“If you just want the gist, here’s the TL;DR version: [Elliptical Curve Crytography,] ECC is the next generation of public key cryptography, and based on currently understood mathematics, it provides a significantly more secure foundation than first-generation public key cryptography systems like RSA. If you’re worried about ensuring the highest level of security while maintaining performance, ECC makes sense to adopt. If you’re interested in the details, read on.”
—Nick Sullivan. ” A (relatively easy to understand) primer on elliptic curve cryptography.” Ars Technica. October 24, 2013.