4 ms·

Regarding ECC, whenever a "constant" appears in a cryptographic algorithm, BEWARE. Yes, I know about "nothing up my sleeve" numbers... However the ones used fo

Regarding ECC, whenever a "constant" appears in a cryptographic algorithm, BEWARE. Yes, I know about "nothing up my sleeve" numbers... However the ones used for ECC are not of this type. What a sad state of affairs.

nullc 13y ago
> However the ones used for ECC are not of this type. Yes they are. The P-XXXr curves which are used by most of the ECC using web (and Tor, when it uses ECC) were generated by using a deterministic strongly pseudo-random procedure which you can repeat for yourself. See page 187 of FIPS 186-3.
- pfortuny 13y ago
  Ab, OK, my bad. Thanks for the pointer and sorry for the mistake.
- nullc 13y ago
  Okay, I need to eat my words here. I went to review the deterministic procedure because I wanted to see if I could reproduce 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).
  - reirob 13y ago
    Thank you very much for going into such detail. But what does it mean for the ECC or RSA methods? Which can be considered as secure? I guess we need more expertise, right?
    - nullc 13y ago
      There is no specific reason to believe ECC methods on these standard curves (or other well selected parameters) to be insecure. But at the same time if someone argues that the NSA could have influenced the parameters to create hidden weaknesses predicated on math unknown to the public, that argument can't be denied. It might even be supported on the basis that they could have easily closed off that argument but did not. (OTOH, equally I could argue that the same selection may have been used to make it stronger). As far as RSA goes, their usage involves fewer free parameters... but to get RSA up to the security levels people are demanding today require increasingly large and slow keys. If the crypto is too burdensome it'll get avoided, and thats worse than whatever speculation you might have about NSA weakening of EC systems.
  - tptacek 13y ago
    You consider it smoking evidence of "cooked" parameters that he hashed a random string? This is the technique suggested by all of IEEE-1363 2000, and in every place its suggested, the standard is at pains to suggest that it provides only a degree of assurance; the standard authors go as far as to repeat the exact same text in each location (RSA, DSA, ECC, &c) while customizing the "what could go wrong" part of the paragraph. 1363 predates the NIST curves, for whatever that's worth.
    - pfortuny 13y ago
      I am now rather confused in my ignorance... So the parameters (for ECC) were chosen "at random" theoretically (and so here lies the trust inherent to these?). Thanks for the info.
      - nullc 13y ago

nullc 13y ago

> However the ones used for ECC are not of this type. Yes they are. The P-XXXr curves which are used by most of the ECC using web (and Tor, when it uses ECC) were generated by using a deterministic strongly pseudo-random procedure which you can repeat for yourself. See page 187 of FIPS 186-3.

pfortuny 13y ago
Ab, OK, my bad. Thanks for the pointer and sorry for the mistake.
nullc 13y ago
Okay, I need to eat my words here. I went to review the deterministic procedure because I wanted to see if I could reproduce 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).
- reirob 13y ago
  Thank you very much for going into such detail. But what does it mean for the ECC or RSA methods? Which can be considered as secure? I guess we need more expertise, right?
  - nullc 13y ago
    There is no specific reason to believe ECC methods on these standard curves (or other well selected parameters) to be insecure. But at the same time if someone argues that the NSA could have influenced the parameters to create hidden weaknesses predicated on math unknown to the public, that argument can't be denied. It might even be supported on the basis that they could have easily closed off that argument but did not. (OTOH, equally I could argue that the same selection may have been used to make it stronger). As far as RSA goes, their usage involves fewer free parameters... but to get RSA up to the security levels people are demanding today require increasingly large and slow keys. If the crypto is too burdensome it'll get avoided, and thats worse than whatever speculation you might have about NSA weakening of EC systems.
- tptacek 13y ago
  You consider it smoking evidence of "cooked" parameters that he hashed a random string? This is the technique suggested by all of IEEE-1363 2000, and in every place its suggested, the standard is at pains to suggest that it provides only a degree of assurance; the standard authors go as far as to repeat the exact same text in each location (RSA, DSA, ECC, &c) while customizing the "what could go wrong" part of the paragraph. 1363 predates the NIST curves, for whatever that's worth.
  - pfortuny 13y ago
    I am now rather confused in my ignorance... So the parameters (for ECC) were chosen "at random" theoretically (and so here lies the trust inherent to these?). Thanks for the info.
    - nullc 13y ago