diff --git a/src/plonk/permutation/keygen.rs b/src/plonk/permutation/keygen.rs
index d5bfadd..c7d9274 100644
--- a/src/plonk/permutation/keygen.rs
+++ b/src/plonk/permutation/keygen.rs
@@ -30,6 +30,9 @@ impl Assembly {
             columns.push((0..params.n).map(|j| (i, j as usize)).collect());
         }
 
+        // Before any equality constraints are applied, every cell in the permutation is
+        // in a 1-cycle; therefore mapping and aux are identical, because every cell is
+        // its own distinguished element.
         Assembly {
             mapping: columns.clone(),
             aux: columns,
@@ -53,9 +56,12 @@ impl Assembly {
             return Err(Error::BoundsFailure);
         }
 
+        // See book/src/design/permutation.md for a description of this algorithm.
+
         let mut left_cycle = self.aux[left_column][left_row];
         let mut right_cycle = self.aux[right_column][right_row];
 
+        // If left and right are in the same cycle, do nothing.
         if left_cycle == right_cycle {
             return Ok(());
         }
@@ -64,6 +70,7 @@ impl Assembly {
             std::mem::swap(&mut left_cycle, &mut right_cycle);
         }
 
+        // Merge the right cycle into the left one.
         self.sizes[left_cycle.0][left_cycle.1] += self.sizes[right_cycle.0][right_cycle.1];
         let mut i = right_cycle;
         loop {