[Cfrg] Complete additon for cofactor 1 short Weierstrass curve?

Dan Brown <dbrown@certicom.com> Thu, 04 December 2014 22:17 UTC

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From: Dan Brown <dbrown@certicom.com>
To: "cfrg@irtf.org" <cfrg@irtf.org>
Thread-Topic: Complete additon for cofactor 1 short Weierstrass curve?
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Date: Thu, 4 Dec 2014 22:17:04 +0000
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Subject: [Cfrg] Complete additon for cofactor 1 short Weierstrass curve?
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Hi CFRG list,

I've interpreted some statements on the CFRG list, and some external web sites, as suggesting the cofactor 1 short Weierstrass curves like P256 do not have (known?) "complete" addition laws (and then, that this would be a potential security problem?).  Possibly, I'm misunderstanding these statements because I think that there is a complete addition law for such curves, described below (if this existence is known, then just skip to the end of this message).

First, Figure 1.2 from http://eprint.iacr.org/2009/580.pdf adapted from Bosma and Lenstra, gives two formulas, which I'll abbreviate to (D:E:F) and (G:H:I), for the sum of two points (in terms of their standard projective coordinates).  These two laws form a complete set, which means, I think, that, for any two input points, (a) at least one of them does not evaluate to (0:0:0), and (b) any evaluation, other than (0:0:0), is the correct value, i.e. a projective representation of the sum of the points.  (Actually, I'm not totally sure about (b): can one of these laws evaluate an incorrect point different from (0:0:0)?  The elementary argument below _assumes_ that this worse kind of failure cannot happen.)

Second, let t be an element of degree 2 over F_p with t^2 = u, for u a non-quadratic residue.  Then the law (D+tG : E+tH : F+tI) is another valid addition law, except possibly where it takes value (0:0:0), because it on the line between (D:E:F) and (G:H:I), and thus represents the same point.  When adding two points with F_p coordinates, this never results (0:0:0), because then both (D:E:F) and (G:H:I) would evaluate to (0:0:0).

Third, if the curve has "cofactor 1", then there is no F_p-rational point of order 2, which means that E+tH never evaluates to zero when the output point has coordinates in F_p.  (And the point at infinity is (0:1:0), so it too has a nonzero value for E+tH.)  So, we can multiply by the previous law by the conjugate E-tH of the Y coordinate, and then drop any remaining t terms from X and Z expressions, because their coefficients must vanish on F_p rational points once the Y coordinate is scaled to be in F_p, to get the law (DE - uGH : E^2 - uH^2 : FE - uHI). This should be complete in the sense that, for F_p rational input points, it never results in (0:0:0), because it is just a nonzero scaling of the previous law, (or because E^2 - uH^2 is the norm of E+tH, which is nonzero on the points of interests).

Well, I haven't really studied these kinds of things before: I more often think of elliptic curves as generic groups, so I could easily be mistaken (e.g. maybe the incorrect outputs really can differ from (0:0:0), or something really wrong in the logic above).  So, I ask the experts here on this list: Is this addition law correct?  Is it complete in the same sense used for Edwards curves?

If this is all correct, then I would suggest that cofactor 1 short Weierstrass do not have a security problem compared to Edwards curves (e.g. cofactor 4), in the sense of lacking a complete addition law, but rather, just an efficiency problem, in the sense of not having any (known) efficient complete law.

Best regards,

Daniel Brown

Research In Motion Limited

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