Is there a way to define modulo division (or functions of modular arithmetic in general) as superposition of (elementary?) functions?
For example, the multiplication is first introduced as summation, the exponentiation - as multiplication, etc. The trigonometric functions can be expressed through other elementary functions. Moreover, a wide range of "continuous" functions can be expressed as infinite series of different nature, whether we are talking about complete set of functions in $L_p$ space, or about derivatives of infinitely differentiable function combined in Taylor series.
I wonder if it is possible to define functions from modulo arithmetics as closed-form expressions of functions which belong to classes listed in the table below. I have no doubts that it is possible to do so via infinite series, although I do not know how. However, I am particularly interested in closed-form expressions from the table
I find this table from this wikipedia article extremely interesting, although it is devoted to the closed-form expressions only and missing modulo arithmetic.