In everyday life, we are constantly required to prove things. For example, whenever you sign into your email account, you prove to the email server that you know your loging password. These proofs are very different from the proofs we saw in elementary school (e.g., of Pythagoras’s theorem) – they might involve interaction, randomization, and are often required to be super efficient in terms of communication and\or computation. Cryptographic proof systems guarantee that false claims are accepted only with small probability, while also offering other features that are not possible for standard proofs, such as maintaining the secrecy of the prover’s private information, allowing super-efficient verification of the proof, and more.