diff --git a/src/curve.rs b/src/curve.rs
index 200dd19..7431d6e 100644
--- a/src/curve.rs
+++ b/src/curve.rs
@@ -80,6 +80,7 @@
 use core::fmt::Debug;
 use core::iter::Iterator;
 use core::ops::{Add, Sub, Neg, Index};
+use core::cmp::{PartialEq, Eq};
 
 use constants;
 use field::FieldElement;
@@ -168,6 +169,48 @@ impl CompressedEdwardsY {
     }
 }
 
+/// A point serialized using Mike Hamburg's Decaf scheme.
+///
+/// XXX think about how this API should work
+#[derive(Copy, Clone, Eq, PartialEq)]
+pub struct DecafPoint(pub [u8; 32]);
+
+impl DecafPoint {
+    /// View this `DecafPoint` as an array of bytes.
+    pub fn to_bytes(&self) -> [u8;32] {
+        self.0
+    }
+
+    /// Attempt to decompress to an `ExtendedPoint`.
+    pub fn decompress(&self) -> Option<ExtendedPoint> {
+        // XXX should decoding be CT ?
+        // XXX should reject unless s = |s|
+        // XXX need to check that xy is nonnegative and reject otherwise
+        let s = FieldElement::from_bytes(&self.0);
+        let ss = s.square();
+        let X = &s + &s;                    // X = 2s
+        let Z = &FieldElement::one() - &ss; // Z = 1+as^2
+        let u = &(&Z * &Z) - &(&constants::d4 * &ss); // u = Z^2 - 4ds^2
+        let uss = &u * &ss;
+        let mut v = match uss.invsqrt() {
+            Some(v) => v,
+            None => return None,
+        };
+        // Now v = 1/sqrt(us^2) if us^2 is a nonzero square, 0 if us^2 is zero.
+        let uv = &v * &u;
+        if uv.is_negative_decaf() == 1u8 {
+            v.negate();
+        }
+        let mut two_minus_Z = -&Z; two_minus_Z[0] += 2;
+        let mut w = &v * &(&s * &two_minus_Z);
+        w.conditional_assign(&FieldElement::one(), s.is_zero());
+        let Y = &w * &Z;
+        let T = &w * &X;
+
+        Some(ExtendedPoint{ X: X, Y: Y, Z: Z, T: T })
+    }
+}
+
 // ------------------------------------------------------------------------
 // Internal point representations
 // ------------------------------------------------------------------------
@@ -425,6 +468,75 @@ impl ExtendedPoint {
         self.to_projective().compress()
     }
 
+    /// Compress in Decaf format.
+    pub fn compress_decaf(&self) -> DecafPoint {
+        // Q: Do we want to encode twisted or untwisted?
+        //
+        // Notes: 
+        // Recall that the twisted Edwards curve E_{a,d} is of the form
+        //
+        //     ax^2 + y^2 = 1 + dx^2y^2. 
+        //
+        // Internally, we operate on the curve with a = -1, d =
+        // -121665/121666, a.k.a., the twist.  But maybe we would like
+        // to use Decaf on the untwisted curve with a = 1, d =
+        // 121665/121666.  (why? interop?)
+        //
+        // Fix i, a square root of -1 (mod p).
+        //
+        // The map x -> ix is an isomorphism from E_{a,d} to E_{-a,-d}. 
+        // Its inverse is x -> -ix.
+        // let untwisted_X = &self.X * &constants::MSQRT_M1;
+        // etc.
+
+        // Step 0: pre-rotation, needed for Decaf with E[8] = Z/8
+
+        let mut X = self.X;
+        let mut Y = self.Y;
+        let mut XY = self.T;
+
+        // If y nonzero and xy nonnegative, continue.
+        // Otherwise, add Q_6 = (i,0) = constants::EIGHT_TORSION[6]
+        // (x,y) + Q_6 = (iy,ix)
+        // (X:Y:Z:T) + Q_6 = (iY:iX:Z:-T)
+
+        // XXX it should be possible to avoid this inversion, but
+        // let's make sure the code is correct first
+        let xy = &XY * &self.Z.invert();
+        let is_neg_mask = 1u8 & !(Y.is_nonzero() & xy.is_nonnegative_decaf());
+        let iX = &X * &constants::SQRT_M1;
+        let iY = &Y * &constants::SQRT_M1;
+        X.conditional_assign(&iY, is_neg_mask);
+        Y.conditional_assign(&iX, is_neg_mask);
+        let minus_XY = -&XY;
+        XY.conditional_assign(&minus_XY, is_neg_mask);
+
+        // Step 1: Compute r = 1/sqrt((a-d)(Z+Y)(Z-Y))
+        let Z_plus_Y  = &self.Z + &Y;
+        let Z_minus_Y = &self.Z - &Y;
+        let t = &constants::a_minus_d * &(&Z_plus_Y * &Z_minus_Y);
+        // t should always be square (why?)
+        // XXX is it safe to use option types here?
+        let mut r = t.invsqrt().unwrap();
+
+        // Step 2: Compute u = (a-d)r
+        let u = &constants::a_minus_d * &r;
+
+        // Step 3: Negate r if -2uZ is negative.
+        let uZ = &u * &self.Z;
+        let minus_r = -&r;
+        let m2uZ = -&(&uZ + &uZ);
+        let mask = m2uZ.is_negative_decaf();
+        r.conditional_assign(&minus_r, mask);
+
+        // Step 4: Compute s = |u(r(aZX - dYT)+Y)/a|
+        let minus_ZX = -&(&self.Z * &X);
+        let dYT = &constants::d * &(&Y * &XY);
+        let mut s = &u * &(&(&r * &(&minus_ZX - &dYT)) + &Y);
+        s.negate();
+        DecafPoint(s.abs_decaf().to_bytes())
+    }
+
     /// Dehomogenize to a PreComputedPoint.
     /// Mainly for testing.
     pub fn to_precomputed(&self) -> PreComputedPoint {
@@ -856,6 +968,12 @@ impl ExtendedPoint {
 // Debug traits
 // ------------------------------------------------------------------------
 
