<Li> "All instances of an NP - complete problem are difficult ." Often some instances, or even most instances, may be easy to solve within polynomial time . However, unless P = NP, any polynomial - time algorithm must asymptotically be wrong on more than polynomially many of the exponentially many inputs of a certain size . </Li> <Li> "If P = NP, all cryptographic ciphers can be broken ." A polynomial - time problem can be very difficult to solve in practice if the polynomial's degree or constants are large enough . For example, ciphers with a fixed key length, such as Advanced Encryption Standard, can all be broken in constant time (and are thus already in P), though with current technology that constant may exceed the age of the universe . </Li> <P> Viewing a decision problem as a formal language in some fixed encoding, the set NPC of all NP - complete problems is not closed under: </P> <Ul> <Li> union </Li> <Li> intersection </Li> <Li> concatenation </Li> <Li> Kleene star </Li> </Ul>

Examples of p np np complete np hard problems