1) Elliptic curve cryptography uses elliptic curves over finite fields to provide a method for constructing cryptographic groups. 2) The security of elliptic curve cryptography relies on the difficulty of solving the elliptic curve discrete logarithm problem. 3) Elliptic curve cryptography provides the same security level as other cryptosystems like RSA but with smaller key sizes, making it advantageous for applications with limited computational power or space.