For example, we have a definition of a cryptosystem:
We define a cryptosystem as tuple $\langle M, K, C\rangle$ where $M$ is a set of texts that are not ciphered, $K$ set of keys, $C$ set of texts that are ciphered. We also define two operations $$ e(\cdot|K_e) :M\rightarrow C$$ $$ d(\cdot|K_d): C \rightarrow M$$ Where $$\langle K_e, K_d\rangle \in K\ \mbox{such that}\ d(e(M|K_e)|K_d)=M$$
Well, definition is irrelevant, what I am interested is notation used to define functions, I haven't seen it before. How do I need to interpret the part in parenthesis $(\cdot|K_e)$? Is it simply a two-argument function? Does the dot signify anything besides "here goes an argument from a set $M$"?
Putting the $\cdot$ allows you to consider it as a one variable function, as an example let $f(x,y):\mathbb R^2\to\mathbb R$ now $$f(\cdot,y):\mathbb R\to\mathbb R$$ $$f(x,\cdot):\mathbb R\to\mathbb R$$
To give an example of how this can be useful, lets consider the map $B:H\times H\to \mathbb R$ (the spaces are irrelevant), but $B$ is a Bilinear form if both $$B(\cdot,v):H\to\mathbb R$$ and $$B(u,\cdot):H\to\mathbb R$$ are linear maps.
Sadly I don't know much about cryptography, but generally $|$ is a mathematical notion for "given", i.e. $P(A|B)$ is the probablility of $A$ given $ B$ has happened.
In this case we have $e(⋅|K_e):M→C$, so it seems that the input is irrelevant, because you already have enough information (this is just my guess of what this means)