Why Ed25519 encodes y-coordinates rather than x-coordinates

Raoul722

4/14/24, 4:10 PM

As stated in RFC8032 and FIPS 186-5, Ed25519 signatures uses the following encoding method:

A curve point (x,y), with coordinates in the range 0 <= x,y < p, is coded as follows. First, encode the y-coordinate as a little-endian string of 32 octets. The most significant bit of the final octet is always zero. To form the encoding of the point, copy the least significant bit of the x-coordinate to the most significant bit of the final octet.

Is there any benefit to encode the y-coordinate rather than the x-coordinate here?

My question comes from the fact that the SEC standard encodes x-coordinates, so it would have been less confusing to follow the same approach in my opinion.

765

1 + 1

ed25519

fgrieu

4/14/24, 4:42 PM

The equation of Edwards curves has $x$ and $y$ interchangeable, thus it seems to me that the other choice would have worked just as well. I don't know a rationale for the choice made.

Score:12

Crypto

bk2204

4/14/24, 8:16 PM

This was a concern for patent reasons. As fgrieu mentions, you can recover one coordinate from the other on Edwards curves, so it doesn't matter which you encode. However, as this answer mentions, there were two patents, which may very well have been invalid due to prior art, which patented encoding the entire x-coordinate and one bit of the y-coordinate.

However, nobody thought to patent doing it the other way around, and so EdDSA encoded the y-coordinate. This is easier than trying to invalidate various patents around the world and avoids people derailing adoption of standards by claiming IP.

+ 1

fgrieu

4/16/24, 11:50 AM

Bernstein is (rightly) careful of avoiding patents (he [lists these and counterarguments](https://cr.yp.to/patents.html)). However swapping $x$ and $y$ to work around a patent on point compression would IMHO not have been a clear-cut defense against a patent **_if_** it had been valid, due to the [doctrine of equivalents](https://www.law.cornell.edu/wex/doctrine_of_equivalents). I'm sure Bernstein understands that. But a feeble argument is better than nothing in a patent fight, and (as this answer shows) some would _think_ this is a working legal workaround, which can only help adoption.

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