P versus NP is defined in the Turing Machine:

Formally, the P is the class of problems that solved by a Deterministic Turing machine in polynomial a time. NP is the class of problems that were resolved by a non-deterministic Turing machine a polynomial time, which means that it can be solved by brute force algorithm in exponential time. We know that every P problem is also an NP one. In fact, the fast algorithm that solves P problems can be quickly authentication algorithm. The question is, can we find an NP problem which did not decide on a fast algorithm solve it. If so then NP is different from P. If no then NP is identical to P. 3