diff --git a/src/derive/curve.rs b/src/derive/curve.rs
index 66d673dabb..4aa569e485 100644
--- a/src/derive/curve.rs
+++ b/src/derive/curve.rs
@@ -833,6 +833,10 @@ macro_rules! new_curve_impl {
         impl<'a, 'b> Mul<&'b $scalar> for &'a $name {
             type Output = $name;
 
+            // This is a simple double-and-add implementation of point
+            // multiplication, moving from most significant to least
+            // significant bit of the scalar.
+
             fn mul(self, other: &'b $scalar) -> Self::Output {
                 let mut acc = $name::identity();
                 for bit in other
@@ -935,21 +939,17 @@ macro_rules! new_curve_impl {
             type Output = $name;
 
             fn mul(self, other: &'b $scalar) -> Self::Output {
-                // TODO: make this faster
-
                 let mut acc = $name::identity();
 
                 // This is a simple double-and-add implementation of point
                 // multiplication, moving from most significant to least
                 // significant bit of the scalar.
-                //
-                // NOTE: We skip the leading bit because it's always unset.
+
                 for bit in other
                     .to_repr()
                     .iter()
                     .rev()
                     .flat_map(|byte| (0..8).rev().map(move |i| Choice::from((byte >> i) & 1u8)))
-                    .skip(1)
                 {
                     acc = acc.double();
                     acc = $name::conditional_select(&acc, &(acc + self), bit);
diff --git a/src/secp256k1/curve.rs b/src/secp256k1/curve.rs
index 51c2340007..64f48a5af7 100644
--- a/src/secp256k1/curve.rs
+++ b/src/secp256k1/curve.rs
@@ -75,3 +75,62 @@ impl CurveAffineExt for Secp256k1Affine {
 fn test_curve() {
     crate::tests::curve::curve_tests::<Secp256k1>();
 }
+
+#[test]
+fn ecdsa_example() {
+    use crate::group::Curve;
+    use crate::{CurveAffine, FieldExt};
+    use rand_core::OsRng;
+
+    fn mod_n(x: Fp) -> Fq {
+        let mut x_repr = [0u8; 32];
+        x_repr.copy_from_slice(x.to_repr().as_ref());
+        let mut x_bytes = [0u8; 64];
+        x_bytes[..32].copy_from_slice(&x_repr[..]);
+        Fq::from_bytes_wide(&x_bytes)
+    }
+
+    let g = Secp256k1::generator();
+
+    for _ in 0..1000 {
+        // Generate a key pair
+        let sk = Fq::random(OsRng);
+        let pk = (g * sk).to_affine();
+
+        // Generate a valid signature
+        // Suppose `m_hash` is the message hash
+        let msg_hash = Fq::random(OsRng);
+
+        let (r, s) = {
+            // Draw arandomness
+            let k = Fq::random(OsRng);
+            let k_inv = k.invert().unwrap();
+
+            // Calculate `r`
+            let r_point = (g * k).to_affine().coordinates().unwrap();
+            let x = r_point.x();
+            let r = mod_n(*x);
+
+            // Calculate `s`
+            let s = k_inv * (msg_hash + (r * sk));
+
+            (r, s)
+        };
+
+        {
+            // Verify
+            let s_inv = s.invert().unwrap();
+            let u_1 = msg_hash * s_inv;
+            let u_2 = r * s_inv;
+
+            let v_1 = g * u_1;
+            let v_2 = pk * u_2;
+
+            let r_point = (v_1 + v_2).to_affine().coordinates().unwrap();
+            let x_candidate = r_point.x();
+            let r_candidate = mod_n(*x_candidate);
+
+            assert_eq!(r, r_candidate);
+        }
+    }
+}