Elliptic Curve Cryptography: Key Exchange (ECDH), Digital Sig (ECDSA/EdDSA) and Encryption (ECIES)

Elliptic Curve Cryptography (ECC) was co-invented by Neal Koblitz and Victor Miller and scaled our discrete log method (g^x) into a multiplicative space (x.G). So the DH (Diffie-Hellman) key exchange method became ECDH and DSA (Digital Signature Algorithm) become ECDSA, and with a tweak, we can also encrypt with ECIES.
Some links on background theory and practice are:
asecuritysite.com/ecc
asecuritysite.com/ecdsa
asecuritysite.com/eddsa
asecuritysite.com/ecies

[@hanskessock3941]

I'm not sure I understand how the symmetric key derived in the ECIES explanation is hidden from Eve. The only secret used by Alice is the random number R, and all the other information she uses is computable by Eve because it only requires G and Bob's public key. You say that Alice sends R to Bob, meaning Eve sees R. Eve has everything that Alice used to derive the key. Did I miss something in your explanation?

[@BillBuchanan]

In ECC, if we have R=r.G, it is extremely difficult to discover r, even if we have R and G.