How do I add two group elements?

[benma]

If I have two edwards25519.ExtendedGroupElement, how do I add them on the curve? I don't see a function for it, but it would be very useful.

Here is an implementation of it in Python:
https://github.com/vbuterin/ed25519/blob/5b4f873bc88de51a3999994e30cb3a3759948308/ed25519.py#L170

I would port it, but I don't know what D is.

[benma]

This seems to be working:

func Add(p, q *edwards25519.ExtendedGroupElement) *edwards25519.ExtendedGroupElement {
    var r edwards25519.CompletedGroupElement
    var qCached edwards25519.CachedGroupElement
    q.ToCached(&qCached)
    edwards25519.GeAdd(&r, p, &qCached)
    var result edwards25519.ExtendedGroupElement
    r.ToExtended(&result)
    edwards25519.FeNeg(&result.X, &result.X)
    return &result
}

I have no idea why the negation of the X is necessary in the end, but without it, the linearity does not hold:

sk1*B + sk2*B = (sk1+sk2)*B

(while taking care that sk1+sk2 has still the first three and the last two bits set correctly).