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I know that the algorithm used to generate the Brainpool curves and the NIST curves is published. The algorithm should be this one. From what it looks like it's rather slow to find secure parameters.

After some research I've also found this paper (in fact it's a PhD thesis I think) on the generation of secure elliptic curves. I've also googled around and found a tool that actually generates such curves (using this algorithm I think). From an efficiency point of view this algorithm looks faster than the NIST's one.

Now my question:
Are the above two algorithms the only ones (published) for generating safe elliptic curves or are there other ones?

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Of course there are others. Of interest might be the paper 'Efficient ephemeral elliptic curve cryptographic keys' by Mieli and Lenstra, which claims to generate fresh Elliptic Curves sufficiently quickly that they can be created on the fly for a single ECDH exchange, and then discarded.

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really nice and recent paper :) May I ask if this was pre-Logjam? –  SEJPM Jun 16 at 19:01
    
@SOJPM: I suspect that the ideas within the paper predated Logjam; however the paper itself is obviously recent (it references a paper published on ePrint in April). –  poncho Jun 17 at 2:59

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