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The Million Dollar Issue That Might Break Cryptography

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P versus NP

Usually, you can confirm a problem’s solution. Math teachers advise you to check your work using your solution in every math lesson in school, whether it be by substituting multiplication for division or by putting the result into a variable.

What if you can simply check a solution? Is it equally simple to find a solution for that solution?

This is the P vs NP problem, which qualifies for the millions of dollars Prize if it is successfully solved.

What are P vs NP?

The effectiveness of algorithms is crucial in computer science.

Most algorithms are regarded as “quick” if they can be solved in what is known as polynomial time.

When a problem can be solved in steps increased by a factor of a polynomial depending on the complexity of the input, this is known as polynomial time.

Therefore, if the complexity of the input is n, a polynomial time algorithm can solve the issue in n^k steps.

P vs NP basically asks the following question: Are problems whose solutions can be verified in polynomial time likewise solvable in polynomial time?

NP- Completeness

An Euler Diagram showing the cases for NP-Completeness for P ≠ NP and P = NP. Credit: Behnam Esfahbod, Wikimedia Commons (CC-BY 3.0)

NP-complete issues are one of the most well-known subproblems. It is possible to easily verify NP-complete issues and use them to simulate every other NP-complete problem. To solve P vs NP, it is therefore much enhanced if one of these problems can be solved in polynomial time. These puzzles range from well-known theoretical issues like the travelling salesman problem to games like Battleship and the best way to solve a NxNxN Rubik’s Cube. A generic solution for NP-complete issues can be discovered if one of these problems has a solution.

Impact

There could be significant drawbacks and advantages if P is shown to equal NP.

Public-key cryptography would be completely altered, and many cryptographic algorithms could be broken, making cybersecurity a major problem.

However, there would also be advancements in research of protein structure predictive model and overall computing due to better integer programming as well as the solving of the travelling salesman problem.

There would be almost no disadvantages or advantages if P is shown not to equal NP.

A generic solution to every NP-complete problem would then receive less attention from researchers, but little would really change.

Closing

The crucial unsolved computer science problem of P vs NP could have major consequences.

Despite the fact that there is a great consensus that P is not equal to NP, any publicly acknowledged proof would surprise the scientific community.

The findings were published on Clay Math.

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