diff --git a/elliptic-curve/src/hash2curve/osswu.rs b/elliptic-curve/src/hash2curve/osswu.rs
index 60796ebe5..3c3669ac3 100644
--- a/elliptic-curve/src/hash2curve/osswu.rs
+++ b/elliptic-curve/src/hash2curve/osswu.rs
@@ -1,9 +1,11 @@
 //! Optimized simplified Shallue-van de Woestijne-Ulas methods.
 //!
-//! <https://eprint.iacr.org/2009/340.pdf>
+//! <https://www.ietf.org/archive/id/draft-irtf-cfrg-hash-to-curve-16.html#straightline-sswu>
 
 use ff::Field;
 use subtle::Choice;
+use subtle::ConditionallySelectable;
+use subtle::ConstantTimeEq;
 
 /// The Optimized Simplified Shallue-van de Woestijne-Ulas parameters
 pub struct OsswuMapParams<F>
@@ -38,50 +40,91 @@ pub trait OsswuMap: Field + Sgn0 {
     /// should be for isogeny where A≠0 and B≠0.
     const PARAMS: OsswuMapParams<Self>;
 
+    /// Optimized sqrt_ratio for q = 3 mod 4.
+    fn sqrt_ratio_3mod4(u: Self, v: Self) -> (Choice, Self) {
+        // 1. tv1 = v^2
+        let tv1 = v.square();
+        // 2. tv2 = u * v
+        let tv2 = u * v;
+        // 3. tv1 = tv1 * tv2
+        let tv1 = tv1 * tv2;
+        // 4. y1 = tv1^c1
+        let y1 = tv1.pow_vartime(Self::PARAMS.c1);
+        // 5. y1 = y1 * tv2
+        let y1 = y1 * tv2;
+        // 6. y2 = y1 * c2
+        let y2 = y1 * Self::PARAMS.c2;
+        // 7. tv3 = y1^2
+        let tv3 = y1.square();
+        // 8. tv3 = tv3 * v
+        let tv3 = tv3 * v;
+        // 9. isQR = tv3 == u
+        let is_qr = tv3.ct_eq(&u);
+        // 10. y = CMOV(y2, y1, isQR)
+        let y = ConditionallySelectable::conditional_select(&y2, &y1, is_qr);
+        // 11. return (isQR, y)
+        (is_qr, y)
+    }
+
     /// Convert this field element into an affine point on the elliptic curve
     /// returning (X, Y). For Weierstrass curves having A==0 or B==0
     /// the result is a point on an isogeny.
     fn osswu(&self) -> (Self, Self) {
-        let tv1 = self.square(); // u^2
-        let tv3 = Self::PARAMS.z * tv1; // Z * u^2
-        let mut tv2 = tv3.square(); // tv3^2
-        let mut xd = tv2 + tv3; // tv3^2 + tv3
-        let x1n = Self::PARAMS.map_b * (xd + Self::ONE); // B * (xd + 1)
-        xd *= -Self::PARAMS.map_a; // -A * xd
-
-        let tv = Self::PARAMS.z * Self::PARAMS.map_a;
-        xd.conditional_assign(&tv, xd.is_zero());
-
-        tv2 = xd.square(); //xd^2
-        let gxd = tv2 * xd; // xd^3
-        tv2 *= Self::PARAMS.map_a; // A * tv2
-
-        let mut gx1 = x1n * (tv2 + x1n.square()); //x1n *(tv2 + x1n^2)
-        tv2 = gxd * Self::PARAMS.map_b; // B * gxd
-        gx1 += tv2; // gx1 + tv2
-
-        let mut tv4 = gxd.square(); // gxd^2
-        tv2 = gx1 * gxd; // gx1 * gxd
-        tv4 *= tv2;
-
-        let y1 = tv4.pow_vartime(Self::PARAMS.c1) * tv2; // tv4^C1 * tv2
-        let x2n = tv3 * x1n; // tv3 * x1n
-
-        let y2 = y1 * Self::PARAMS.c2 * tv1 * self; // y1 * c2 * tv1 * u
-
-        tv2 = y1.square() * gxd; //y1^2 * gxd
-
-        let e2 = tv2.ct_eq(&gx1);
-
-        // if e2 , x = x1, else x = x2
-        let mut x = Self::conditional_select(&x2n, &x1n, e2);
-        // xn / xd
-        x *= xd.invert().unwrap();
-
-        // if e2, y = y1, else y = y2
-        let mut y = Self::conditional_select(&y2, &y1, e2);
-
-        y.conditional_assign(&-y, self.sgn0() ^ y.sgn0());
+        // 1.  tv1 = u^2
+        let tv1 = self.square();
+        // 2.  tv1 = Z * tv1
+        let tv1 = Self::PARAMS.z * tv1;
+        // 3.  tv2 = tv1^2
+        let tv2 = tv1.square();
+        // 4.  tv2 = tv2 + tv1
+        let tv2 = tv2 + tv1;
+        // 5.  tv3 = tv2 + 1
+        let tv3 = tv2 + Self::ONE;
+        // 6.  tv3 = B * tv3
+        let tv3 = Self::PARAMS.map_b * tv3;
+        // 7.  tv4 = CMOV(Z, -tv2, tv2 != 0)
+        let tv4 = ConditionallySelectable::conditional_select(
+            &Self::PARAMS.z,
+            &-tv2,
+            !Field::is_zero(&tv2),
+        );
+        // 8.  tv4 = A * tv4
+        let tv4 = Self::PARAMS.map_a * tv4;
+        // 9.  tv2 = tv3^2
+        let tv2 = tv3.square();
+        // 10. tv6 = tv4^2
+        let tv6 = tv4.square();
+        // 11. tv5 = A * tv6
+        let tv5 = Self::PARAMS.map_a * tv6;
+        // 12. tv2 = tv2 + tv5
+        let tv2 = tv2 + tv5;
+        // 13. tv2 = tv2 * tv3
+        let tv2 = tv2 * tv3;
+        // 14. tv6 = tv6 * tv4
+        let tv6 = tv6 * tv4;
+        // 15. tv5 = B * tv6
+        let tv5 = Self::PARAMS.map_b * tv6;
+        // 16. tv2 = tv2 + tv5
+        let tv2 = tv2 + tv5;
+        // 17.   x = tv1 * tv3
+        let x = tv1 * tv3;
+        // 18. (is_gx1_square, y1) = sqrt_ratio(tv2, tv6)
+        let (is_gx1_square, y1) = Self::sqrt_ratio_3mod4(tv2, tv6);
+        // 19.   y = tv1 * u
+        let y = tv1 * self;
+        // 20.   y = y * y1
+        let y = y * y1;
+        // 21.   x = CMOV(x, tv3, is_gx1_square)
+        let x = ConditionallySelectable::conditional_select(&x, &tv3, is_gx1_square);
+        // 22.   y = CMOV(y, y1, is_gx1_square)
+        let y = ConditionallySelectable::conditional_select(&y, &y1, is_gx1_square);
+        // 23.  e1 = sgn0(u) == sgn0(y)
+        let e1 = self.sgn0().ct_eq(&y.sgn0());
+        // 24.   y = CMOV(-y, y, e1)
+        let y = ConditionallySelectable::conditional_select(&-y, &y, e1);
+        // 25.   x = x / tv4
+        let x = x * tv4.invert().unwrap();
+        // 26. return (x, y)
         (x, y)
     }
 }