Which cryptography approach is based on elliptic curves over finite fields?

Elliptic Curve Cryptography (ECC) is specifically designed around the mathematical properties of elliptic curves over finite fields. This approach utilizes the structure of elliptic curves to create secure cryptographic keys that are smaller than those used in traditional methods, yet equally secure. The main advantage of ECC is that it provides a higher level of security with shorter key lengths, making it efficient in terms of both speed and processing power. This characteristic is particularly beneficial in environments with limited computational resources, such as mobile devices or IoT systems.

In contrast, symmetric key cryptography relies on a single key for both encryption and decryption, while asymmetric key cryptography employs a pair of keys (public and private) but is not specific to elliptic curves. Hash functions serve a different purpose by producing fixed-size hashes from variable-length data to ensure data integrity, rather than being designed for encryption purposes. Thus, the unique feature of ECC being explicitly related to elliptic curves makes it the correct choice in this context.