Is it easy to factorize a number of the form $n = t^{2} \cdotp p$?

Question

Is it easy to factorize a number of the form $n = t^{2} \cdotp p$, where $t$ and $p$ are large primes?

Answer 1

Integers of the form $n = p\cdot q^2$, $p$ and $q$ prime, are the basis of a few public-key cryptosystems, including ESIGN, Okamoto-Uchiyama, cryptosystems based around the hardness of finding the class number of quadratic fields. As such, being able to efficiently factor numbers of this form would immediately break a number of cryptosystems. This would be a big deal. Additionally, this is also related to a classic problem in number theory, which would be of independent interest.

As far as concrete factorization algorithms targeting this type of integers, there are various strands:

• Peralta and Okamoto slightly optimized ECM against this form, but the result is not significantly superior to regular ECM.
• Boneh, Durfee, and Howgrave-Graham used Coppersmith-style root-finding techniques to factor $n$ in approximately $n^{1/9}$ operations when $p$ and $q$ are similarly-sized. While this may sound impressive, it is not better than the number field sieve for numbers of cryptographically-relevant size.
• Castagnos, Joux, Laguillaumie, and Nguyen observed that, given some extra information about $n$ it becomes possible to factor $n$ in polynomial time. But this is not the general case and only applies to special cases, like NICE.