Re: [Cfrg] Extra ECC desideratum: hard static DH and q-strong DH problems
Watson Ladd <watsonbladd@gmail.com> Fri, 08 August 2014 19:28 UTC
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Date: Fri, 08 Aug 2014 12:28:48 -0700
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From: Watson Ladd <watsonbladd@gmail.com>
To: Dan Brown <dbrown@certicom.com>
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Subject: Re: [Cfrg] Extra ECC desideratum: hard static DH and q-strong DH problems
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On Aug 8, 2014 12:17 PM, "Dan Brown" <dbrown@certicom.com> wrote: > > Three reasons that static ECDH key agreement should still be considered in > the CFRG curve recommendation: > > 1. Existing versions of TLS allow static DH for sever authentication, and > allow ephemeral re-use in DHE, and if I understand, these allowance would > carry if implemented with ECC. Also, IKE allows ephemeral reuse. > 2. Other IETF protocols using ECC, e.g. CMS, may require static DH keys, due > to less interaction being available. Perhaps JOSE uses static ECDH too? > 3. Perhaps future versions of TLS will drop signature uses for > authentication, and instead use static DH keys for authentication. > (Something like in MQV or some other key agreement scheme.) Or, maybe IETF > will later adopt protocols that use the esoteric blinding properties of raw > static DH. > > So, I think it desirable for the choice of curve to be one that fares well > static DH keys are used. > This was mentioned prior to the CFRG interim. As I pointed out at that meeting, hashing points renders the paper of Cheon irrelevent. > In particular, there are two variants of the Diffie--Hellman problem related > to static DH keys whose difficulty potentially varies with the curve: > > 1. The static DHP from (http://eprint.iacr.org/2004/306). If the static DHP > is easy for a given target public key, then the static DH applications above > are insecure for the target public key. > 2. A variant (*) of the q-strong DHP from (http://eprint.iacr.org/2004/171 ). > If the q-strong DHP is hard, then static DH applications above resist a > total break attack in the sense that it is infeasible the adversary to > extract the static DH private key. > > (*) the variant I have in mind is one in which that adversary succeeds when > it recovers the static private key, rather than performing some special > operation with it. > > An algorithm from the 2004/306 eprint, and extended by Cheon, has opposite > effects on these two problem: making the q-strong easier, but providing > evidence that the static DHP is hard. This effectiveness algorithm depends > on the factorization of n-1 and n+1 where n is the large prime factor in the > order of the group. I think it is desirable for both of these problem to be > hard, and I think that if n has the right properties, then we can get good > assurances for both problems. > > If I had to choose which of the two problems to accommodate, I'd choose the > q-strong DHP, because one can just make a strong, but plausible assumption > that the static DHP is hard. In other words, instead of assuming that the > DHP is hard, as usual, we further assume that the DHP is hard for every > single key, i.e. the DHP has no negligible fraction weak keys. Assuming > this covers that case that your static DHP has enough entropy to be > unguessable, but not enough to escape falling into a subset that is > negligible fraction of all keys. E.g. suppose your 256-bit key has only > 128-bits of entropy (but is pseudorandom of course). > > I also add that one of the security proof for TLS > (http://eprint.iacr.org/2013/339) relies on some assumptions PRF-ODH and a > similarly-named problem Strong DH from (http://eprint.iacr.org/1999/007) > which may have may relationship with the q-strong DH problem. For example, > I've got a hunch that if we assume the variant of the q-strong DHP is hard, > then I think that imply the hardness of PRF-ODH and the ABR StrongDH > problems. Because of the short time frame, I sharing this thought before > really confirming it. > > Of course, this desideratum is a theoretical one, but some of the other > desiderata seem to be better-safe-than-sorry or helpful to make > implementations more robust, so I wanted to add it the list. > > Also, Certicom has some IPR around the method. > > Best regards, > > Daniel Brown > Research In Motion Limited > > > _______________________________________________ > Cfrg mailing list > Cfrg@irtf.org > http://www.irtf.org/mailman/listinfo/cfrg >