Check out the why and how of the Public Key Signature Scheme for the Cardano Network.
"The abstract goal is always the same. There exists a hard problem that cannot be solved unless someone has a secret piece of knowledge. The holder of this piece of knowledge is said to be the owner of the keypair and should be the only entity that has the ability to use it."
Also see how Cardano is planning on integrating quantum computer resistant signatures into their network with view of the successful development of quantum computers being a possibility in the future.
Check out Cardano's introduction to key signatures here
