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Elliptic Curve
Cryptography relies on the difficulty of solving mathematical problems, such as the factor of large integers composed of two large prime numbers and the discrete logarithm of a random elliptic curve. This module provides low-level features of the latter. Elliptic Curve protocols can provide the same security with smaller keys.
There are 2 forms of elliptic curves, Fp and F2^m. These curves use irreducible
trinomial or pentanomial. Being a generic interface to a wide range of algorithms,
the curves are generally referenced by EcGroup. There are many built-in groups
found in Nid.
OpenSSL Wiki explains the fields and curves in detail at Elliptic Curve Cryptography.