RSA How to find the private key to a given public key when n doesn't consist of two primes?

Question

I have the homework to find to the public key (120, 3) the private key. I guess 120 is n and 3 will be e. So that $\lfloor\sqrt{120}\rfloor=10$ I can't find a matching prime number. So it gets more complicated. I also know $\phi(n)=(p-1)(q-1)$ and $3\cdot d\equiv 1\mod (p-1)(q-1)$ But here I stuck, could somebody help me from this point on?

Answer 1

I also know $\phi(n)=(p-1)(q-1)$

This actually only holds if $n=pq$ and if $p$ and $q$ are both primes.

When $n$ doesn't factor as nicely you'll need the more general definition of $\phi(n)$ which can be computed from the following three axioms (given the prime factorization of $n$):

• If $\gcd(n,m)=1$ for any $n,m$ then $\phi(n\cdot m)=\phi(n)\phi(m)$
• If $p$ is prime and $k\geq 1$ then $\phi(p^k)=p^{k-1}(p-1)$

So in your case $$\phi(120)=\phi(2^3\cdot 3\cdot 5)=\phi(2^3)\phi(3)\phi(5)=(2^2\cdot1)\cdot2\cdot4=32$$

does multi-prime RSA make any sense in practical use?

Yes, using more than two primes can make sense if you use the chinese remainder theorem (CRT) which yields a speed-up of $k^2/4$ for $k$ primes compared to using only $k=2$. See fgrieu's excellent answer for a discussion of why one wants that and what one has to look out for when actually deploying multi-prime RSA and the table in DW's answer to the same question for an overview of how many primes to use for each modulus size.