Bounded Quadratic Congruence Problem Variant (for some specific Residue) Given: 3 positive integers $a,b,L$. Problem: Is there a positive integer $x&lt;L$ such that $x^2≡ a(mod\ b)$?

The above problem is NPComplete (as mentioned in G&amp;J) even if we have the factorization of $b$ given. My query is the following:

Is there some specific value of $a$ (say $a_0$) such that, if we limit the original problem (the residue) to only $a_0$, the problem ceases to be NPComplete (and much easily solvable than the generic case) ? 

My guess is for any/each specific $a_0$ the problem still remains NPComplete but I am not certain. Anyone please ?
Yes.  If $a=1$, the problem is no longer NP-complete; the answer to the problem is always "yes, there is such a positive integer".