A Connection Between Breaking RSA and Factoring Integers

Nikita's public key is $(n,e)$. If we compute the factorization of $n=pq$, then we can compute $\varphi (n)$ and hence deduce her secret decoder number $d$.

It is no easier to $\varphi (n)$ than to factor $n$:
Suppose $n=pq$. Given $\varphi (n)$, it is very easy to compute $p$ and $q$. We have

$$\varphi (n) = (p-1)(q-1) = pq-(p+q)+1,$$

so we know $pq=n$ and $p+q = n+1 - \varphi (n)$. Thus we know the polynomial

$$x^2 - (p+q)x + pq = (x-p)(x-q)$$

whose roots are $p$ and $q$.

There is also a more complicated ``probabilistic algorithm'' to find $p$ and $q$ given the secret decoding number $d$. I might describe it in the next lecture.