[Cryptography] Why is ECC secure?

[Viktor Dukhovni]

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.

And yes, many of the major advances in the arithmetic of Elliptic
Curves are then made in the early to mid 20th centries.

Significant attention to and progress in factoring algorithms
(quadratic sieve, ECM, and GNFS) is quite recent.

So just because the concept of prime number dates back to antiquity,
while Elliptic curves do not, it is I think a false meme that
therefore we have a multi-millenium lead on understanding RSA vs.
ECC.  (It is amusing in this context to note the role of ECM in
factoring composites).

> Prime numbers came before modern machine-assisted cryptanalysis.

This seems irrelevant, elliptic curves date back to the mid 18th
century, so what?

-- 
	Viktor.