Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?

Question

http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate polynomials which work provably and have been shown to break $RSA$ system if half of low significant bits of either $P$ or $Q$ are known.

The equation that comes out is $$(2^ku+v)(2^ku'+v')=PQ$$ where if we assume $v$ is known. Then $vv'\equiv PQ\bmod 2^k$ gives $v'$.

So we have a quadratic diophantine equation $$2^kuu'+(uv'+u'v)=\frac{PQ-vv'}{2^k}.$$

Why do I need Coppersmith's method to solve this? Can't a regular diophantine solver work here and so are there explicit polynomials where Coppersmith is better than standard solver in bivariate quadratic case?

Answer 1

Partial answer about "regular diophantine solver".

Finding points on conics in general require integer factorization.

Several papers deal with points on $a x^2+b y^2=c z^2$ and they require factorization of $a,b,c$.

Another example is $x^2 - a y^2=n z^2$. Solving it will compute the square root of $a$ modulo $n$.

Coppersmith method avoid factorization at the cost of small solutions.