The Waldo example shows the concept, but to make zero-knowledge proofs work mathematically, we need circuits - a way to convert any secret knowledge into math that computers can verify.

Think of circuits as translating your claim into pure mathematics. Instead of "I know where Waldo is," you create math equations that are only true if you really know the secret.

How Circuits Work:

• Gates: Basic operations (addition, multiplication)

• Wires: Connect gates and carry values

• Constraints: Mathematical rules that must be satisfied

Every ZK proof system works the same way: convert your secret knowledge into a circuit, then prove the circuit works without revealing your secret inputs. This is the mathematical foundation that makes zero-knowledge proofs possible!