satisfied."
+        );
+
+        let equal_gate = ComparisonGate::<F, D> {
+            num_bits,
+            num_chunks,
+            _phantom: PhantomData,
+        };
+        let equal_vars = EvaluationVars {
+            local_constants: &[],
+            local_wires: &get_wires(first_input, first_input),
+            public_inputs_hash: &HashOut::rand(),
+        };
+        assert!(
+            equal_gate
+                .eval_unfiltered(equal_vars)
+                .iter()
+                .all(|x| x.is_zero()),
             "Gate constraints are not satisfied."
         );
     }
diff --git a/src/gates/switch.rs b/src/gates/switch.rs
index 1df5fea9..2f6b7122 100644
--- a/src/gates/switch.rs
+++ b/src/gates/switch.rs
@@ -236,20 +236,25 @@ impl<F: RichField + Extendable<D>, const D: usize> SwitchGenerator<F, D> {
 
         let get_local_wire = |input| witness.get_wire(local_wire(input));
 
-        for e in 0..self.gate.chunk_size {
-            let switch_bool_wire = local_wire(self.gate.wire_switch_bool(self.copy));
-            let first_input = get_local_wire(self.gate.wire_first_input(self.copy, e));
-            let second_input = get_local_wire(self.gate.wire_second_input(self.copy, e));
-            let first_output = get_local_wire(self.gate.wire_first_output(self.copy, e));
-            let second_output = get_local_wire(self.gate.wire_second_output(self.copy, e));
+        let switch_bool_wire = local_wire(self.gate.wire_switch_bool(self.copy));
 
-            if first_output == first_input && second_output == second_input {
-                out_buffer.set_wire(switch_bool_wire, F::ZERO);
-            } else if first_output == second_input && second_output == first_input {
-                out_buffer.set_wire(switch_bool_wire, F::ONE);
-            } else {
-                panic!("No permutation from given inputs to given outputs");
-            }
+        let mut first_inputs = Vec::new();
+        let mut second_inputs = Vec::new();
+        let mut first_outputs = Vec::new();
+        let mut second_outputs = Vec::new();
+        for e in 0..self.gate.chunk_size {
+            first_inputs.push(get_local_wire(self.gate.wire_first_input(self.copy, e)));
+            second_inputs.push(get_local_wire(self.gate.wire_second_input(self.copy, e)));
+            first_outputs.push(get_local_wire(self.gate.wire_first_output(self.copy, e)));
+            second_outputs.push(get_local_wire(self.gate.wire_second_output(self.copy, e)));
+        }
+
+        if first_outputs == first_inputs && second_outputs == second_inputs {
+            out_buffer.set_wire(switch_bool_wire, F::ZERO);
+        } else if first_outputs == second_inputs && second_outputs == first_inputs {
+            out_buffer.set_wire(switch_bool_wire, F::ONE);
+        } else {
+            panic!("No permutation from given inputs to given outputs");
         }
     }
 
@@ -261,12 +266,12 @@ impl<F: RichField + Extendable<D>, const D: usize> SwitchGenerator<F, D> {
 
         let get_local_wire = |input| witness.get_wire(local_wire(input));
 
+        let switch_bool = get_local_wire(self.gate.wire_switch_bool(self.copy));
         for e in 0..self.gate.chunk_size {
             let first_output_wire = local_wire(self.gate.wire_first_output(self.copy, e));
             let second_output_wire = local_wire(self.gate.wire_second_output(self.copy, e));
             let first_input = get_local_wire(self.gate.wire_first_input(self.copy, e));
             let second_input = get_local_wire(self.gate.wire_second_input(self.copy, e));
-            let switch_bool = get_local_wire(self.gate.wire_switch_bool(self.copy));
 
             let (first_output, second_output) = if switch_bool == F::ZERO {
                 (first_input, second_input)