diff --git a/elliptic-curve/src/lib.rs b/elliptic-curve/src/lib.rs
index c5cf968db..c829740fd 100644
--- a/elliptic-curve/src/lib.rs
+++ b/elliptic-curve/src/lib.rs
@@ -24,6 +24,8 @@ extern crate std;
 pub mod error;
 pub mod oid;
 pub mod ops;
+pub mod point;
+pub mod scalar;
 pub mod secret_key;
 
 // TODO(tarcieri): other curve forms
@@ -41,9 +43,12 @@ pub use rand_core;
 #[cfg(feature = "zeroize")]
 pub use zeroize;
 
-use core::{fmt::Debug, ops::Add};
+use core::{
+    fmt::Debug,
+    ops::{Add, Mul},
+};
 use generic_array::{typenum::Unsigned, ArrayLength, GenericArray};
-use subtle::ConditionallySelectable;
+use subtle::{ConditionallySelectable, ConstantTimeEq};
 
 #[cfg(feature = "rand_core")]
 use rand_core::{CryptoRng, RngCore};
@@ -67,10 +72,13 @@ pub trait Curve: Clone + Debug + Default + Eq + Ord + Send + Sync {
 /// Elliptic curve with curve arithmetic support
 pub trait Arithmetic: Curve {
     /// Scalar type for a given curve
-    type Scalar: ConditionallySelectable + Default + secret_key::FromSecretKey<Self>;
+    type Scalar: ConditionallySelectable
+        + ConstantTimeEq
+        + Default
+        + secret_key::FromSecretKey<Self>;
 
     /// Affine point type for a given curve
-    type AffinePoint: ConditionallySelectable + ops::MulBase<Scalar = Self::Scalar>;
+    type AffinePoint: ConditionallySelectable + Mul<scalar::NonZeroScalar<Self>> + point::Generator;
 }
 
 /// Associate an object identifier (OID) with a curve
diff --git a/elliptic-curve/src/ops.rs b/elliptic-curve/src/ops.rs
index 6e763c544..09f811d7e 100644
--- a/elliptic-curve/src/ops.rs
+++ b/elliptic-curve/src/ops.rs
@@ -10,14 +10,3 @@ pub trait Invert {
     /// Invert a field element.
     fn invert(&self) -> CtOption<Self::Output>;
 }
-
-/// Fixed-base scalar multiplication.
-///
-/// This trait is intended to be implemented on a point type.
-pub trait MulBase: Sized {
-    /// Scalar type
-    type Scalar;
-
-    /// Multiply scalar by the generator point for the elliptic curve
-    fn mul_base(scalar: &Self::Scalar) -> CtOption<Self>;
-}
diff --git a/elliptic-curve/src/point.rs b/elliptic-curve/src/point.rs
new file mode 100644
index 000000000..67a810a93
--- /dev/null
+++ b/elliptic-curve/src/point.rs
@@ -0,0 +1,7 @@
+//! Traits for elliptic curve points
+
+/// Obtain the generator point.
+pub trait Generator {
+    /// Get the generator point for this elliptic curve
+    fn generator() -> Self;
+}
diff --git a/elliptic-curve/src/scalar.rs b/elliptic-curve/src/scalar.rs
new file mode 100644
index 000000000..0237e8cf3
--- /dev/null
+++ b/elliptic-curve/src/scalar.rs
@@ -0,0 +1,51 @@
+//! Scalar types
+
+use crate::{Arithmetic, Curve};
+use subtle::{ConstantTimeEq, CtOption};
+
+#[cfg(feature = "zeroize")]
+use zeroize::Zeroize;
+
+/// Non-zero scalar type.
+///
+/// This type ensures that its value is not zero, ala `core::num::NonZero*`.
+/// To do this, the generic `S` type must impl both `Default` and
+/// `ConstantTimeEq`, with the requirement that `S::default()` returns 0.
+///
+/// In the context of ECC, it's useful for ensuring that scalar multiplication
+/// cannot result in the point at infinity.
+pub struct NonZeroScalar<C: Curve + Arithmetic> {
+    scalar: C::Scalar,
+}
+
+impl<C> NonZeroScalar<C>
+where
+    C: Curve + Arithmetic,
+{
+    /// Create a [`NonZeroScalar`] from a scalar, performing a constant-time
+    /// check that it's non-zero.
+    pub fn new(scalar: C::Scalar) -> CtOption<Self> {
+        let is_zero = scalar.ct_eq(&C::Scalar::default());
+        CtOption::new(Self { scalar }, !is_zero)
+    }
+}
+
+impl<C> AsRef<C::Scalar> for NonZeroScalar<C>
+where
+    C: Curve + Arithmetic,
+{
+    fn as_ref(&self) -> &C::Scalar {
+        &self.scalar
+    }
+}
+
+#[cfg(feature = "zeroize")]
+impl<C> Zeroize for NonZeroScalar<C>
+where
+    C: Curve + Arithmetic,
+    C::Scalar: Zeroize,
+{
+    fn zeroize(&mut self) {
+        self.scalar.zeroize();
+    }
+}
diff --git a/elliptic-curve/src/weierstrass/public_key.rs b/elliptic-curve/src/weierstrass/public_key.rs
index 1092912ef..796a0b866 100644
--- a/elliptic-curve/src/weierstrass/public_key.rs
+++ b/elliptic-curve/src/weierstrass/public_key.rs
@@ -5,9 +5,14 @@ use super::{
     point::{CompressedPoint, CompressedPointSize, UncompressedPoint, UncompressedPointSize},
     Curve,
 };
-use crate::{ops::MulBase, secret_key::FromSecretKey, Arithmetic, Error, SecretKey};
-use core::fmt::{self, Debug};
-use core::ops::Add;
+use crate::{
+    point::Generator, scalar::NonZeroScalar, secret_key::FromSecretKey, Arithmetic, Error,
+    SecretKey,
+};
+use core::{
+    fmt::{self, Debug},
+    ops::{Add, Mul},
+};
 use generic_array::{
     typenum::{Unsigned, U1},
     ArrayLength, GenericArray,
@@ -107,6 +112,7 @@ where
 impl<C> PublicKey<C>
 where
     C: Curve + Arithmetic,
+    C::AffinePoint: Mul<NonZeroScalar<C>, Output = C::AffinePoint>,
     C::ElementSize: Add<U1>,
     <C::ElementSize as Add>::Output: Add<U1>,
     CompressedPoint<C>: From<C::AffinePoint>,
@@ -118,14 +124,13 @@ where
     ///
     /// The `compress` flag requests point compression.
     pub fn from_secret_key(secret_key: &SecretKey<C>, compress: bool) -> Result<Self, Error> {
-        let ct_option =
-            C::Scalar::from_secret_key(&secret_key).and_then(|s| C::AffinePoint::mul_base(&s));
+        let ct_option = C::Scalar::from_secret_key(&secret_key).and_then(NonZeroScalar::new);
 
         if ct_option.is_none().into() {
             return Err(Error);
         }
 
-        let affine_point = ct_option.unwrap();
+        let affine_point = C::AffinePoint::generator() * ct_option.unwrap();
 
         if compress {
             Ok(PublicKey::Compressed(affine_point.into()))