diff --git a/Cargo.toml b/Cargo.toml
index 1fb2e67c..828299e8 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -32,7 +32,7 @@ subtle = "2.3"
 
 [dependencies.halo2]
 git = "https://github.com/zcash/halo2.git"
-rev = "6acacf1aca12f34fc311aa59056e40adc0e6d8bd"
+rev = "cae6f6af725cf1f5bc94e126a0b41e9ac602a302"
 
 [dependencies.pasta_curves]
 git = "https://github.com/zcash/pasta_curves.git"
diff --git a/src/circuit/gadget/ecc.rs b/src/circuit/gadget/ecc.rs
index 049c6b9e..31d252e3 100644
--- a/src/circuit/gadget/ecc.rs
+++ b/src/circuit/gadget/ecc.rs
@@ -1,140 +1,378 @@
 //! Gadgets for elliptic curve operations.
 
-use std::fmt;
+use ff::Field;
+use std::fmt::Debug;
 
 use halo2::{
-    arithmetic::CurveAffine,
+    arithmetic::{CurveAffine, FieldExt},
     circuit::{Chip, Layouter},
     plonk::Error,
 };
 
-/// Trait allowing circuit's fixed points to be enumerated.
-pub trait FixedPoints<C: CurveAffine>: Clone + fmt::Debug {}
-
 /// The set of circuit instructions required to use the ECC gadgets.
-pub trait EccInstructions<C: CurveAffine>: Chip<Field = C::Base> {
-    /// Variable representing an element of the elliptic curve's scalar field.
-    type Scalar: Clone + fmt::Debug;
+pub trait EccInstructions<C: CurveAffine>: Chip<C::Base> {
+    /// Variable representing an element of the elliptic curve's base field, that
+    /// is used as a scalar in variable-base scalar mul.
+    ///
+    /// It is not true in general that a scalar field element fits in a curve's
+    /// base field, and in particular it is untrue for the Pallas curve, whose
+    /// scalar field `Fq` is larger than its base field `Fp`.
+    ///
+    /// However, the only use of variable-base scalar mul in the Orchard protocol
+    /// is in deriving diversified addresses `[ivk] g_d`,  and `ivk` is guaranteed
+    /// to be in the base field of the curve. (See non-normative notes in
+    /// https://zips.z.cash/protocol/nu5.pdf#orchardkeycomponents.)
+    type ScalarVar: Clone + Debug;
+    /// Variable representing a full-width element of the elliptic curve's
+    /// scalar field, to be used for fixed-base scalar mul.
+    type ScalarFixed: Clone + Debug;
+    /// Variable representing a signed short element of the elliptic curve's
+    /// scalar field, to be used for fixed-base scalar mul.
+    ///
+    /// A `ScalarFixedShort` must be in the range [-(2^64 - 1), 2^64 - 1].
+    type ScalarFixedShort: Clone + Debug;
     /// Variable representing an elliptic curve point.
-    type Point: Clone + fmt::Debug;
+    type Point: Clone + Debug;
+    /// Variable representing the affine short Weierstrass x-coordinate of an
+    /// elliptic curve point.
+    type X: Clone + Debug;
     /// Variable representing the set of fixed bases in the circuit.
-    type FixedPoints: FixedPoints<C>;
+    type FixedPoints: Clone + Debug;
+    /// Variable representing the set of fixed bases to be used in scalar
+    /// multiplication with a short signed exponent.
+    type FixedPointsShort: Clone + Debug;
     /// Variable representing a fixed elliptic curve point (constant in the circuit).
-    type FixedPoint: Clone + fmt::Debug;
+    type FixedPoint: Clone + Debug;
+    /// Variable representing a fixed elliptic curve point (constant in the circuit)
+    /// to be used in scalar multiplication with a short signed exponent.
+    type FixedPointShort: Clone + Debug;
 
-    /// Witnesses the given scalar as a private input to the circuit.
-    fn witness_scalar(
-        layouter: &mut impl Layouter<Self>,
+    /// Witnesses the given base field element as a private input to the circuit
+    /// for variable-base scalar mul.
+    fn witness_scalar_var(
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
+        value: Option<C::Base>,
+    ) -> Result<Self::ScalarVar, Error>;
+
+    /// Witnesses the given full-width scalar as a private input to the circuit
+    /// for fixed-base scalar mul.
+    fn witness_scalar_fixed(
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
         value: Option<C::Scalar>,
-    ) -> Result<Self::Scalar, Error>;
+    ) -> Result<Self::ScalarFixed, Error>;
+
+    /// Witnesses the given signed short scalar as a private input to the circuit
+    /// for fixed-base scalar mul.
+    fn witness_scalar_fixed_short(
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
+        value: Option<C::Scalar>,
+    ) -> Result<Self::ScalarFixedShort, Error>;
 
