In computational complexity a fairly popular claim seen in most textbooks equates class P with practically feasible algorithms, and everything not known to be P (but usually within NP, "not known to be P" is an awkward thing we need to say until P!=NP is finally proven, even though we all know it to be true already) with not practically feasible algorithms. While it's often noted that it might be untrue for some cases, with some P algorithms being impractical, and some not known to be P algorithms to be practical, such disclaimers usually consider these cases to be contrived and not worth any further consideration.
Here's a really simple example of a genuinely useful problem in P (linear even) that's not practical at all.
Utterly impractical O(n) problem
Imagine problem of falsifying digital signature for a message in some reasonable digital signature scheme. Inputs are public key of constant length, and message of length n. The algorithm is as follows:
- Extract private key from the public key by brute force, or some other algorithm (constant time)
- Sign the message (linear time in n required just to read the message, and no more than linear time necessary)
Yet, unless someone breaks the public key system and provides vastly faster method of extracting private key based on the public key, we won't be able to solve it even for n=1.
This problem is of great practical importance - the entire online banking system would fail, and billions of dollars of fraud would happen if someone could practically solve it. So it's not fair to call utterly unfeasible P problems contrieved.