Fpt problem in quadratic residues

[smslca]

In wikipedia source: http://en.wikipedia.org/wiki/Quadratic_residue

under "composite modulus" section

I found the line
"On the other hand, if we want to know if there is a solution for x less than some given limit c, this problem is NP-complete;however, this is a fixed-parameter tractable problem, where c is the parameter."

what does it mean by "given limit c , and fixed parameter tractable with c as parameter". Does this mean regardless of large values are given for c as a limit , we can solve the quadratic congruence without knowing the factorization? or does it has any other meaning?
what is the limit of c such that we cannot solve quadratic congruence using fpt

If i am wrong or obscure in my question, please notify me.

 

[fresh_42]

In wikipedia source: http://en.wikipedia.org/wiki/Quadratic_residue

under "composite modulus" section

I found the line
"On the other hand, if we want to know if there is a solution for x less than some given limit c, this problem is NP-complete;however, this is a fixed-parameter tractable problem, where c is the parameter."

what does it mean by "given limit c , and fixed parameter tractable with c as parameter".

It simply means, that the problem is dependent of (a fixed parameter) $c$ and the module $n$. This means there are basically two input parameters in which runtime can be expressed.

Does this mean regardless of large values are given for c as a limit , we can solve the quadratic congruence without knowing the factorization?

No. It means: We do not know, whether the quadratic residue problem (QRP) is NP-hard or not. But it is NP-hard if we introduce the parameter $c$ as the upper bound of a possible solution into the problem. This indicates that QRP is indeed NP-hard even though we have no formal proof.

... or does it has any other meaning?
what is the limit of c such that we cannot solve quadratic congruence using fpt

If i am wrong or obscure in my question, please notify me.

Factorization is only meant as a comparison here. And of course, if the module is factorized, then QRP can be solved in P.