P versus NP (polynomial versus nondeterministic polynomial)

P versus NP (polynomial versus nondeterministic polynomial) refers to a theoretical question presented in 1971 by Leonid Levin and Stephen Cook, concerning mathematical problems that are easy to solve (P type) as opposed to problems that are difficult to solve (NP type).

Any P type problem can be solved in "polynomial time." (A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient.) A P type problem is a polynomial in the number of bits that it takes to describe the instance of the problem at hand. An example of a P type problem is finding the way from point A to point B on a map. An NP type problem requires vastly more time to solve than it takes to describe the problem. An example of an NP type problem is breaking a 128-bit digital cipher. The P versus NP question is important in communications, because it may ultimately determine the effectiveness (or ineffectiveness) of digital encryption methods.

An NP problem defies any brute-force approach at solution, because finding the correct solution would take trillions of years or longer even if all the supercomputers in the world were put to the task. Some mathematicians believe that this obstacle can be surmounted by building a computer capable of trying every possible solution to a problem simultaneously. This hypothesis is called P equals NP. Others believe that such a computer cannot be developed (P is not equal to NP). If it turns out that P equals NP, then it will become possible to crack the key to any digital cipher regardless of its complexity, thus rendering all digital encryption methods worthless.

This was last updated in September 2005

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