Why do we focus on polynomial time, rather than other kinds of time? Polynomial time seems to be mentioned quite frequently on this site. It often forms a threshold between two possible outcomes like being secure or an attack's validity.  I know what $\mathcal{O}(n^c)$ means mathematically, but why do cryptographers not use other kinds of time for so many thresholds?

I don't think that I'm asking quite the same thing as What does &quot;running in polynomial time&quot; really mean?
The focus on polynomial time comes from cryptography's historical origin as a branch of computational complexity.


It makes sense for a theoretical field to develop technology-independent ways of measuring efficiency. Actual clock time or number of clock cycles are technology-dependent. Talking about running time in an asymptotic sense is convenient because it makes specific technology irrelevant. No matter how fast machine A is compared to machine B, if machine A runs a $\Theta(n \log n)$ algorithm and machine B runs a $\Theta(n)$ algorithm, then eventually (for large enough $n$) machine B will be faster in terms of actual time.
Algorithms that run in time $\Theta(n)$ are objectively fast/scalable/"efficient".
Suppose an "efficient" algorithm makes calls to a subroutine (we don't count the cost of the subroutine against this algorithm), and the subroutine can also be realized by an "efficient" algorithm. Then, intuitively, the overall system (accounting for both the calling program &amp; subroutine) should be "efficient" too. This is a basic composition property, and without it you have a very messy theory. Polynomial-time is the minimal way to define "efficient" that contains running time $\Theta(n)$ and enjoys this composition property.


It is for these reasons that "polynomial time" is synonymous with "efficient" in computational complexity. Its minimal nature makes it a natural and well-motivated definition.

However, just because it's well-motivated and natural doesn't mean it's sacred. Mihir Bellare points out asymptotic complexity as one of the arbitrary aspects of cryptographic research, to be critically examined. 

As others have said here, there is plenty of work that follows a concrete (rather than asymptotic) style. There is nothing wrong with working in this style. The main differences are: (1) things are a little messier; you can't hide all the mess inside an unspecified polynomial; (2) you forfeit the (very convenient) composability property.