[curves] F2m curve compression

[lvh]

Hi,

> On Dec 4, 2016, at 1:06 PM, zaki at manian.org wrote:
> 
> It appears Ofek got the help he needed here.
> https://github.com/pyca/cryptography/pull/3287 <https://github.com/pyca/cryptography/pull/3287>

I’m responsible for that (hopefully not too terrible given the use case) explanation. The SECG document just has the high-level description in terms of Z=Y/X, but doesn’t seem to mention Frobenius, trace or half-trace in that section. (I haven’t read the rest of the document to see if they eventually do; I’m currently traveling but will be skimming that later. A quick search seems to suggest the string Frobenius does not occur in that document.)

If anyone has any good documentation for how you actually _implement_ that (in code, not just the mathematical derivation), I’d be much obliged; even if it’s only a way to check my work. OpenSSL has an implementation but it’s very OpenSSL so perhaps not the best for instruction; looks like Golang only implements P-x curves, and not the SECG F2^m curves. (I found a Golang implementation on GitHub, but I’m less likely to trust it than the Golang stdlib, and it didn’t implement point compression.)

On a related note: how bad would it be to have a “default” compression for these curves? My understanding is that there are a few options, and while I’ve never seen anything but the SECG method for the SECG curves, I don’t know if the other compression methods are often used in the wild for these or other F2^m curves.

lvh

> On Sat, Dec 3, 2016 at 6:08 PM Ofek Lev <ofekmeister at gmail.com <mailto:ofekmeister at gmail.com>> wrote:
> I understand for prime curves it is just `bytes(0x02 + flag) + bytes(x)` where flag is the LSB of y. For the F2m curves I cannot make out how to do it.
> 
> IEEE P1363 <http://grouper.ieee.org/groups/1363/IBC/material/P1363.3-D1-200805.pdf%20section%205.6.6.1.2> section 5.6.6.1.2 appears to say flag is '1 if y of point > y of inverse point else 0' which I think just means `if y > x`.
> 
> these slides <http://cs.ucsb.edu/~koc/ccs130h/projects/03-ecc-protocols/Julio_Slides.pdf> (slide 15) by Julio Lopez and Ricardo Dahab appear to suggest my interpretation of the IEEE method is off (I think).
> 
> http://www.secg.org/sec1-v2.pdf <http://www.secg.org/sec1-v2.pdf%20section%202.3.3%20part%202.2.2> (which I think is the standard reference) section 2.3.3 part 2.2.2 has yet another notation that I do not understand.
> 
> I was told there are multiple ways. Can someone please explain the most *standard* (or easiest) way requiring size m + 1, preferably from a programmer's perspective? This math is beyond me :)
> 
> Any insight would be greatly appreciated.
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