Quadratic Congruences on Average Case

2018-04-10T01:41:54Z (GMT) by Frank Vega
P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? This question was first mentioned in a letter written by John Nash to the National Security Agency in 1955. However, a precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is NP-complete. To attack the P versus NP question the concept of NP-completeness has been very useful. The Quadratic Congruences is a known NP-complete problem. We show this problem can be solved in polynomial time for the average case. It is true that Hamilton cycle and some NP-complete problems could be solved in average case over inputs. However, this algorithm, in the same way as Quicksort, is polynomial for a large amount of inputs because of the infinite set of elements that cannot be solved in polynomial time is infinitesimal.