Elliptic Curve Cryptography (ECC) is the powerful method that protects the value in the Bitcoin network.  It uses special arithmetic  to produce the familiar private/public key pairs  used in Bitcoin and other cryptocurrencies, and provides a very clever method of allowing someone to sign a message using their private key, without ever revealing their private key to the verifier.  This is the link to one episode of James De Angelo’s wonderful Bitcoin 101 blackboard series, where he explains how the process works.