Re: [Cfrg] New names for draft-ladd-safecurves

Mike Hamburg <mike@shiftleft.org> Tue, 21 January 2014 05:41 UTC

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From: Mike Hamburg <mike@shiftleft.org>
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To: Robert Ransom <rransom.8774@gmail.com>
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Cc: "cfrg@irtf.org" <cfrg@irtf.org>, Jon Callas <jon@callas.org>
Subject: Re: [Cfrg] New names for draft-ladd-safecurves
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On Jan 20, 2014, at 9:06 PM, Robert Ransom <rransom.8774@gmail.com> wrote:
> I would suggest using the Montgomery-form x coordinate with the sign
> bit of the Edwards-form x coordinate.  (In fact, I *did* suggest that:
> <http://www.ietf.org/mail-archive/web/cfrg/current/msg03868.html>
> <http://www.ietf.org/mail-archive/web/cfrg/current/msg03870.html>)

Amusingly, if you say that "sign" is "Jacobi symbol" (when p==3 mod 4), the two are the same for q-torsion points.  But either way, the difference is a couple of field muls vs a slightly messier spec.

> And yes, the Brier-Joye formulas to recover Montgomery-form y after
> the Montgomery ladder would be faster than Brauer's algorithm on
> Edwards-form points for variable-base single-scalar multiplication.


Is this still true for large curves?  I don't think it's true asymptotically, and if people switch from inverse-by-exp to inverse-by-blind-and-EGCD, then you lose your free square root and the tradeoff might happen before 521 bits.  You might get your freebie back again if you spec that sign is Jacobi symbol and use a blind EGCD inverse-and-Jacobi-symbol, but again this only works for p==3 mod 4.

Cheers,
-- Mike