Return to Question

Given a quaternary quadratic equation of form $$Q(a,b,c,d)=m$$ in $\Bbb Z[a,b,c,d]$ with coefficient sizes and $|m|$ bounded in magnitude by $B\in\Bbb N$ where $m\neq0$ if we are looking for solutions modulo $q$ where $q$ is either a prime power or a composite what is the complexity with which we can solve this in $O(\log^\alpha (Bq))$ time at a fixed $\alpha>0$?

Over $\Bbb Z$ this is at least as hard as factoring ($ab=m$).

Given a quaternary quadratic equation of form $$Q(a,b,c,d)=m$$ in $\Bbb Z[a,b,c,d]$ with coefficient sizes and $|m|$ bounded in magnitude by $B\in\Bbb N$ where $m\neq0$ if we are looking for solutions modulo $q$ where $q$ is either a prime power or a composite then can the complexity with which we can solve this be $O(\log^\alpha (Bq))$ time at a fixed $\alpha>0$?

Given a quaternary quadratic equation of form $Q(a,b,c,d)=m$ in $\Bbb Z[a,b,c,d]$ with coefficient size bounded in magnitude by $B$ if we are looking for solutions modulo $q$ where $q$ is either a prime power or a composite what is the complexity with which we can solve this in $O(\log^\alpha (Bq))$ time at a fixed $\alpha>0$?

Given a quaternary quadratic equation of form $$Q(a,b,c,d)=m$$ in $\Bbb Z[a,b,c,d]$ with coefficient sizes and $|m|$ bounded in magnitude by $B\in\Bbb N$ where $m\neq0$ if we are looking for solutions modulo $q$ where $q$ is either a prime power or a composite what is the complexity with which we can solve this in $O(\log^\alpha (Bq))$ time at a fixed $\alpha>0$?

Over $\Bbb Z$ this is at least as hard as factoring ($ab=m$).

Given a quaternary quadratic form $Q(a,b,c,d)\in\Bbb Z[a,b,c,d]$ with coefficient size bounded in magnitude by $B$ if we are looking for solutions modulo $q$ where $q$ is either a prime power or a composite what is the complexity with which we can solve this in $O(\log^\alpha (Bq))$ time at a fixed $\alpha>0$?

Given a quaternary quadratic equation of form $Q(a,b,c,d)=m$ in $\Bbb Z[a,b,c,d]$ with coefficient size bounded in magnitude by $B$ if we are looking for solutions modulo $q$ where $q$ is either a prime power or a composite what is the complexity with which we can solve this in $O(\log^\alpha (Bq))$ time at a fixed $\alpha>0$?