diff --git a/ml-dsa/src/hint.rs b/ml-dsa/src/hint.rs
index 20ced9f..7c24c45 100644
--- a/ml-dsa/src/hint.rs
+++ b/ml-dsa/src/hint.rs
@@ -7,27 +7,31 @@ use module_lattice::utils::Truncate;
 use crate::algebra::{AlgebraExt, BaseField, Decompose, Elem, Field, Polynomial, Vector};
 use crate::param::{EncodedHint, SignatureParams};
 
+/// Algorithm 39 `MakeHint`: computes hint bit indicating whether adding `z` to `r` alters the high
+/// bits of `r`.
 fn make_hint<TwoGamma2: Unsigned>(z: Elem, r: Elem) -> bool {
     let r1 = r.high_bits::<TwoGamma2>();
     let v1 = (r + z).high_bits::<TwoGamma2>();
     r1 != v1
 }
 
-// The method only deals with public data, so we don't need to worry that / and % are not
-// constant-time.
-#[allow(clippy::integer_division_remainder_used)]
+/// Algorithm 40 `UseHint`: returns the high bits of `r` adjusted according to hint `h`.
+#[allow(clippy::integer_division_remainder_used, reason = "params are public")]
 fn use_hint<TwoGamma2: Unsigned>(h: bool, r: Elem) -> Elem {
     let m: u32 = (BaseField::Q - 1) / TwoGamma2::U32;
     let (r1, r0) = r.decompose::<TwoGamma2>();
     let gamma2 = TwoGamma2::U32 / 2;
-    if h && r0.0 <= gamma2 {
-        Elem::new((r1.0 + 1) % m)
-    } else if h && r0.0 >= BaseField::Q - gamma2 {
-        Elem::new((r1.0 + m - 1) % m)
-    } else if h {
-        // We use the Elem encoding even for signed integers.  Since r0 is computed
-        // mod+- 2*gamma2 (possibly minus 1), it is guaranteed to be in [-gamma2, gamma2].
-        unreachable!();
+
+    if h {
+        if r0.0 > 0 && r0.0 <= gamma2 {
+            Elem::new((r1.0 + 1) % m)
+        } else if (r0.0 == 0) || (r0.0 >= BaseField::Q - gamma2) {
+            Elem::new((r1.0 + m - 1) % m)
+        } else {
+            // We use the Elem encoding even for signed integers.  Since r0 is computed
+            // mod+- 2*gamma2 (possibly minus 1), it is guaranteed to be in [-gamma2, gamma2].
+            unreachable!();
+        }
     } else {
         r1
     }
@@ -227,6 +231,18 @@ mod test {
         panic!("Could not find suitable test value");
     }
 
+    #[test]
+    fn use_hint_r0_is_zero() {
+        type TwoGamma2 = <MlDsa65 as ParameterSet>::TwoGamma2;
+        let m = (BaseField::Q - 1) / TwoGamma2::U32;
+        let r = Elem::new(0);
+        let (r1, r0) = r.decompose::<TwoGamma2>();
+        assert_eq!(r0.0, 0);
+
+        let result = use_hint::<TwoGamma2>(true, r);
+        assert_eq!(result, Elem::new((r1.0 + m - 1) % m));
+    }
+
     #[test]
     fn use_hint_threshold() {
         type TwoGamma2 = <MlDsa65 as ParameterSet>::TwoGamma2;