I'm a RSA n00b when it comes to the mathematics for RSA. After spending some time reading, watching lectures, etc. I pretty much have everything down, except for how to figure out the equation for determining the private key $d$.
Reading this very Q/A forum, a person said, "There are better ways to find $d$ from $e$ if you know $\varphi(n)$. But if you don't, you're in trouble, because you need to factorize $n$ to do that."
Taking that comment to heart, what is the "better way" to find "$d$" if you know $\varphi(n)$?
$m=42$ Message to be encrypted
$p=61$ Prime 1
$q=53$ Prime 2
$e=17$ Random Exponent greater than 2
I know that $m^e = c \mod n$ is to encrypt and you get $2557$. To decrypt the equation is $c^d = m \mod n$, but my hangup is figuring out $d$. Can somebody spell out the equation, process, etc on figuring out $d$?