Wikipedia states that the time complexity of the [General Number Field Sieve (GNFS)][1] is
$$\exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\log N)^{\frac{1}{3}}(\log \log N)^{\frac{2}{3}}\right),$$
where $N$ is the number to be factored.

The same site also mentions that [quasi-polynomial time][2] is
$$\mbox{QP} = \bigcup_{c \in \mathbb{N}} \mbox{DTIME} \left(2^{\log^c n}\right),$$
where $n$ is the length of the input, in other words, $n=\log N$.

Now my question is whether GNFS is in quasi-polynomial time. It appears not, because of the existence of the $(\log N)^{1/3}$ exponent.

How about [sub-exponential][3]? Wikipedia says
$$\text{SUBEXP}=\bigcap_{\varepsilon>0} \text{DTIME}\left(2^{n^\varepsilon}\right).$$
I think the answer is still negative, because $\varepsilon=1/3$ in this case and it can’t be smaller as the intersection requires.

So the running time is exponential. Am I right?

**PS**

ChatGPT said the following when I asked:

> Is  factoring  in  quasi-polynomial  time?
>
> Yes, factoring is believed to be solvable in quasi-polynomial time. In 2019, a team of researchers led by Martin R. Albrecht proposed a new algorithm called the “Number Field Sieve in Quasi Polynomial Time” (NFQPT) for factoring integers. The algorithm has a time complexity of $\exp(O(\log(N)^{1/3}\log(\log(N))^{2/3}))$. This means that the algorithm runs in time that is quasi-polynomial in the number of digits of $N$, which is much faster than the current best-known algorithm for factoring, the General Number Field Sieve, which has a time complexity of $\exp(O(\log(N)^{1/2}\log(\log(N))^{1/2}))$ and is considered to be sub-exponential. However, it should be noted that NFQPT has not been implemented or tested extensively and is still in the research phase.

I couldn’t find the mentioned research result. Can you?

  [1]: https://en.wikipedia.org/wiki/General_number_field_sieve
  [2]: https://en.wikipedia.org/wiki/Time_complexity#Quasi-polynomial_time
  [3]: https://en.wikipedia.org/wiki/Time_complexity#Sub-exponential_time