     /// Witnesses the given point as a private input to the circuit.
     fn witness_point(
-        layouter: &mut impl Layouter<Self>,
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
         value: Option<C>,
     ) -> Result<Self::Point, Error>;
 
-    /// Gets a fixed point into the circuit.
-    fn get_fixed(
-        layouter: &mut impl Layouter<Self>,
-        fixed_points: Self::FixedPoints,
-    ) -> Result<Self::FixedPoint, Error>;
+    /// Extracts the x-coordinate of a point.
+    fn extract_p(point: &Self::Point) -> &Self::X;
 
-    /// Performs point addition, returning `a + b`.
+    /// Returns a fixed point that had been previously loaded into the circuit.
+    /// The pre-loaded cells are used to set up equality constraints in other
+    /// parts of the circuit where the fixed base is used.
+    fn get_fixed(&self, fixed_points: Self::FixedPoints) -> Result<Self::FixedPoint, Error>;
+
+    /// Returns a fixed point to be used in scalar multiplication with a signed
+    /// short exponent.
+    fn get_fixed_short(
+        &self,
+        fixed_points: Self::FixedPointsShort,
+    ) -> Result<Self::FixedPointShort, Error>;
+
+    /// Performs incomplete point addition, returning `a + b`.
+    ///
+    /// This returns an error in exceptional cases.
+    fn add_incomplete(
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
+        a: &Self::Point,
+        b: &Self::Point,
+    ) -> Result<Self::Point, Error>;
+
+    /// Performs complete point addition, returning `a + b`.
     fn add(
-        layouter: &mut impl Layouter<Self>,
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
         a: &Self::Point,
         b: &Self::Point,
     ) -> Result<Self::Point, Error>;
 
     /// Performs point doubling, returning `[2] a`.
-    fn double(layouter: &mut impl Layouter<Self>, a: &Self::Point) -> Result<Self::Point, Error>;
+    fn double(
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
+        a: &Self::Point,
+    ) -> Result<Self::Point, Error>;
 
     /// Performs variable-base scalar multiplication, returning `[scalar] base`.
     fn mul(
-        layouter: &mut impl Layouter<Self>,
-        scalar: &Self::Scalar,
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
+        scalar: &Self::ScalarVar,
         base: &Self::Point,
     ) -> Result<Self::Point, Error>;
 
-    /// Performs fixed-base scalar multiplication, returning `[scalar] base`.
+    /// Performs fixed-base scalar multiplication using a full-width scalar, returning `[scalar] base`.
     fn mul_fixed(
-        layouter: &mut impl Layouter<Self>,
-        scalar: &Self::Scalar,
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
+        scalar: &Self::ScalarFixed,
         base: &Self::FixedPoint,
     ) -> Result<Self::Point, Error>;
+
+    /// Performs fixed-base scalar multiplication using a short signed scalar, returning `[scalar] base`.
+    fn mul_fixed_short(
+        &self,
+        layouter: &mut impl Layouter<C::Base>,
+        scalar: &Self::ScalarFixedShort,
+        base: &Self::FixedPointShort,
+    ) -> Result<Self::Point, Error>;
 }
 
-/// An element of the given elliptic curve's scalar field.
+/// An element of the given elliptic curve's base field, that is used as a scalar
+/// in variable-base scalar mul.
+///
+/// It is not true in general that a scalar field element fits in a curve's
+/// base field, and in particular it is untrue for the Pallas curve, whose
+/// scalar field `Fq` is larger than its base field `Fp`.
+///
+/// However, the only use of variable-base scalar mul in the Orchard protocol
+/// is in deriving diversified addresses `[ivk] g_d`,  and `ivk` is guaranteed
+/// to be in the base field of the curve. (See non-normative notes in
+/// https://zips.z.cash/protocol/nu5.pdf#orchardkeycomponents.)
 #[derive(Debug)]
-pub struct Scalar<C: CurveAffine, EccChip: EccInstructions<C>> {
-    inner: EccChip::Scalar,
+pub struct ScalarVar<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> {
+    chip: EccChip,
+    inner: EccChip::ScalarVar,
 }
 