+impl Debug for DecafPoint {
+    fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
+        write!(f, "DecafPoint: {:?}", &self.0[..])
+    }
+}
+
 impl Debug for ExtendedPoint {
     fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
         write!(f, "ExtendedPoint(\n\tX: {:?},\n\tY: {:?},\n\tZ: {:?},\n\tT: {:?}\n)",
@@ -898,6 +1016,8 @@ impl Debug for CachedPoint {
 #[cfg(test)]
 mod test {
     use test::Bencher;
+    use rand::OsRng;
+
     use field::FieldElement;
     use scalar::Scalar;
     use subtle::CTAssignable;
@@ -1162,6 +1282,67 @@ mod test {
         assert!(ExtendedPoint::identity().is_identity());
     }
 
+    #[test]
+    fn test_decaf_decompress_id() {
+        let compressed_id = DecafPoint([0u8; 32]);
+        let id = compressed_id.decompress().unwrap();
+        // This should compress (as ed25519) to the following:
+        let mut bytes = [0u8; 32]; bytes[0] = 1;
+        assert_eq!(id.compress(), CompressedEdwardsY(bytes));
+    }
+
+    #[test]
+    fn test_decaf_compress_id() {
+        let id = ExtendedPoint::identity();
+        assert_eq!(id.compress_decaf(), DecafPoint([0u8; 32]));
+    }
+
+    #[test]
+    fn test_decaf_basepoint_roundtrip() {
+        // XXX fix up this test
+        let bp = BASE_CMPRSSD.decompress().unwrap();
+        let bp_decaf = bp.compress_decaf();
+        let bp_recaf = bp_decaf.decompress().unwrap();
+        let diff = &bp - &bp_recaf;
+        let diff2 = diff.double();
+        let diff4 = diff2.double();
+        //println!("bp {:?}",       bp);
+        //println!("bp_decaf {:?}", bp_decaf);
+        //println!("bp_recaf {:?}", bp_recaf);
+        //println!("diff {:?}", diff.compress());
+        //println!("diff2 {:?}", diff2.compress());
+        //println!("diff4 {:?}", diff4.compress());
+        assert_eq!(diff4.compress(), ExtendedPoint::identity().compress());
+    }
+
+    #[test]
+    fn test_decaf_four_torsion_basepoint() {
+        //println!("");
+        let bp = BASE_CMPRSSD.decompress().unwrap();
+        let bp_decaf = bp.compress_decaf();
+        //println!("orig, {:?}", bp.compress_decaf());
+        for i in (0..8).filter(|x| x % 2 == 0) {
+            let Q = &bp + &constants::EIGHT_TORSION[i];
+            //println!("{}, {:?}", i, Q.compress_decaf());
+            assert_eq!(Q.compress_decaf(), bp_decaf);
+        }
+    }
+
+    #[test]
+    fn test_decaf_four_torsion_random() {
+        //println!("");
+        let mut rng = OsRng::new().unwrap();
+        let s = Scalar::random(&mut rng);
+        let P = ExtendedPoint::basepoint_mult(&s);
+        let P_decaf = P.compress_decaf();
+        //println!("orig, {:?}", P.compress_decaf());
+        for i in (0..8).filter(|x| x % 2 == 0) {
+            let Q = &P + &constants::EIGHT_TORSION[i];
+            //println!("{}, {:?}", i, Q.compress_decaf());
+            assert_eq!(Q.compress_decaf(), P_decaf);
+        }
+    }
+
     #[bench]
     fn bench_basepoint_mult(b: &mut Bencher) {
         b.iter(|| ExtendedPoint::basepoint_mult(&A_SCALAR));