diff --git a/Cargo.lock b/Cargo.lock
index 6f524ee64..e40a12328 100644
--- a/Cargo.lock
+++ b/Cargo.lock
@@ -502,8 +502,7 @@ checksum = "48c757948c5ede0e46177b7add2e67155f70e33c07fea8284df6576da70b3719"
 [[package]]
 name = "elliptic-curve"
 version = "0.14.0-rc.32"
-source = "registry+https://github.com/rust-lang/crates.io-index"
-checksum = "cda94f31325c4275e9706adecbb6f0650dee2f904c915a98e3d81adaaaa757aa"
+source = "git+http://github.com/RustCrypto/traits.git#95838a01bd186da138211685a338cac7185f950e"
 dependencies = [
  "base16ct",
  "crypto-bigint",
@@ -1473,7 +1472,7 @@ source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "32497e9a4c7b38532efcdebeef879707aa9f794296a4f0244f6f69e9bc8574bd"
 dependencies = [
  "fastrand",
- "getrandom 0.4.2",
+ "getrandom 0.3.4",
  "once_cell",
  "rustix",
  "windows-sys",
diff --git a/Cargo.toml b/Cargo.toml
index dfd952412..e8808e0c0 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -27,4 +27,5 @@ hash2curve = { path = "hash2curve" }
 primefield = { path = "primefield" }
 primeorder = { path = "primeorder" }
 
+elliptic-curve = { git = "http://github.com/RustCrypto/traits.git" }
 rustcrypto-group = { git = "https://github.com/RustCrypto/group" }
diff --git a/p256/src/arithmetic.rs b/p256/src/arithmetic.rs
index c02987ad4..b045539a1 100644
--- a/p256/src/arithmetic.rs
+++ b/p256/src/arithmetic.rs
@@ -68,4 +68,13 @@ impl PrimeCurveParams for NistP256 {
     fn mul_by_generator_vartime(k: &Scalar) -> ProjectivePoint {
         tables::BASEPOINT_TABLE_VARTIME.mul(k)
     }
+
+    #[cfg(all(feature = "alloc", feature = "precomputed-tables"))]
+    fn mul_by_generator_and_mul_add_vartime(
+        a: &Scalar,
+        b_scalar: &Scalar,
+        b_point: &ProjectivePoint,
+    ) -> ProjectivePoint {
+        tables::BASEPOINT_TABLE_VARTIME.lincomb(a, &[(*b_point, *b_scalar)])
+    }
 }
diff --git a/p384/src/arithmetic.rs b/p384/src/arithmetic.rs
index 1554bfb80..ccc1cdf3c 100644
--- a/p384/src/arithmetic.rs
+++ b/p384/src/arithmetic.rs
@@ -72,4 +72,13 @@ impl PrimeCurveParams for NistP384 {
     fn mul_by_generator_vartime(k: &Scalar) -> ProjectivePoint {
         tables::BASEPOINT_TABLE_VARTIME.mul(k)
     }
+
+    #[cfg(all(feature = "alloc", feature = "precomputed-tables"))]
+    fn mul_by_generator_and_mul_add_vartime(
+        a: &Scalar,
+        b_scalar: &Scalar,
+        b_point: &ProjectivePoint,
+    ) -> ProjectivePoint {
+        tables::BASEPOINT_TABLE_VARTIME.lincomb(a, &[(*b_point, *b_scalar)])
+    }
 }
diff --git a/p521/src/arithmetic.rs b/p521/src/arithmetic.rs
index 71fa8aa39..57e384e1b 100644
--- a/p521/src/arithmetic.rs
+++ b/p521/src/arithmetic.rs
@@ -77,4 +77,13 @@ impl PrimeCurveParams for NistP521 {
     fn mul_by_generator_vartime(k: &Scalar) -> ProjectivePoint {
         tables::BASEPOINT_TABLE_VARTIME.mul(k)
     }
+
+    #[cfg(all(feature = "alloc", feature = "precomputed-tables"))]
+    fn mul_by_generator_and_mul_add_vartime(
+        a: &Scalar,
+        b_scalar: &Scalar,
+        b_point: &ProjectivePoint,
+    ) -> ProjectivePoint {
+        tables::BASEPOINT_TABLE_VARTIME.lincomb(a, &[(*b_point, *b_scalar)])
+    }
 }
diff --git a/primeorder/src/lib.rs b/primeorder/src/lib.rs
index 55c5344f5..dcbe89a10 100644
--- a/primeorder/src/lib.rs
+++ b/primeorder/src/lib.rs
@@ -29,7 +29,7 @@ pub use elliptic_curve::{
 