-impl<C: CurveAffine, EccChip: EccInstructions<C>> Scalar<C, EccChip> {
-    /// Constructs a new point with the given value.
+impl<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> ScalarVar<C, EccChip> {
+    /// Constructs a new ScalarVar with the given value.
     pub fn new(
-        mut layouter: impl Layouter<EccChip>,
+        chip: EccChip,
+        mut layouter: impl Layouter<C::Base>,
+        value: Option<C::Base>,
+    ) -> Result<Self, Error> {
+        chip.witness_scalar_var(&mut layouter, value)
+            .map(|inner| ScalarVar { chip, inner })
+    }
+}
+
+/// A full-width element of the given elliptic curve's scalar field, to be used for fixed-base scalar mul.
+#[derive(Debug)]
+pub struct ScalarFixed<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> {
+    chip: EccChip,
+    inner: EccChip::ScalarFixed,
+}
+
+impl<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> ScalarFixed<C, EccChip> {
+    /// Constructs a new ScalarFixed with the given value.
+    pub fn new(
+        chip: EccChip,
+        mut layouter: impl Layouter<C::Base>,
         value: Option<C::Scalar>,
     ) -> Result<Self, Error> {
-        EccChip::witness_scalar(&mut layouter, value).map(|inner| Scalar { inner })
+        chip.witness_scalar_fixed(&mut layouter, value)
+            .map(|inner| ScalarFixed { chip, inner })
+    }
+}
+
+/// A signed short element of the given elliptic curve's scalar field, to be used for fixed-base scalar mul.
+#[derive(Debug)]
+pub struct ScalarFixedShort<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> {
+    chip: EccChip,
+    inner: EccChip::ScalarFixedShort,
+}
+
+impl<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq>
+    ScalarFixedShort<C, EccChip>
+{
+    /// Constructs a new ScalarFixedShort with the given value.
+    ///
+    /// # Panics
+    ///
+    /// The short scalar must be in the range [-(2^64 - 1), (2^64 - 1)].
+    pub fn new(
+        chip: EccChip,
+        mut layouter: impl Layouter<C::Base>,
+        value: Option<C::Scalar>,
+    ) -> Result<Self, Error> {
+        // Check that the scalar is in the range [-(2^64 - 1), (2^64 - 1)]
+        if let Some(value) = value {
+            let mut sign = C::Scalar::one();
+
+            // T = (p-1) / 2
+            let t = (C::Scalar::zero() - C::Scalar::one()) * C::Scalar::TWO_INV;
+
+            if value > t {
+                sign = -sign;
+            }
+            let magnitude = value * sign;
+            assert!(magnitude < C::Scalar::from_u128(1 << 64));
+        }
+
+        chip.witness_scalar_fixed_short(&mut layouter, value)
+            .map(|inner| ScalarFixedShort { chip, inner })
     }
 }
 
 /// An elliptic curve point over the given curve.
 #[derive(Debug)]
-pub struct Point<C: CurveAffine, EccChip: EccInstructions<C>> {
+pub struct Point<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> {
+    chip: EccChip,
     inner: EccChip::Point,
 }
 
-impl<C: CurveAffine, EccChip: EccInstructions<C>> Point<C, EccChip> {
+impl<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> Point<C, EccChip> {
     /// Constructs a new point with the given value.
-    pub fn new(mut layouter: impl Layouter<EccChip>, value: Option<C>) -> Result<Self, Error> {
-        EccChip::witness_point(&mut layouter, value).map(|inner| Point { inner })
+    pub fn new(
+        chip: EccChip,
+        mut layouter: impl Layouter<C::Base>,
+        value: Option<C>,
+    ) -> Result<Self, Error> {
+        let point = chip.witness_point(&mut layouter, value);
+        point.map(|inner| Point { chip, inner })
     }
 
-    /// Returns `self + other`.
-    pub fn add(&self, mut layouter: impl Layouter<EccChip>, other: &Self) -> Result<Self, Error> {
-        EccChip::add(&mut layouter, &self.inner, &other.inner).map(|inner| Point { inner })
+    /// Extracts the x-coordinate of a point.
+    pub fn extract_p(&self) -> X<C, EccChip> {
+        X::from_inner(self.chip.clone(), EccChip::extract_p(&self.inner).clone())
     }
 
-    /// Returns `[2] self`.
-    pub fn double(&self, mut layouter: impl Layouter<EccChip>) -> Result<Self, Error> {
-        EccChip::double(&mut layouter, &self.inner).map(|inner| Point { inner })
+    /// Wraps the given point (obtained directly from an instruction) in a gadget.
+    pub fn from_inner(chip: EccChip, inner: EccChip::Point) -> Self {
+        Point { chip, inner }
+    }
+
+    /// Returns `self + other` using complete addition.
+    pub fn add(&self, mut layouter: impl Layouter<C::Base>, other: &Self) -> Result<Self, Error> {
+        assert_eq!(self.chip, other.chip);
+        self.chip
+            .add(&mut layouter, &self.inner, &other.inner)
+            .map(|inner| Point {
+                chip: self.chip.clone(),
+                inner,
+            })
+    }
+
+    /// Returns `self + other` using incomplete addition.
+    pub fn add_incomplete(
+        &self,
+        mut layouter: impl Layouter<C::Base>,
+        other: &Self,
+    ) -> Result<Self, Error> {
+        assert_eq!(self.chip, other.chip);
+        self.chip
+            .add_incomplete(&mut layouter, &self.inner, &other.inner)
+            .map(|inner| Point {
+                chip: self.chip.clone(),
+                inner,
+            })
     }
 
