Is logging a valid word do describe the process of authenticating in an computer system?

Considering that the controller controls the login process, I would go for LoginController. In software developement a Logging Controller sounds like a controller responsible for managing logs (system outputs). A LoggingInController would convey the same meaning as a LoginController, since a login describes the action of logging in, but the simplicity of a simple LoginController makes it a more obvious choice.

Notice the difference between log in (a verb) and login or log-in (a noun).

to Log in: to connect to a computer system by putting in a particular set of letters or numbers

log in: the set of letters and numbers that you write on a computer screen in order to connect to a computer system

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edit: To answer the question directly, yes "logging" is a valid word to use. You generally "log in" but while you are in the process of that act it would be called "logging in".

In the case of how you know logging to be used, it is the verb form of the noun log as in a segment of a tree. Computing borrowed the term "log" in reference to logbooks that are used to track events. Originally when you would "log in" or "log on" (which historically was two different things but nowadays "log on" is not used as much) this action was recorded much like in logbooks since administrators needed to manage time that users were on the system. Logging also refers to any output that is generated internally within the system rather than being sent back to the user and is often stored in a "log file" which is often shortened to just "log" or the plural "logs."

Just in case anyone's interested in "in" versus "on" it referred to how much access was given to a system. Logging "on" meant that you had specific actions you could make on the system where as logging "in" meant that you had root (admin) access and could perform any function. Somewhere along the line "in" pretty much became the standard and the distinction blurred. When terminating a session where you logged in you would log out and where you logged on you would log off.