[Cryptography] Why is ECC secure?
Viktor Dukhovni
cryptography at dukhovni.org
Wed Sep 30 11:40:12 EDT 2015
On Tue, Sep 29, 2015 at 09:47:01PM -0700, Bill Cox wrote:
> A few weeks ago, I managed to prove what I'm sure is already well known:
> that for Edwards curves, Finv(a) is just sn(a, k), where sn is the Jacobi
> Elliptic sine function. The whole Edwards curve addition rule, at least in
> one quadrant, can be restated (in Wolfram Alpha language) as:
>
> x3 = JacobiSN[EllipticF[ArcSin[x1], d] + EllipticF[ArcSin[x2], d], d]
>
> or more simply in regular notation:
>
> x3 = sn(F(arcsin(x1), d) + sn(F(arcsin(x2), d), d)
The existence of the "exponential map" for compact one-dimensional
Lie-groups (such as Edwards curves, at least for d < 0) is not at
all surprising. The "exponential map" exists for *all* Lie-groups,
and yields a group homorphism from the tangent vector space at the
identity under addition into the group.
In the special case of compact one-dimensional Lie-groups the
exponential map is necessarily a group isomorphism with the real
circle (the exponential map is periodic with some period T).
It is likely feasible to compute a local inverse of the exponential
map (near the identity element) with enough precision to make
discrete logarithms practical on Edwards curves over the real
numbers (find $n$ given $nP$ for some base point $P$).
But this applies only to curves over the reals, which are not
terribly relevant to cryptography. It does not carry over to curves
over prime fields (or Galois extensions).
It still seems like you're ignoring the lack of a generic correspondence
between the continuous and discrete cases. Yes *some* things work
the same way, but important distinctions remain.
--
Viktor.
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