$\varphi(n)$ is a [multiplicative function][1]: it is computed by the formula

$$ \varphi(n) = n \prod_{p \mid n} \frac{p-1}{p} $$

or something equivalent. This is basically the *only* method of computation known that remains feasible when $n$ is not small.

Thus, if $N$ is the modulus you want to use for RSA, you need to know its prime factorization so that you can compute $\varphi(N)$. And you don't want anyone else to know it's prime factorization, otherwise *they* could compute $\varphi(N)$.


  [1]: https://en.wikipedia.org/wiki/Multiplicative_function