     /// Returns `[by] self`.
     pub fn mul(
         &self,
-        mut layouter: impl Layouter<EccChip>,
-        by: &Scalar<C, EccChip>,
+        mut layouter: impl Layouter<C::Base>,
+        by: &ScalarVar<C, EccChip>,
     ) -> Result<Self, Error> {
-        EccChip::mul(&mut layouter, &by.inner, &self.inner).map(|inner| Point { inner })
+        assert_eq!(self.chip, by.chip);
+        self.chip
+            .mul(&mut layouter, &by.inner, &self.inner)
+            .map(|inner| Point {
+                chip: self.chip.clone(),
+                inner,
+            })
     }
 }
 
-/// A constant elliptic curve point over the given curve, for which scalar multiplication
-/// is more efficient.
+/// The affine short Weierstrass x-coordinate of an elliptic curve point over the
+/// given curve.
 #[derive(Debug)]
-pub struct FixedPoint<C: CurveAffine, EccChip: EccInstructions<C>> {
+pub struct X<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> {
+    chip: EccChip,
+    inner: EccChip::X,
+}
+
+impl<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> X<C, EccChip> {
+    /// Wraps the given x-coordinate (obtained directly from an instruction) in a gadget.
+    pub fn from_inner(chip: EccChip, inner: EccChip::X) -> Self {
+        X { chip, inner }
+    }
+}
+
+/// A constant elliptic curve point over the given curve, for which window tables have
+/// been provided to make scalar multiplication more efficient.
+#[derive(Clone, Debug)]
+pub struct FixedPoint<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> {
+    chip: EccChip,
     inner: EccChip::FixedPoint,
 }
 
-impl<C: CurveAffine, EccChip: EccInstructions<C>> FixedPoint<C, EccChip> {
+impl<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> FixedPoint<C, EccChip> {
     /// Gets a reference to the specified fixed point in the circuit.
-    pub fn get(
-        mut layouter: impl Layouter<EccChip>,
-        point: EccChip::FixedPoints,
-    ) -> Result<Self, Error> {
-        EccChip::get_fixed(&mut layouter, point).map(|inner| FixedPoint { inner })
+    pub fn get(chip: EccChip, point: EccChip::FixedPoints) -> Result<Self, Error> {
+        chip.get_fixed(point)
+            .map(|inner| FixedPoint { chip, inner })
     }
 
     /// Returns `[by] self`.
     pub fn mul(
         &self,
-        mut layouter: impl Layouter<EccChip>,
-        by: &Scalar<C, EccChip>,
+        mut layouter: impl Layouter<C::Base>,
+        by: &ScalarFixed<C, EccChip>,
     ) -> Result<Point<C, EccChip>, Error> {
-        EccChip::mul_fixed(&mut layouter, &by.inner, &self.inner).map(|inner| Point { inner })
+        assert_eq!(self.chip, by.chip);
+        self.chip
+            .mul_fixed(&mut layouter, &by.inner, &self.inner)
+            .map(|inner| Point {
+                chip: self.chip.clone(),
+                inner,
+            })
+    }
+}
+
+/// A constant elliptic curve point over the given curve, used in scalar multiplication
+/// with a short signed exponent
+#[derive(Clone, Debug)]
+pub struct FixedPointShort<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> {
+    chip: EccChip,
+    inner: EccChip::FixedPointShort,
+}
+
+impl<C: CurveAffine, EccChip: EccInstructions<C> + Clone + Debug + Eq> FixedPointShort<C, EccChip> {
+    /// Gets a reference to the specified fixed point in the circuit.
+    pub fn get(chip: EccChip, point: EccChip::FixedPointsShort) -> Result<Self, Error> {
+        chip.get_fixed_short(point)
+            .map(|inner| FixedPointShort { chip, inner })
+    }
+
+    /// Returns `[by] self`.
+    pub fn mul(
+        &self,
+        mut layouter: impl Layouter<C::Base>,
+        by: &ScalarFixedShort<C, EccChip>,
+    ) -> Result<Point<C, EccChip>, Error> {
+        assert_eq!(self.chip, by.chip);
+        self.chip
+            .mul_fixed_short(&mut layouter, &by.inner, &self.inner)
+            .map(|inner| Point {
+                chip: self.chip.clone(),
+                inner,
+            })
     }
 }