Function for encryption/decryption - What is $n \phi(n)$?

Question

In my notes there are the following functions of encryption/decryption:

$$E_k(x)=x+k$$

$$D_k(y)=y-k$$

($E_k : \mathbb{Z}_n \rightarrow \mathbb{Z}_n$)

($D_k : \mathbb{Z}_n \rightarrow \mathbb{Z}_n$)

$$E_k(x)=ax+b, a\in \mathbb{Z}_n^{\star} , b \in \mathbb{Z}_n$$

$$D_k(y)=a^{-1}(y-b)$$

($E_k : \mathbb{Z}_n \rightarrow \mathbb{Z}_n$)

($D_k : \mathbb{Z}_n \rightarrow \mathbb{Z}_n$)

where $k$ is the key, $k\in \mathbb{Z}$

$n \phi(n)$

$$y_1=ax_1+b \\ y_2=ax_2+b$$

$$\Rightarrow y_1-y_2=a(x_1-x_2) \\ \Rightarrow a=(y_1-y_2)(x_1-x_2)^{-1}$$

For example $$y_1=a(x+1)+b \\ y_2=ax+b$$

Could you explain to me what $n \phi(n)$ is??

In my notes there is only the notation, there isn't any explanation for what it stands...

Answer 1

Maybe this will helpful Euler function, page 4. I have the course of cryptography so this function seems to me as an Euler function.

Answer 2

I guess the idea is that in the second example the key is the pair $(b, a)$, where $b \in \mathbb{Z}_{n}$, and $a \in \mathbb{Z}_{n}^{\star}$, that is, $a$ in an invertible element in $\mathbb{Z}_{n}$.

Now $\mathbb{Z}_{n}$ has $n$ elements, and $\mathbb{Z}_{n}^{\star}$ has $\varphi(n)$ elements, where $\varphi$ is Euler's $\varphi$ function, so all in all you have $n \, \varphi(n)$ keys here.