The recent ‘Logjam’ attack shows that a well-funded intelligence agency might be able to crack 1024-bit Diffie Hellman keys (at least if the same group is used by many systems). When using RSA, cracking 1024-bit keys may not be beyond the most powerful adversaries either.
There are two solutions to this problem.
The first is to simply use longer keys. There is nothing wrong with algorithms like RSA and Diffie-Hellman in themselves: the maths isn’t broken. It’s just that as our computational power increases, we need to use longer keys to provide the same level of security; we could just do that.
The second solution is to use a different kind of cryptography, where that level of security is provided by shorter keys. There is such a thing—it's called elliptic curve cryptography and it uses some pretty advanced maths.
Advanced maths tends to impress people (I know, as I was a mathematician one day, and it always made me feel rather awkward). But when it comes to cryptography, it is actually a big weakness.
Ideally, we would use encryption algorithms that could be easily understood by anyone who could do a bit of programming. That would, for instance, mean that a lot of people would be able to find bugs in crypto libraries, whether they have been inserted accidentally or deliberately (“backdoors”).
The curious case of Dual_EC_DRBG – a random-number generator using elliptic curve cryptography that is generally assumed to contain an NSA backdoor – is a good example of why this matters. If the algorithm had used more basic maths, the backdoor (which was warned against as early as 2007) may have been known beyond a few cryptographers, which could have prevented the wide scale use of the algorithm.
And did you know that the elliptic curve you likely use to connect to a web server using HTTPS contains some unexplained variables?
If you’re really paranoid (which in this case, I don’t think you should be), this could be evidence of a deliberately inserted weakness that exploits a not publicly known attack.
I do think that, on balance, the advantage of the shorter keys outweighs the disadvantage of the advanced maths. But, as elliptic curve cryptography is being adopted widely, we really need more people to study the maths.
