it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture"). The modularity theorem involved elliptic curves, which was also Wiles's own specialist area.[15][16]

From the Introduction to Ed25519, there are some speed benefits, and some security benefits. One of the more interesting security benefits is that it is immune to several side channel attacks: No secret array indices. The software never reads or writes data from secret addresses in RAM; the pattern of addresses is completely predictable. The software is therefore immune to cache-timing attacks, hyperthreading attacks, and other side-channel attacks that rely on leakage of addresses through the CPU cache. No secret branch conditions. The software never performs conditional branches based on secret data; the pattern of jumps is completely predictable. The software is therefore immune to side-channel attacks that rely on leakage of information through the branch-prediction unit. For comparison, there have been several real-world cache-timing attacks demonstrated on various algorithms. http://en.wikipedia.org/wiki/Timing_attack

More "sales pitch" comes from this IETF draft: While the NIST curves are advertised as being chosen verifiably at random, there is no explanation for the seeds used to generate them. In contrast, the process used to pick these curves is fully documented and rigid enough so that independent verification has been done. This is widely seen as a security advantage, since it prevents the generating party from maliciously manipulating the parameters. – ATo Aug 21, 2016 at 7:25