Encoding from Extended Coordinates

To encode a Ristretto point represented by the point \((X:Y:Z:T)\) in extended coordinates, an implementation proceeds as follows.

  1. \(u_1 \gets (Z_0 + Y_0)(Z_0 - Y_0) \).
  2. \(u_2 \gets X_0 Y_0 \).
  3. \(I \gets \mathrm{invsqrt}(u_1 u_2^2) \).
  4. \(D_1 \gets u_1 I \).
  5. \(D_2 \gets u_2 I \).
  6. \(Z_{inv} \gets D_1 D_2 T_0 \).
  7. If \( T_0 Z_{inv} \) is negative:
    1. \( (X, Y) \gets (Y_0 (\pm 1/\sqrt{a}), X_0 (\mp \sqrt{a})) \).
    2. \( D \gets D_1 / \sqrt{a-d} \).
  8. Otherwise:
    1. \( (X, Y) \gets (X_0, Y_0) \).
    2. \( D \gets D_2 \).
  9. If \( X Z_{inv} \) is negative, set \( Y \gets - Y\).
  10. Compute \( s \gets |\sqrt{-a} (Z - Y) D| \), i..e, compute \(\sqrt{-a} (Z - Y) D\) and negate it if it is negative.
  11. Return the canonical byte encoding of \( s \).

Since \( a = \pm 1 \), implementations can replace multiplications by \(a\) with sign changes, as appropriate.

When \(a = -1\), (10) becomes \(s \gets |(Z-Y)D|\), and choosing \( Q_4 = (i, 0) \) is convenient since it simplifies (7.1) to \( (X,Y) \gets (iY_0, iX_0) \).

The inverse square root in step 3 always exists when \((X:Y:Z:T)\) is a valid representative.