Re: [Cfrg] ECC mod 8^91+5
David Jacobson <dmjacobson@sbcglobal.net> Wed, 17 May 2017 04:28 UTC
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To: Dan Brown <danibrown@blackberry.com>, "cfrg@irtf.org" <cfrg@irtf.org>
References: <810C31990B57ED40B2062BA10D43FBF501B181DA@XMB116CNC.rim.net>
From: David Jacobson <dmjacobson@sbcglobal.net>
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Date: Tue, 16 May 2017 21:25:27 -0700
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Subject: Re: [Cfrg] ECC mod 8^91+5
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8^91+5 is provably prime. In Mathematica I loaded the PrimalityProving module, and checked whether 8^91+5 was prime, and got True, along with a "Certificate" that is about 2 pages long. I don't know the math of certificates, but if anyone in interested I can post the certificate. --David On 5/16/17 10:35 AM, Dan Brown wrote: > 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 > > > > > _______________________________________________ > Cfrg mailing list > Cfrg@irtf.org > https://www.irtf.org/mailman/listinfo/cfrg >
- [Cfrg] ECC mod 8^91+5 Dan Brown
- Re: [Cfrg] ECC mod 8^91+5 David Jacobson
- Re: [Cfrg] ECC mod 8^91+5 Dan Brown
- Re: [Cfrg] ECC mod 8^91+5 Dan Brown
- Re: [Cfrg] ECC mod 8^91+5 Thomas Garcia
- Re: [Cfrg] ECC mod 8^91+5 Ilari Liusvaara
- Re: [Cfrg] ECC mod 8^91+5 Dan Brown
- Re: [Cfrg] ECC mod 8^91+5 Dan Brown
- Re: [Cfrg] ECC mod 8^91+5 Ilari Liusvaara
- Re: [Cfrg] ECC mod 8^91+5 D. J. Bernstein
- Re: [Cfrg] ECC mod 8^91+5 Samuel Neves
- Re: [Cfrg] ECC mod 8^91+5 D. J. Bernstein
- Re: [Cfrg] ECC mod 8^91+5 Dan Brown
- Re: [Cfrg] ECC mod 8^91+5 Dan Brown
- Re: [Cfrg] ECC mod 8^91+5 Paterson, Kenny
- Re: [Cfrg] ECC mod 8^91+5 Hanno Böck
- Re: [Cfrg] ECC mod 8^91+5 Salz, Rich
- Re: [Cfrg] ECC mod 8^91+5 Stephen Farrell
- Re: [Cfrg] ECC mod 8^91+5 Dan Brown