What is elliptic curve cryptography and why is it important to Monero?
Please provide both a basic description and an in depth description of the type found on Moneropedia
Elliptic curve cryptography (ECC) is a form of public-key cryptography where mathematical properties of elliptic curves are used to ensure security of cryptographic methods used. This allows to perform the following in a secure way:
It is important to Monero, because it facilitates important operations such as key-pair creation (used for generating Monero addresses), Elliptic curve Diffie-Hellman (used in the Monero stealth-address scheme), and signing and verifying transactions. The curve used by Monero is one of the SafeCurves, and is the Curve25519, widely used for many other secure applications.
For further reading on ECC, here's my go-to link.