diff --git a/src/fri/recursive_verifier.rs b/src/fri/recursive_verifier.rs
index 0e7fa381..046a54ae 100644
--- a/src/fri/recursive_verifier.rs
+++ b/src/fri/recursive_verifier.rs
@@ -320,6 +320,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
             let g_ext = self.convert_to_ext(g);
             let phi_ext = self.convert_to_ext(phi);
             let zero = self.zero_extension();
+            // `subgroup_x = g*phi, vanish_zeta = g*phi - zeta`
             let tmp = self.double_arithmetic_extension(
                 F::ONE,
                 F::NEG_ONE,
diff --git a/src/util/reducing.rs b/src/util/reducing.rs
index 8e8fa661..d2f4fcf2 100644
--- a/src/util/reducing.rs
+++ b/src/util/reducing.rs
@@ -215,6 +215,8 @@ impl<const D: usize> ReducingFactorTarget<D> {
         builder.mul_extension(exp, x)
     }
 
+    /// Returns `(self.shift(x), a*b)`.
+    /// Used to take advantage of the second arithmetic operation in the `ArithmeticExtensionGate`.
     pub fn shift_and_mul<F>(
         &mut self,
         x: ExtensionTarget<D>,