NIST curves backdoored

Gregory Maxwell on the Tor-talk list has found that NIST approved curves, which is to say NSA approved curves, were not generated by the claimed procedure, which is a very strong indication that if you use NIST curves in your cryptography, NSA can read your encrypted data.

So don’t use anything NIST approved.

NIST curves are supposedly verifiably random, which, if true, would be proof that they are not backdoored. Gregory Maxwell attempted to replicate this claim, found that they were verifiably nonrandom.

From: Gregory Maxwell <gmaxwell@gmail.com>
To: "This mailing list is for all discussion about theory, design, and development of Onion Routing."
	<tor-talk@lists.torproject.org>
Subject: Re: [tor-talk] NIST approved crypto in Tor?
Reply-To: tor-talk@lists.torproject.org

On Sat, Sep 7, 2013 at 4:08 PM, anonymous coward
<anonymous.coward@posteo.de> wrote:

Bruce Schneier recommends *not* to use ECC. It is safe to assume he knows what he says.

I believe Schneier was being careless there. The ECC parameter sets commonly used on the internet (the NIST P-xxxr ones) were chosen using a published deterministically randomized procedure. I think the notion that these parameters could have been maliciously selected is a remarkable claim which demands remarkable evidence.

On Sat, Sep 7, 2013 at 8:09 PM, Gregory Maxwell <gmaxwell@gmail.com> wrote:

Okay, I need to eat my words here.

I went to review the deterministic procedure because I wanted to see if I could repoduce the SECP256k1 curve we use in Bitcoin. They don’t give a procedure for the Koblitz curves, but they have far less design freedom than the non-koblitz so I thought perhaps I’d stumble into it with the “most obvious” procedure.

The deterministic procedure basically computes SHA1 on some seed and uses it to assign the parameters then checks the curve order, etc.. wash rinse repeat.

Then I looked at the random seed values for the P-xxxr curves. For example, P-256r’s seed is c49d360886e704936a6678e1139d26b7819f7e90.

_No_ justification is given for that value. The stated purpose of the “veritably random” procedure “ensures that the parameters cannot be predetermined. The parameters are therefore extremely unlikely to be susceptible to future special-purpose attacks, and no trapdoors can have been placed in the parameters during their generation”.

Considering the stated purpose I would have expected the seed to be some small value like … “6F” and for all smaller values to fail the test. Anything else would have suggested that they tested a large number of values, and thus the parameters could embody any undisclosed mathematical characteristic whos rareness is only bounded by how many times they could run sha1 and test.

I now personally consider this to be smoking evidence that the parameters are cooked. Maybe they were only cooked in ways that make them stronger? Maybe????

SECG also makes a somewhat curious remark:

“The elliptic curve domain parameters over (primes) supplied at each security level typically consist of examples of two different types of parameters — one type being parameters associated with a Koblitz curve and the other type being parameters chosen verifiably at random — although only verifiably random parameters are supplied at export strength and at extremely high strength.”

The fact that only “verifiably random” are given for export strength would seem to make more sense if you cynically read “verifiably random” as backdoored to all heck. (though it could be more innocently explained that the performance improvements of Koblitz wasn’t so important there, and/or they considered those curves weak enough to not bother with the extra effort required to produce the Koblitz curves).