# Efficient way of knowing large factors of $\phi(n)$ given small prime factors and $n$

Knowing large prime factor$$(r > n^{1/4})$$ of $$\phi(n)$$ can easily factorize n and hence learn $$\phi(n)$$.

If we have knowledge on all small prime factors $$(2< r_i << n^{1/4})$$ of $$\phi(n)$$ then, is there any efficient way to factorize $$n$$ and $$\phi(n)$$?

Note: $$n$$ is an RSA number

• What is pi(n)? Normally $\pi(n)$ denotes the number of primes $\le n.$ Do you mean $\varphi(n)?$ Since a RSA number $n$ is often written as $n=pq,$ it would be less confusing if you use another symbol for your factor. Sep 27, 2018 at 10:17
• In general, no; if $p, q$ are both safe-primes, then the product all the small prime factors of $\phi(n)$ is $4$; that knowledge doesn't help very much... Sep 27, 2018 at 17:53