diff --git a/ml-dsa/src/algebra.rs b/ml-dsa/src/algebra.rs
index b89bd283..85de202e 100644
--- a/ml-dsa/src/algebra.rs
+++ b/ml-dsa/src/algebra.rs
@@ -210,3 +210,53 @@ impl<K: ArraySize> AlgebraExt for Vector<K> {
         )
     }
 }
+
+#[cfg(test)]
+mod test {
+    use super::*;
+
+    use crate::{MlDsa65, ParameterSet};
+
+    type Mod = <MlDsa65 as ParameterSet>::TwoGamma2;
+    const MOD: u32 = Mod::U32;
+    const MOD_ELEM: Elem = Elem::new(MOD);
+
+    #[test]
+    fn mod_plus_minus() {
+        for x in 0..MOD {
+            // BaseField::Q {
+            let x = Elem::new(x);
+            let x0 = x.mod_plus_minus::<Mod>();
+
+            // Outputs from mod+- should be in the half-open interval (-gamma2, gamma2]
+            let positive_bound = x0.0 <= MOD / 2;
+            let negative_bound = x0.0 > BaseField::Q - MOD / 2;
+            assert!(positive_bound || negative_bound);
+
+            // The output should be equivalent to the input, mod 2 * gamma2.  We add 2 * gamma2
+            // before comparing so that both values are "positive", avoiding interactions between
+            // the mod-Q and mod-M operations.
+            let xn = x + MOD_ELEM;
+            let x0n = x0 + MOD_ELEM;
+            assert_eq!(xn.0 % MOD, x0n.0 % MOD);
+        }
+    }
+
+    #[test]
+    fn decompose() {
+        for x in 0..MOD {
+            let x = Elem::new(x);
+            let (x1, x0) = x.decompose::<Mod>();
+
+            // The low-order output from decompose() is a mod+- output, optionally minus one.  So
+            // they should be in the closed interval [-gamma2, gamma2].
+            let positive_bound = x0.0 <= MOD / 2;
+            let negative_bound = x0.0 >= BaseField::Q - MOD / 2;
+            assert!(positive_bound || negative_bound);
+
+            // The low-order and high-order outputs should combine to form the input.
+            let xx = (MOD * x1.0 + x0.0) % BaseField::Q;
+            assert_eq!(xx, x.0);
+        }
+    }
+}
diff --git a/ml-dsa/src/hint.rs b/ml-dsa/src/hint.rs
index df9f321e..ced3cf34 100644
--- a/ml-dsa/src/hint.rs
+++ b/ml-dsa/src/hint.rs
@@ -22,18 +22,18 @@ fn use_hint<TwoGamma2: Unsigned>(h: bool, r: Elem) -> Elem {
     let gamma2 = TwoGamma2::U32 / 2;
     if h && r0.0 <= gamma2 {
         Elem::new((r1.0 + 1) % m)
-    } else if h && r0.0 > BaseField::Q - gamma2 {
+    } 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, it is guaranteed to be in (gamma2, gamma2].
+        // mod+- 2*gamma2 (possibly minus 1), it is guaranteed to be in [-gamma2, gamma2].
         unreachable!();
     } else {
         r1
     }
 }
 
-#[derive(Clone, PartialEq)]
+#[derive(Clone, PartialEq, Debug)]
 pub struct Hint<P>(pub Array<Array<bool, U256>, P::K>)
 where
     P: SignatureParams;
@@ -116,7 +116,7 @@ where
     }
 
     fn monotonic(a: &[usize]) -> bool {
-        a.iter().enumerate().all(|(i, x)| i == 0 || a[i - 1] < *x)
+        a.iter().enumerate().all(|(i, x)| i == 0 || a[i - 1] <= *x)
     }
 
     pub fn bit_unpack(y: &EncodedHint<P>) -> Option<Self> {
diff --git a/ml-dsa/src/lib.rs b/ml-dsa/src/lib.rs
index 3efe91cf..96d856bf 100644
--- a/ml-dsa/src/lib.rs
+++ b/ml-dsa/src/lib.rs
@@ -89,7 +89,7 @@ pub use crate::util::B32;
 pub use signature::Error;
 
 /// An ML-DSA signature
-#[derive(Clone, PartialEq)]
+#[derive(Clone, PartialEq, Debug)]
 pub struct Signature<P: MlDsaParams> {
     c_tilde: Array<u8, P::Lambda>,
     z: Vector<P::L>,
@@ -899,4 +899,42 @@ mod test {
         sign_verify_round_trip_test::<MlDsa65>();
         sign_verify_round_trip_test::<MlDsa87>();
     }
+
+    fn many_round_trip_test<P>()
+    where
+        P: MlDsaParams,
+    {
+        use rand::Rng;
+
+        const ITERATIONS: usize = 1000;
+
+        let mut rng = rand::thread_rng();
+        let mut seed = B32::default();
+
+        for _i in 0..ITERATIONS {
+            let seed_data: &mut [u8] = seed.as_mut();
+            rng.fill(seed_data);
+
+            let kp = P::key_gen_internal(&seed);
+            let sk = kp.signing_key;
+            let vk = kp.verifying_key;
+
+            let M = b"Hello world";
+            let rnd = Array([0u8; 32]);
+            let sig = sk.sign_internal(&[M], &rnd);
+
+            let sig_enc = sig.encode();
+            let sig_dec = Signature::<P>::decode(&sig_enc).unwrap();
+
+            assert_eq!(sig_dec, sig);
+            assert!(vk.verify_internal(&[M], &sig_dec));
+        }
+    }
+
+    #[test]
+    fn many_round_trip() {
+        many_round_trip_test::<MlDsa44>();
+        many_round_trip_test::<MlDsa65>();
+        many_round_trip_test::<MlDsa87>();
+    }
 }