 use elliptic_curve::{
     CurveArithmetic, Generate,
-    ops::{Invert, MulVartime},
+    ops::{Invert, LinearCombination, MulVartime},
     subtle::CtOption,
 };
 
@@ -73,6 +73,19 @@ pub trait PrimeCurveParams:
     fn mul_by_generator_vartime(k: &Scalar<Self>) -> ProjectivePoint<Self> {
         ProjectivePoint::GENERATOR.mul_vartime(k)
     }
+
+    /// Multiply `a` by the generator of the prime-order subgroup, adding the result to the point
+    /// `P` multiplied by the scalar `b`, i.e. compute `aG + bP`.
+    fn mul_by_generator_and_mul_add_vartime(
+        a: &Scalar<Self>,
+        b_scalar: &Scalar<Self>,
+        b_point: &ProjectivePoint<Self>,
+    ) -> ProjectivePoint<Self> {
+        ProjectivePoint::<Self>::lincomb_vartime(&[
+            (ProjectivePoint::GENERATOR, *a),
+            (*b_point, *b_scalar),
+        ])
+    }
 }
 
 /// Trait which allows curves to specify a variable-time basepoint table.
diff --git a/primeorder/src/projective.rs b/primeorder/src/projective.rs
index 0c169b71b..0baacc57e 100644
--- a/primeorder/src/projective.rs
+++ b/primeorder/src/projective.rs
@@ -970,16 +970,13 @@ where
         C::mul_by_generator_vartime(scalar)
     }
 
-    // When we're guaranteed *not* to have basepoint tables available (because they need `alloc`)
-    // use linear combinations for this computation, but they're slower when they are available
-    // TODO(tarcieri): `WnafBase::multiscalar_mul` w\ basepoint table when `alloc` *is* available
-    #[cfg(not(feature = "alloc"))]
+    #[inline]
     fn mul_by_generator_and_mul_add_vartime(
         a: &Self::Scalar,
         b_scalar: &Self::Scalar,
         b_point: &Self,
     ) -> Self {
-        Self::lincomb(&[(Self::GENERATOR, *a), (*b_point, *b_scalar)])
+        C::mul_by_generator_and_mul_add_vartime(a, b_scalar, b_point)
     }
 }
 
diff --git a/sm2/src/arithmetic.rs b/sm2/src/arithmetic.rs
index 0a75d2d9b..da34f25d9 100644
--- a/sm2/src/arithmetic.rs
+++ b/sm2/src/arithmetic.rs
@@ -68,4 +68,13 @@ impl PrimeCurveParams for Sm2 {
     fn mul_by_generator_vartime(k: &Scalar) -> ProjectivePoint {
         tables::BASEPOINT_TABLE_VARTIME.mul(k)
     }
+
+    #[cfg(all(feature = "alloc", feature = "precomputed-tables"))]
+    fn mul_by_generator_and_mul_add_vartime(
+        a: &Scalar,
+        b_scalar: &Scalar,
+        b_point: &ProjectivePoint,
+    ) -> ProjectivePoint {
+        tables::BASEPOINT_TABLE_VARTIME.lincomb(a, &[(*b_point, *b_scalar)])
+    }
 }