diff --git a/src/edwards.rs b/src/edwards.rs
new file mode 100644
index 0000000..11dd05c
--- /dev/null
+++ b/src/edwards.rs
@@ -0,0 +1,153 @@
+//! Point arithmetic for the doppio curve.
+//!
+//! This implements the equations for point arithmetic from
+//! HWCD: https://eprint.iacr.org/2008/522.pdf
+
+#![allow(non_snake_case)]
+
+use core::ops::{Add, Neg, Sub};
+
+use crate::field::{FieldElement, EDWARDS_D};
+use crate::Ristretto255Scalar;
+
+// ------------------------------------------------------------------------
+// Internal point representations
+// ------------------------------------------------------------------------
+
+/// An `EdwardsPoint` represents a point on the Edwards form of the Doppio curve.
+#[derive(Copy, Clone)]
+#[allow(missing_docs)]
+pub struct EdwardsPoint {
+    pub(crate) X: FieldElement,
+    pub(crate) Y: FieldElement,
+    pub(crate) Z: FieldElement,
+    pub(crate) T: FieldElement,
+}
+
+// ------------------------------------------------------------------------
+// Constructors
+// ------------------------------------------------------------------------
+
+impl Default for EdwardsPoint {
+    fn default() -> EdwardsPoint {
+        EdwardsPoint {
+            X: FieldElement::zero(),
+            Y: FieldElement::one(),
+            Z: FieldElement::one(),
+            T: FieldElement::zero(),
+        }
+    }
+}
+
+// ------------------------------------------------------------------------
+// Doubling
+// ------------------------------------------------------------------------
+
+impl EdwardsPoint {
+    /// Add this point to itself.
+    // TODO: add a test that A.double() = A * A
+    fn double(&self) -> EdwardsPoint {
+        let two: FieldElement = Ristretto255Scalar::from(2u8).into();
+
+        let A = self.X * self.X;
+        let B = self.Y * self.Y;
+        // Is it cheaper to add Z*Z + Z*Z or multiply 2*Z*Z?
+        let C = two * self.Z * self.Z;
+        // a = -1
+        let D = -A;
+        let E = (self.X + self.Y) * (self.X + self.Y) - A - B;
+        let G = D + B;
+        let F = G - C;
+        let H = D - B;
+
+        EdwardsPoint {
+            X: E * F,
+            Y: G * H,
+            Z: F * G,
+            T: E * H,
+        }
+    }
+}
+
+// ------------------------------------------------------------------------
+// Addition and Subtraction
+// ------------------------------------------------------------------------
+
+impl<'a, 'b> Add<&'b EdwardsPoint> for &'a EdwardsPoint {
+    type Output = EdwardsPoint;
+
+    fn add(self, other: &'b EdwardsPoint) -> EdwardsPoint {
+        // k = 2d'. d' = -d/a and a = -1, so k = 2d.
+        let two: FieldElement = Ristretto255Scalar::from(2u8).into();
+        let k = EDWARDS_D * two;
+
+        let A = (self.Y - self.X) * (other.Y - other.X);
+        let B = (self.Y + self.X) * (other.Y + other.X);
+        let C = k * self.T * other.T;
+        let D = two * self.Z * other.Z;
+        let E = B - A;
+        let F = D - C;
+        let G = D + C;
+        let H = B + A;
+
+        EdwardsPoint {
+            X: E * F,
+            Y: G * H,
+            Z: F * G,
+            T: E * H,
+        }
+    }
+}
+
+impl<'a, 'b> Sub<&'b EdwardsPoint> for &'a EdwardsPoint {
+    type Output = EdwardsPoint;
+
+    // TODO: add a test that A - B = A + -B
+    fn sub(self, other: &'b EdwardsPoint) -> EdwardsPoint {
+        // The same equation as addition, except other.X and other.T are negated.
+        // k = 2d'. d' = -d/a and a = -1, so k = 2d.
+        let two = Ristretto255Scalar::from(2u8).into();
+        let k = EDWARDS_D * two;
+
+        let A = (self.Y - self.X) * (other.Y + other.X);
+        let B = (self.Y + self.X) * (other.Y - other.X);
+        let C = k * self.T * other.T;
+        let D = two * self.Z * other.Z;
+        let E = B - A;
+        let F = D + C;
+        let G = D - C;
+        let H = B + A;
+
+        EdwardsPoint {
+            X: E * F,
+            Y: G * H,
+            Z: F * G,
+            T: E * H,
+        }
+    }
+}
+
+// ------------------------------------------------------------------------
+// Negation
+// ------------------------------------------------------------------------
+
+impl<'a> Neg for &'a EdwardsPoint {
+    type Output = EdwardsPoint;
+
+    fn neg(self) -> EdwardsPoint {
+        EdwardsPoint {
+            X: -self.X,
+            Y: self.Y,
+            Z: self.Z,
+            T: -self.T,
+        }
+    }
+}
+
+impl Neg for EdwardsPoint {
+    type Output = EdwardsPoint;
+
+    fn neg(self) -> EdwardsPoint {
+        -&self
+    }
+}
diff --git a/src/field.rs b/src/field.rs
index a43f0c3..288b1f5 100644
--- a/src/field.rs
+++ b/src/field.rs
@@ -8,10 +8,14 @@
 //! implementation contributed to `curve25519-dalek` by Andrew Moon.
 
 use std::default::Default;
-use std::ops::{Add, Mul, Sub};
+use std::ops::{Add, Mul, Neg, Sub};
 
 use crate::Ristretto255Scalar;
 
+/// Edwards `d` value, equal to `-86649/86650 mod p`.
+/// TODO: actually generate the right value here
+pub(crate) const EDWARDS_D: FieldElement = FieldElement([0; 5]);
+
 /// A field element modulo \\(2\^{252} +
 /// 27742317777372353535851937790883648493\\), the ground field for
 /// the doppio curve and the scalar field for the ristretto255 group.
@@ -182,6 +186,14 @@ impl Into<Ristretto255Scalar> for FieldElement {
     }
 }
 
+impl Neg for FieldElement {
+    type Output = Self;
+
+    fn neg(self) -> Self {
+        unimplemented!()
+    }
+}
+
 impl FieldElement {
     pub fn zero() -> FieldElement {
         FieldElement([0; 5])
diff --git a/src/lib.rs b/src/lib.rs
index 937ea92..76567a8 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -1,5 +1,6 @@
 pub type Ristretto255Scalar = curve25519_dalek::scalar::Scalar;
 
+mod edwards;
 mod field;
 
 #[cfg(test)]