RSA How to select a value K such that $e d = 1 + k \varphi(N)$ holds I am learning RSA cryptography. The part I am stuck on is understanding how k and the public exponent $e$ is selected.

Given the formulas;

Decrypting: $c^d \bmod N = (m^e)^d \bmod N$
Which is equal to m the message

Encrypting:
$m^e \bmod N = c$

My question is when generating the private exponent how do we find what the public exponent should be and what k should be in the formula

$$d = (1 + k*\varphi(N))/e$$

I understand it has something to do with the inverse of modules but I don
t get the math behind it.
You apply the extended Euclidean algorithm to $e$ and $\phi(N)$ (which have to have gcd equal to $1$) and we get $x,y \in \mathbb{Z}$ such that

$$xe + y\phi(N) = 1$$

The $x$ (taken modulo $\phi(N)$, if needed) is the $e$ you are looking for. The $k$ is totally irrelevant for encryption/decryption, but it's actually the $y$ in the above equation (rewrite your equation and see).