Typically proofs don’t merely show that something is true, but they also show why it is true. If I wanted to demonstrate evolution to you, I would show you the fossil record; if I wanted to convict a murderer, I would exhibit a smoking gun; and if I wanted to prove a theorem to you, I would write down a rigorous mathematical argument that you could verify step by step. In essence, to really prove something, it seems like you must give some kind of explanation, the “why” or the “how.”
Related: Can God Be Proved Mathematically?
But it turns out that might not always be the case. Theoretical computer scientists began to challenge this notion in the 1980s by asking if it is ever possible to prove a claim without revealing any additional information beyond the mere truth of the claim. These are called zero-knowledge proofs, and they sound impossible. In a seminal 1989 paper, researchers demonstrated that they are very real.
Here’s how a zero-knowledge proof can help with my Waldo dilemma: I take an opaque cloth that’s much larger than the book, and I cover the page with it. I shift the book around under the cloth to a random location so you lose track of where it is. Then, I cut a small silhouette of Waldo’s face in the cloth so that you can see him (and only him) peeking through. You’re now convinced that I had indeed found Waldo, but that’s all you’ve learned. You already knew Waldo was somewhere on the page, and you have no clue what part of the book you’re looking at, so you’ve merely learned the truth of my claim—I know where Waldo is hiding.
Our second example will more closely resemble how zero-knowledge proofs typically work in theory and practice. For th
