Re: [Cfrg] [CFRG] Safecurves v Brainpool / Rigid v Pseudorandom

Paul Lambert <> Tue, 14 January 2014 01:22 UTC

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From: Paul Lambert <>
To: Watson Ladd <>, Dan Brown <>
Date: Mon, 13 Jan 2014 17:21:51 -0800
Thread-Topic: [Cfrg] [CFRG] Safecurves v Brainpool / Rigid v Pseudorandom
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Subject: Re: [Cfrg] [CFRG] Safecurves v Brainpool / Rigid v Pseudorandom
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> Furthermore, what is your position on the question of ECC choices?
> Instead of going around in circles of increasing irrelevance, let's
> have it between these options:
> 0: Nothing
> 1: NIST
> 2: Brainpool
> 3: Chicago
> 4: NIST+Brainpool
> 5: Brainpool+Chicago
> 6: NIST+Chicago
> 7: NIST+Brainpool+Chicago
> Each of these options has costs. Each has advantages. And I've not even
> scratched the surface: maybe P256 stays, along with Ed448-Goldilocks,
> and P521. So far I think that 2 alone is terrible, 1 alone okay, 3
> alone superior (but anything sans NIST is a nonstarter for DOD), so 6
> seems good, because no one will pick brainpool from 7.

It's naïve to think that we can make 'choices' here in the CFRG or IETF.
I am not sure of your prior experience in the standards 
making process, but it is a consensus building effort that builds on
clear documentation.  The best we can do is to clearly document new
options and fairly show the potential benefits.

Individual products, industry consortiums, Governments may mandate
specific choices.  Even then, it is impossible to rapidly remove
any type of standard once fielded - even if there are clear security
problems (SHA-1, WEP, etc.)

There are no known significant attacks against the NIST/SECG curves,
only a fashionable suspicion of NIST and the NSA.  They will
be in use for a long time.  Fashion does have its place and
we do need to consider and document the real level of confidence
that we have as an industry in any algorithm.

Curves based on 'new ECC math' for Edwards curves shows significant
potential for more efficient and easier implementations (I include
side channel protection under easier implementations).  A major
challenge we have is to provide clarity that we are introducing new
math ways to make calculations on elliptic curves.  The Small
Weierstrass curves have been so dominate in literature and 
standards that they are described as 'the' elliptic curve.

I am very supportive of our introduction of these families of
new curves.  Speed and ease of correct/secure implementation
are significant benefits of the 'new math'.

So ... on choices, we can strongly recommend based on the benefits.
Suite B will live on for a long time based on the slow pace 
of Government initiatives.

Brainpool, the previous fashionable alternative seems less fashionable today.

Edwards based curves seem an optimal choice where new 
curves are able to be introduced. They are both fashionable (not NIST)
and have demonstrable benefits.

Now ... the current draft does seem confusing to me in this context on
benefits and clear curve recommendations.  There are a few too many and
we could trim or clarify when each would be used.


> >
> > Also, you suggested that I was referring to the field choices. I did
> not intend to, but I should clarify.
> >
> > For a given field, pseudorandom curve better resists secret attacks
> than rigidity.

