Is there a theoretical bound on how fast a classical integer factorization algorithm (i.e., non-quantum algorithm) can be?
$$\begin{smallmatrix} {\text {Factorization method}} & {\text {Time complexity}} & {\text {Approx. discovery}} \\ {\text {Trial division}} & O(\sqrt n) & n.d.\\ {\text {Kraitchik}} & O(\sqrt n) & 1920 \\ {\text {Pollard $p-1$}} & O(p) & 1974\\ {\text {Pollard Rho}} & O(n^{1/4} ({\log n})^2) & 1975 \\ {\text {Morrison & Brillhart, CFRAC}} & O(n^{(2 + o(1)) \log \log n / \log n} ) & 1975 \\ {\text {Williams $p+1$}} & O(p) & 1982\\ {\text {Pomerance, Quadratic sieve}} & L_n\bigg[\frac 1 2, \sqrt[3] \frac {64} 9\bigg] & 1985 \\ {\text {Montgomery, Multipolynomial Quadratic sieve}} & ? & 1985 \\ {\text {Lenstra, Elliptic Curve method}} & L_p\bigg[\frac 1 2, \sqrt 2\bigg] & 1987 \\ {\text {Pollard, Number Field sieve}} & L_n\bigg[\frac 1 3, \sqrt[3] \frac {64} 9\bigg] & 1988 \\ {\text {Shor, Quantum factoring}} & O({(\log n)}^2 {(\log \log n)} ) & 1994 \end{smallmatrix}$$
$p$ refers to the smaller factor of $n$.
L-notation: L-notation is an asymptotic notation analogous to big-$O$ notation, denoted as $L_{n}[\alpha ,c]$ for a bound variable $n$ tending to $\infty$. It is defined as $$L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}$$ where $c$ is a positive constant, and $\alpha$ is a constant $ 0 \leq \alpha \leq 1$.
Shor's quantum factoring obviously shows a fast algorithm using a quantum computer. So, that establishes a bound. But are there any theoretical obstructions preventing a faster classical algorithm than what is known currently? If yes, please provide references.
References:
Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé, et al.. The State of the Art in Integer Factoring and Breaking Public-Key Cryptography. IEEE Security and Privacy Magazine, 2022, 20 (2), pp.80-86. ff10.1109/MSEC.2022.3141918ff. ffhal-03691141 (Accessed online. Feb 8, 2023): https://hal.science/hal-03691141