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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

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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:

  • Public key encryption, where the sender can encrypt a message using the recipient's public key, and the recipient can decrypt it using his private key.
  • Signature, where the signer can sign a message using his private key, and anyone can verify the signature using the signer's public key.
  • Shared secret key, where any 2 parties can securely agree upon a shared secret key, which is then normally used as the key for symetric encryption to encrypt any further messages between the 2 parties.

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.

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