I was looking into the predictions by some researchers that RSA and Diffie-Hellman may not be secure in the next few years due to advances in math and being able to calculate the discrete logarithm problem in new ways no longer making them "impossible" to solve backwards (Math Advances Raise the Prospect of an Internet Security Crisis). Would this cause problems with Elliptic Curves? I know they are based on different principles, but could continued advances in this area compromise ECC? Can we consider ECC a replacement if RSA and DH were compromised?
A couple things:
- This article is two years old, so take its predictions with a grain of salt.
- In the two years that have elapsed, the predicted advances have not materialized, and there is little indication they will soon.
The core of those arguments was Joux's 2013 result on the discrete logarithm problem in finite fields of small characteristic. Those methods do not directly apply to any standard cryptographic primitive like RSA or DH, and it is not known how to translate Joux's algorithm into an attack on these primitives. The underlying math is fundamentally different.
It's important to be cautious and conservative when choosing cryptographic primitives, but people have been predicting the death of RSA and DH for many years, and they've lasted this long.
Again, it's not known how to translate Joux's methods into a general-purpose algorithm for the discrete log problem in any other group. Thus, while it's possible for his attack to work in elliptic curve groups, there simply isn't any indication that this is the case.
It can affect several classes of Elliptic curves. In particular, if this curves have pairings or any other way to translate the problem to a finite field (or extension finite field). The tricky part about ECC is that if you do not pick your Elliptic curve properly, you might end up with a weak implementation. Or, with an implementation in which the problem can be reduced to a finite field.
With math and computational advances, for protecting systems we should increase key size. We know that recommended key size for elliptic curves is approximately $2s$ bit, which $s$ is a desired security level.Today $s=80$, but with the advances of math, $s$ is increasing. $s=96$ is believed to provide protection until $2020$. For more detail you can see "Yearly report on algorithms and key sizes".