diff --git a/src/internal_math.rs b/src/internal_math.rs
index 515191c..686b315 100644
--- a/src/internal_math.rs
+++ b/src/internal_math.rs
@@ -59,28 +59,26 @@ impl Barrett {
 ///
 /// * `a` `0 <= a < m`
 /// * `b` `0 <= b < m`
-/// * `m` `1 <= m <= 2^31`
-/// * `im` = ceil(2^64 / `m`)
+/// * `m` `1 <= m < 2^32`
+/// * `im` = ceil(2^64 / `m`) = floor((2^64 - 1) / `m`) + 1
 #[allow(clippy::many_single_char_names)]
 pub(crate) fn mul_mod(a: u32, b: u32, m: u32, im: u64) -> u32 {
     // [1] m = 1
     // a = b = im = 0, so okay
 
     // [2] m >= 2
-    // im = ceil(2^64 / m)
+    // im = ceil(2^64 / m) = floor((2^64 - 1) / m) + 1
     // -> im * m = 2^64 + r (0 <= r < m)
     // let z = a*b = c*m + d (0 <= c, d < m)
     // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
     // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
     // ((ab * im) >> 64) == c or c + 1
-    let mut z = a as u64;
-    z *= b as u64;
+    let z = (a as u64) * (b as u64);
     let x = (((z as u128) * (im as u128)) >> 64) as u64;
-    let mut v = z.wrapping_sub(x.wrapping_mul(m as u64)) as u32;
-    if m <= v {
-        v = v.wrapping_add(m);
+    match z.overflowing_sub(x.wrapping_mul(m as u64)) {
+        (v, true) => (v as u32).wrapping_add(m),
+        (v, false) => v as u32,
     }
-    v
 }
 
 /// # Parameters
@@ -280,6 +278,14 @@ mod tests {
         let b = Barrett::new(2147483647);
         assert_eq!(b.umod(), 2147483647);
         assert_eq!(b.mul(1073741824, 2147483645), 2147483646);
+
+        // test `2^31 < self._m < 2^32` case.
+        let b = Barrett::new(3221225471);
+        assert_eq!(b.umod(), 3221225471);
+        assert_eq!(b.mul(3188445886, 2844002853), 1840468257);
+        assert_eq!(b.mul(2834869488, 2779159607), 2084027561);
+        assert_eq!(b.mul(3032263594, 3039996727), 2130247251);
+        assert_eq!(b.mul(3029175553, 3140869278), 1892378237);
     }
 
     #[test]