[Cryptography] Why is ECC secure?

[Tony Arcieri]

On Thu, Aug 13, 2015 at 8:33 AM, Viktor Dukhovni <cryptography at dukhovni.org>
wrote:

> On Wed, Aug 12, 2015 at 11:34:45PM -0700, Ryan Carboni wrote:
>
> > Quite bluntly, millennia have been spent towards prime numbers.
> >
> > The history of ECC is quite short. The history of post-quantum prime
> > is even shorter.
>
> The history of both RSA and ECC is quite short.  Prime numbers
> indeed go back to antiquity, but deep insights into prime number
> theory start with Fermat, and the foundations of modern number
> theory are laid by Euler, Legendre, Gauss and Dirichlet in the late
> 18th and the 19th century.  Analytic number theory and group theory
> are both 19th century advances.  Around the same time Elliptic
> curves are studied by Abel and Weierstress in the 19th century.


It's also important to note that both RSA and ECC use prime numbers
(specifically prime fields in the latter's case). In many ways I think they
can be seen as special cases of a single bigger problem (which I believe is
the hidden subgroup problem, correct me if I'm wrong)

When it comes to cryptanalysis, the real question is the amount of work
that has gone into breaking things like the RSA trapdoor function as
opposed to just "prime numbers" or factoring. Both ECC and RSA use prime
numbers, and the discrete logarithm problem has probably received a similar
amount of study to factoring (and indeed there are spooky similarities
between these problems too).

-- 
Tony Arcieri
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