> What about the known attacks which Brainpool implementations suffer
> from? There still isn't a side-channel clean one that gets correct
> answers all the time. Contrast with NIST and the new curves. (Yes, we
> need to churn out some code, but it's clear it can be done). I admire
> your persistence in necroequoflagellation,  but as DJB observed the
> previous time this came up, you've simply declared 486662 special, with
> no evidence that the curve is actually special.
> If you noticed a particularly unusual endomorphism, or an interesting
> map to some other scheme, or the Q-lift having unusually large rank, or
> an infinite Tate-Sharavich group, that would be reason to worry or
> consider it special. But the coefficient being small?
> What's so special about that?
> >
> > Field size would on my list of issues, but way, way down on my
> ranking of importance.
> >
> > A couple people have suggested to me (off list) to generate new
> curves with explainable seeds but the same fields as NIST curves.
> These people should not be listened to: the world does not need more
> short Weierstrass curves. Pick rigid Edwards curves over the same
> field, and make a small for speed. I understand the field proliferation
> concern, particularly for hardware, but that's where a lot of speed
> comes from: ask AGL or Rich Salz how much speed they want to give up.
> We have Rich Salz on record as saying that Akamai is held back from PFS
> by the CPU demand, and with Curve25519 that will be solved. In software
> additional fields cost little.
> >
> > The result would seem to be intermediate between Brainpool and
> Chicago.
> >
> > My main reluctance with that would further babble, and yet more
> difficulty with interoperability.
> Interop solution: MTI. Odds are people will pick a hardness, and the
> fastest thing that delivers that hardness will win out. Also, make
> clear standards can exclude curves, even to the extent of picking one
> and saying "This is it!". I think the majority of protocols where
> performance in software is a concern and correctness of implementation
> an issue should consider Curve25519 carefully, but I am not going to
> argue that I know better than the communities around these protocols
> what is appropriate for the hardware they are looking at running on and
> the constraints they face. Some protocols will need to dispense with
> choices because of these limitations I am going to argue that I know
> cryptography better than they do in most cases, and that's how I see
> this question: making a catalog of curves, where you can reach in and
> go "Oh, this one has high software performance, acceptable hardware
> performance, good security, useful for ECDH" or "This one is great in
> hardware, ok in software, moderate security (and some gotchas), all
> protocols" and make a choice easily.
> Sincerely,
> Watson Ladd
> >
> >
> > From: Watson Ladd
> > Sent: Monday, January 13, 2014 4:26 PM
> > To: Dan Brown
> > Cc:
> > Subject: Re: [Cfrg] [CFRG] Safecurves v Brainpool / Rigid v
> > Pseudorandom
> >
> >
> > On Mon, Jan 13, 2014 at 8:55 AM, Dan Brown <>;
> wrote:
> >>
> >> Last week I wrote that I would soon write up my disagreements with
> the safe curves site.
> >>
> >>
> >>
> >> My first disagreement is written up below, but first two editorial
> issues:
> >>
> >>
> >>
> >> - My last email was in the context of Watson’s draft spec for the
> Chicago (Bernstein?) curves, but I would like to detach the issue from
> the I-D. My comment about the reference in Watson’s I-D to a website
> was just an editorial comment.
> >>
> >>
> >>
> >> - These technical issues that I hope CFRG discusses just might be
> resolved and written into a CFRG I-D cataloging of elliptic curves and
> their relative merits and so on, which should be independent of
> Watson’s spec for the Chicago curves. For now, I opted to discuss via
> email list, rather than placing these arguments into my own one-sided
> individual I-D. Please advise me if such an I-D is preferred to email.
> >
> >
> > I think this is an excellent idea (the catalogue of good curves).
> >
> >>
> >>
> >>
> >> Today, I checked that
> >>
> >>
> >>
> >>
> >>
> >> <snip>
> >
> > Rather then engage a peripheral argument, let me lay out the case for
> > the Chicago curves (named after the city in which Bernstein resides,
> > and in a long IETF tradition starting with TCP Reno, Oakley key
> > agreement protocol, and fundamentally of no import: see Shakespeare,
> > Romeo and Juliet, Act 2, Scene 2, or Wittgenstein, Philosophical
> > Investigations for more information) in the fairest light.
> >
> > First efficiency: point addition on an Edwards curve costs 11
> > multiplications, doubling 7 multiplications. This is for a complete,
> > strongly unified addition formula, which requires no work for special
> > cases. By contrast on a short Weierstrass curve addition costs 16
> > multiplications, doubling 8.
> > This is with no assumptions on Z, which if we impose reduce the cost
> > of addition on a short Weierstrass curve to 11 multiplications. But
> if
> > we impose the same condition on the Edwards curve, the cost is 10
> > multiplications.
> >
> > However, I've forced the Edwards curve to represent the identity,
> > which if we are willing to drop, slices off two more multiplications.
> > By contrast the
> > Weierstrass curve has special cases that need additional work to
> > handle in a secure manner. Most people just don't bother. I also let
> > the Weierstrass form be special with a=-3, while keeping the Edwards
> > curve fully general.
> >
> > So given the same prime Edwards has short Weierstrass beat. This is
> > true no matter what algorithm for exponentiation is used. And I even
> > gave short Weierstrass curves a handicap by letting them avoid
> > representing the entire curve and having exceptional cases, and fix
> > some parameters.
> >
> > Montgomery form is even better: differential addition costs 10M, and
> > calculates a doubling for free. Sadly the differential part makes
> > radix-k algorithms tough, but in memory constrained environments
> (like
> > hardware) Montgomery form can't be beat. (And we haven't even used
> > nonuniform differential addition chains yet.. I'm still hopeful).
> >
> > Then there is the prime shape: Brainpool uses a random prime, which
> is
> > a terrible idea from an efficiency perspective. NIST primes are sized
> > for 32 bit machines, and have some hiccups on 64 bit machines. But
> the
> > Chicago primes are primes for all seasons: 2^k-c with small c is nice
> > no matter what radix you adopt. Unsaturated arithmetic makes addition
> > faster because reductions aren't necessary at all times, and the
> > Chicago primes are good from that perspective.
> >
> > Now security: having an efficient complete addition formula makes
> > writing timing, cache, and branch side channel free code a piece of
> > cake. By contrast short Weierstrass curves have all sorts of wrong
> > curve, or invalid point, or somewhere the calculation messes up
> > issues. Usually the defense is a final check on the result being on
> > the code, but that isn't the same as knowing that no attack is
> > possible because your code always generates the right answer.
> >
> > Lastly, many websites want to deploy PFS ciphersuites but are
> deterred
> > by the computation expense involved. Minimizing that expense is
> > essential. In the wider IETF context we are catering to devices from
> > the lowly MSP430, to the latest and greatest supercomputer. What does
> > Brainpool let us do? We should stick to the NIST and Chicago curves:
> > NIST for those who require it, and Chicago for those who require the
> > extra speed.
> >
> > I would like to hear good arguments against this position. Robert
> > Ransom I understand wants twisted Edwards curves for an extra bit of
> > speed, using isogenies to keep them the same as the Chicago curves,
> > but selecting only those with small parameters and particularly nice
> > prime shapes.
> > Dan Brown believes that the Brainpool primes are less special in a
> way
> > that has to do with ECC security, despite this never affecting ECC
> > before.
> >
> > Sincerely,
> > Watson Ladd
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