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

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?

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 $$n$$.