Re: [Cfrg] 25519 naming

Robert Ransom <rransom.8774@gmail.com> Tue, 02 September 2014 08:28 UTC

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Date: Tue, 2 Sep 2014 01:28:26 -0700
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From: Robert Ransom <rransom.8774@gmail.com>
To: Andrey Jivsov <crypto@brainhub.org>
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Subject: Re: [Cfrg] 25519 naming
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On 9/2/14, Andrey Jivsov <crypto@brainhub.org>; wrote:

> X25519 performs 9 F(p) multiplications and squares per each bit of a
> scalar, for the total of 2295 F(p) operations, not counting additions.
> Plus there are also 265 operation for an inversion.
>
> According to the above, conversion from -1-twisted Edwards coordinates to
> X25519 unfortunately involves a F(p) inversion.
>
>     265/2560 ~= 10%
>
> The problem with fixing canonical coordinates seems to be the 10% penalty in
> conversion from Edwards coordinates to Montgomery due to F(p) inversion.  I
> wonder if it is possible to not perform the inversion by starting with z!=1
> in the Montgomery formula?

That would cost an extra 1M per scalar bit.  1I to rescale the input
point is *always* faster.

> This would give us fixed-base optimization and dual key use that comes with
> Edwards formula and the speed of Montgomery for variable-base ECDH.

A Montgomery-form x coordinate (or its reciprocal) provides that.
That's why I've been arguing for a point format based on the
Montgomery-form x coordinate for so long, even though no one seems to
be listening.


Robert Ransom