Books

• Introduction to Modern Cryptography by Jonathan Katz and Yehuda Lindell
• Applied Cryptography by Dan Boneh and Victor Shoup
• Introduction to Modern Cryptography by Mihir Bellare and Phillip Rogaway
• An Intensive Introduction to Cryptography by Boaz Barak
• Yevgeniy Dodis - Lecture Notes

Introduction

Let’s say there’s some tree with a certain number of nodes. Take a friend and blindfold them. With probability 50% take off a leaf from the tree. Tell your friend…something. I didn’t get it.

Determining that two graphs are isomorphic is (at worst) a quasi polynomial problem. ($n^{\mathrm{polylog} n}$)

Let’s say an infinitely powerful prover states that two graphs are isomorphic. It’s trivial to verify it.

What if it says they’re not. It’s not very trivial to verify it.

Interactive Proofs

P suggests $G_0$ and $G_1$ are not isomorphic. V comes up with some permutation $\Pi$ and a bit $B$.

V sends some permutation of $G_0$ and $G_1$, and asks if it’s isomorphic to one of $G_0$ or $G_1$. Over time, it repeats this process and becomes more convinced it’s correct.

Soundness vs Completeness

For some language $L$, a cheating prover cannot convince V except with some negligible probability.

https://en.wikipedia.org/wiki/Probabilistically_checkable_proof