# The Million Dollar Problem That Could Break Cryptography

### P versus NP

Typically, you can review a solution to a problem. Whether you’re using multiplication for division or substituting the result for a variable, math teachers tell you to check your work against your result in every high school math class.

But suppose you can easily verify a solution, is it just as easy to solve that solution?

This is the P-NP problem, a Millennium Prize problem where the solver gets a million dollars if a valid proof is provided.

#### What is P vs NP?

In computer science, the efficiency of algorithms is very important. Most algorithms are considered “fast” if they can be solved in a standard called polynomial time. Polynomial time is when a problem is solvable in steps scaled by a factor of a polynomial, given the complexity of the input. So let’s assume the input complexity is a number n, a polynomial time algorithm can solve a problem in n^{k} Steps.

Essentially, P vs NP asks the question: are problems whose solutions can be verified in polynomial time also be solved in polynomial time?

#### NP-completeness

One of the most well-known subproblems are NP-complete problems. NP-complete problems are those that can be verified quickly and can be used to simulate any other NP-complete problem. Therefore, solving any of these problems in polynomial time is a big boost to solving P versus NP. Some of these problems include games like Battleship and the optimal solution for an NxNxN Rubik’s Cube, but also famous theoretical questions like the traveling salesman problem. If a solution to any of these problems is found, then a general solution to NP-complete problems can also be found.

#### influence

If P is shown to be equal to NP, this could have serious consequences and benefits. Cyber security would be a major concern as public key cryptography could be turned on its head and many ciphers could be cracked. However, there would also be improvements in protein structure prediction research and overall computation due to better integer programming and the solution to the traveling salesman problem.

If P is demonstrably not equal to NP, there would be almost no disadvantages and advantages. Researchers would then focus less on a general solution to all NP-complete problems, which wouldn’t really change much.

#### Conclude

P vs NP is a critical unsolved problem in computer science that could have drastic implications. Although the great consensus is that P does not equal NP, any commonly accepted proof would shake the scientific world.

References:

- “The P vs NP Problem” by Stephen Cook, 2001, claymath.org
- “Computers and Recalcitrance: A Guide to NP-Completeness Theory” by Michael Garey, David S. Johnson, 1979, ISBN 0-7167-1045-5

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