Elliptic Curve Cryptography (ECC) can be used as a tool for encrypting data, creating digital signatures, and
performing key exchanges. Regarding the encryption capability, the first schemes that appeared were either versions of
already existing public-key algorithms (Massey-Omura and ElGamal) or new schemes specified ad hoc (Menezes-
Vanstone). However, all the initial elliptic curve encryption schemes had similar problems, and thus were conveniently
discarded by the academic community. The encryption schemes currently used are known as hybrid cryptosystems, as
they use both symmetric and asymmetric techniques. Among those hybrid cryptosystems based on ECC, the best known
ones are the Elliptic Curve Integrated Encryption Scheme (ECIES), the Provably Secure Elliptic Curve encryption scheme
(PSEC), and the Advanced Cryptographic Engine (ACE).
In this work, we present an extensive review of the basic concepts of elliptic curves, the initial ECC encryption algorithms, and the current ECC hybrid cryptosystems. After that, we provide a comprehensive comparison of ECIES, PSEC, and ACE, highlighting the main differences between them. Finally, we conclude that, with the available data, it can be stated that ECIES is the best ECC encryption scheme from a performance and ease of implementation point of view.