RSA cryptography question

Question

RSA user Alice has a public key ($n_A=pq$,$e_A$), where $p$ and $q$ are different primes such that the least common multiple $l$ of $p-1$ and $q-1$ is relatively small (i.e. $l$ is close to max($p-1,q-1$)). Explain why, even without factoring $n_A$, a cryptanalyst would quickly find a decryption key $d_A$ for Alice's public key.

Help is greatly appreciated. :)

Answer 1

Without giving away the answer (so you can explore it), here is a huge hint:

• Fermat's "Difference of Squares Method"