OT On an integer factoring algorithm based on smooth class number of quadratic fields

[Georgi Guninski <gguninski@gmail.com>]

I asked this on matherovflow at
https://mathoverflow.net/questions/475285/on-an-integer-factoring-algorithm-based-on-smooth-class-number-of-quadratic-fiel

We got an algorithm and toy implementation of integer factoring algorithm
based on smooth class number of quadratic fields.

It is close to the elliptic curve factorization method (ECM) and
succeeds if it can find find $B_1$ smooth integers (all primes
factors are less than $B_1$).

Sketch of the algorithm:

For integers $a,b,c$, let $q$ be the binary quadratic form $Qfb(a,b,c)$
and define $D(q)=b^2-4ac$ (as defined in the pari/gp).

If $D(q)<0$, then $\{ q^k \}$ is abelian group of order
divisor of $h$ where $h$ is the class number of $\mathbb{Q}{\sqrt{D(q)}}$
(not counting square factors).

Elements of order $2$ in the group are related to factoring
$D(q)$.

If $h$ is $B_1$ smooth, write $h=2^k B_2$ with odd $B_2$.
Compute $q_2:=q^{B_2}$ with enumerating the primes one by one.

Then compute $q_2^{2^i}$ hoping to find factorization of $D(q)$.

In case of failure, try a small multiple $m D(q)$.

The algorithm sometimes works for $D(q)>0$.

As a corollary, this algorithm might compute multiple of $h$.

>Q1 Is this algorithm known, references?

If someone is interested in the toy pari/gp implementation,
contact me at my gmail address.