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

Watson Ladd <> Mon, 13 January 2014 21:26 UTC

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Date: Mon, 13 Jan 2014 13:26:10 -0800
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From: Watson Ladd <>
To: Dan Brown <>
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Subject: Re: [Cfrg] [CFRG] Safecurves v Brainpool / Rigid v Pseudorandom
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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

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

Watson Ladd