# NP-Completeness¶

NP-completeness is a concept from computational complexity theory, which is a field that studies the difficulty of solving different computational problems. To explain it in a way that avoids heavy math jargon, though still rigorous, let’s break it down step by step.

Key Concepts:

**Decision Problems**: These are problems that have a simple “yes” or “no” answer. For example, “Is there a subset of numbers that adds up to a target value?”**P (Polynomial time):**A class of decision problems that can be solved quickly (in polynomial time) by a computer. “Quickly” here means that the time it takes to solve the problem scales reasonably as the problem gets bigger (e.g., finding a path in a graph).**NP (Nondeterministic Polynomial time):**A class of problems for which, if someone gives you a solution, you can verify that it’s correct quickly (in polynomial time). You might not be able to solve the problem quickly, but if you had a solution, checking if it’s right is fast. For example, given a solution to a Sudoku puzzle, it’s easy to check if it’s correct, even though solving the puzzle might take a lot of time.**NP-complete:**These are the hardest problems in NP. If you can solve an NP-complete problem quickly (in polynomial time), then you could solve all NP problems quickly. Think of NP-complete problems as a “core” or “representative” set of the most difficult problems that still fit into the NP class.

Example of an NP-complete problem: The traveling salesman problemâ€”finding the shortest route that visits a list of cities exactly once. It’s easy to check if a proposed route is the shortest, but finding that route in the first place is tough.

## Why NP-completeness is important¶

NP-complete problems are important because they define the boundary between problems that are “easily solvable” and those that seem to be hard. Many cryptographic proofs, like Zero-Knowledge (ZK) and systems like SNARKs and STARKs, build on the fact that certain NP-complete problems are hard to solve, which provides security in cryptographic systems.

## Connection to ZK proofs¶

ZK proofs often deal with NP problems because we want to verify that someone knows a solution to a problem without revealing what the solution is. These proofs take advantage of the fact that while solving an NP-complete problem might be hard, verifying a solution is easy.

For example, in a ZK proof, you might want to prove you know the solution to an NP-complete problem without revealing the solution itself, but instead just prove you can verify it efficiently.

## Putting it together with STARKs/SNARKs¶

- SNARKs and STARKs are cryptographic tools that allow for verifying computations efficiently without needing to go through the entire computation process again.
- They are designed to handle NP-complete problems in a way that allows proof without direct computation, which ties back to the hardness of solving these problems while taking advantage of the ease of verification.

To sum up, NP-completeness is about problems where verifying a solution is easy, but finding the solution is hard. In cryptography, we often use the fact that some problems are NP-complete to make systems secure, since solving the problem is hard, but checking the solution is easy, which is a key idea in Zero-Knowledge proofs, SNARKs, and STARKs.