Re: [Cfrg] ECC mod 8^91+5

Thomas Garcia <tgarcia.3141@gmail.com> Wed, 02 August 2017 07:52 UTC

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From: Thomas Garcia <tgarcia.3141@gmail.com>
Date: Wed, 2 Aug 2017 08:52:05 +0100
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To: Dan Brown <danibrown@blackberry.com>
Cc: "cfrg@irtf.org" <cfrg@irtf.org>
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Subject: Re: [Cfrg] ECC mod 8^91+5
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Hi Dan,
The transformation you proposed (x->iu, y->(-i/2)^(1/2)v) does indeed
define an isomorphism between the curve (2y^2=x^3+x) and the curve
(v^2=u^3-u), modulo p=8^91+5. As this transformation is defined over GF(p),
it preserves the group of points modulo p.

Thomas G.

On Tue, Aug 1, 2017 at 10:22 PM, Dan Brown <danibrown@blackberry.com> wrote:

> Hi CFRG,
>
> A minor addition to this topic.
>
> In my first email on this topic, and in my recent IETF 99 presentation, I
> mentioned that the proposed special curve 2y^2=x^3+x was similar to curves
> proposed in Miller's 1985 paper introducing of ECC.
>
> I went back and worked out some more details about this similarity (just
> some basic elliptic curve math).
>
> To prepare, note that there exists a field element i in GF(8^91+5) with
> i^2=-1.
>
> Putting x=iu and y=(-i/2)^(1/2)v defines a map to curve with equation
> v^2=u^3-u.  I believe (99% sure) that this map is an isomorphism (of
> GF(8^91+5)-rational points).  This equation has the same form as Miller's
> proposed equation y^2 = x^3 - ax, with a=1.
>
> However, Miller does suggest that value a should not be a perfect square,
> which rules out a=1 above.
>
> I believe (50% sure) that Miller made this non-square a recommendation
> merely to help keep the group cofactor down (by ensuring a unique point of
> order 2, namely (0,0)).
>
> By contrast 2y^2=x^3+x, has a subgroup of order 8.  (With points O, (0,0),
> (i,0), (-i,0), (1,1), (1,-1), (-1,i), (-1,-i).)  A subgroup of order 4 (or
> 8) is nowadays considered (arguably) an advantage, because of various
> Edwards curves (but I am only 10% sure, since I haven't looked at this in a
> while, please correct me this is wrong).
>
> So, the special curve 2y^2=x^3+x is isomorphic to a curve with an even
> more compact representation: y^2=x^3-x.
>
> Despite the more compact equation, the original form 2y^2=x^3+x is
> slightly preferable because it is already in the convenient Montgomery
> form, so I plan to use the original in the ID.   (But if somebody knows how
> to do the Montgomery ladder math equally as efficient on a Miller curve
> y^2=x^3-x, then I'm all ears :)
>
> Best regards,
>
> Dan
>
> -----Original Message-----
> From: Cfrg [mailto:cfrg-bounces@irtf.org] On Behalf Of Dan Brown
> Sent: Tuesday, May 16, 2017 1:36 PM
> To: cfrg@irtf.org
> Subject: [Cfrg] ECC mod 8^91+5
>
> Hi all,
>
> I'm considering writing an I-D on doing ECC over the field of size
>    8^91+5    (=2^273+5),
> because it:
> - is written in just 6 symbols (=low Kolmogorov complexity, heuristically
> minimizing threat of NOBUS-trapdoor),
> - has easy and fast inversion, Legendre symbols, and square roots,
> - has efficient arithmetic using at most five 64-bit words (use base 2^55),
> - is at least 2^(256-epsilon),
> - is (probably) prime, so not an extension field (has no subfields for
> descent-type attacks on ECDLP).
> Other fields can improve on some of these properties, but might worsen the
> others.
>
> For ECC with this field, I am also considering the special curve
>    2y^2=x^3+x,
> because it:
> - is written in just 10 symbols (similar gains to 6-symbol field),
> - has Montgomery form (and easily converts to Weierstrass),
> - has efficient endomorphism (so it is a GLV curve),
> - is similar to curves already suggested by Miller in 1985 (well-aged),
> - is similar to sect256k1 already used in bitcoin (incentivized),
> - has an small enough cofactor 72 (over field size 8^91+5),
> - avoids the main ECDLP attacks: Pohlig-Hellman, Menezes-Okamoto-Vanstone,
> etc.,
> - is similar to the special curves of Koblitz-Menezes [ia.cr/2008/390,
> Sec 11.1, Example 5] resisting a speculative attack.
> The motivation for this special curve largely matches the motivation for
> the special field.
>
> The curve's risks are at least:
> - CM (endomorphism) makes it potentially weak (after 32 years of being
> safe) (note exactly opposing Koblitz-Menezes rationale),
> - its small coefficients are weak for some unpublished reason (continuing
> trend of weak small-coefficients, y^2=x^3 (singular), supersingular, etc.
> being weak),
> - weak twist order (so, it requires a static ECDH Montgomery ladder to use
> public key validation),
> - weak Cheon resistance (but this is an attack with many queries, much
> computation, and faulty or no KDF).
> - den Boer or Maurer-Wolf reductions are not tight as possible, so perhaps
> it has a big gap between DHP and DLP Other curves (over this field) can
> reduce these risks, but may also lose some of the benefits.
>
> Overall, E(GF(8^91+5)):2y^2=x^3+x might offer competitive efficiency with
> fairly reasonable security (for 128-bit symmetric keys). It is only an
> incremental change over other standard ECC curves, not anything too radical.
>
> I'd be happy to hear what CFRG thinks, or if the CFRG would welcome such
> an I-D as a CFRG work item.  I hope to have this topic presented briefly at
> an upcoming CFRG meeting.
>
> Best regards,
>
> Dan Brown
>
>
>
>
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