diff --git a/src/montgomery.rs b/src/montgomery.rs
index 232fc3254..dc53a9d3f 100644
--- a/src/montgomery.rs
+++ b/src/montgomery.rs
@@ -335,17 +335,23 @@ impl<'a, 'b> Mul<&'b Scalar> for &'a MontgomeryPoint {
             W: FieldElement::one(),
         };
 
-        let bits: [i8; 256] = scalar.bits();
-
-        for i in (0..255).rev() {
-            let choice: u8 = (bits[i + 1] ^ bits[i]) as u8;
+        // Go through the bits from most to least significant, using a sliding window of 2
+        let mut bits = scalar.bits_le().rev();
+        let mut prev_bit = bits.next().unwrap();
+        for cur_bit in bits {
+            let choice: u8 = (prev_bit ^ cur_bit) as u8;
 
             debug_assert!(choice == 0 || choice == 1);
 
             ProjectivePoint::conditional_swap(&mut x0, &mut x1, choice.into());
             differential_add_and_double(&mut x0, &mut x1, &affine_u);
+
+            prev_bit = cur_bit;
         }
-        ProjectivePoint::conditional_swap(&mut x0, &mut x1, Choice::from(bits[0] as u8));
+        // The final value of prev_bit above is scalar.bits()[0], i.e., the LSB of scalar
+        ProjectivePoint::conditional_swap(&mut x0, &mut x1, Choice::from(prev_bit as u8));
+        // Don't leave the bit in the stack
+        prev_bit.zeroize();
 
         x0.as_affine()
     }
diff --git a/src/scalar.rs b/src/scalar.rs
index db4bf433f..3cdf0e300 100644
--- a/src/scalar.rs
+++ b/src/scalar.rs
@@ -847,16 +847,14 @@ impl Scalar {
         ret
     }
 
-    /// Get the bits of the scalar.
-    pub(crate) fn bits(&self) -> [i8; 256] {
-        let mut bits = [0i8; 256];
-        #[allow(clippy::needless_range_loop)]
-        for i in 0..256 {
-            // As i runs from 0..256, the bottom 3 bits index the bit,
-            // while the upper bits index the byte.
-            bits[i] = ((self.bytes[i >> 3] >> (i & 7)) & 1u8) as i8;
-        }
-        bits
+    /// Get the bits of the scalar, in little-endian order
+    pub(crate) fn bits_le(&self) -> impl DoubleEndedIterator<Item = bool> + '_ {
+        (0..256).map(|i| {
+            // As i runs from 0..256, the bottom 3 bits index the bit, while the upper bits index
+            // the byte. Since self.bytes is little-endian at the byte level, this iterator is
+            // little-endian on the bit level
+            ((self.bytes[i >> 3] >> (i & 7)) & 1u8) == 1
+        })
     }
 
     /// Compute a width-\\(w\\) "Non-Adjacent Form" of this scalar.