Solving the equation $X^a \bmod b = N$ Does this equation has a unique solution? Is it solvable?
$$X^a \bmod b = N$$
where
X is unknown
a, b, and N are all known
 A: 
I'm responding to the Comment by the OP, responding to my request for context.  It seems that there is a real context for the Question that has yet to be shared, so I'm marking this as a Community Wiki Answer and hoping that it helps the OP to better understand what kind of context will be useful to willing Readers.

In a "cryptographic challenge" I would expect that you know something important about the "known" values of $a,b,N$.  Whether the equation is solvable (and whether such solutions are unique) is dependent on their specific values, as you will see from doing a few examples.
A mathematical way of writing the equation (in contrast with the "programmer way" of writing it in the body of the Question) is:
$$ x^a \equiv N \bmod b $$
A general approach to solving such equations is the discrete logarithm problem:  

"Several important algorithms in public-key cryptography base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution."

But this does not save us from needing to know what kind of values $a,b,N$ are.
Is $b$ a prime?  Is $N$ coprime to $b$, or perhaps is $N$ zero?  The Question as currently stated does not actually tell us that these are (positive?) integer values.
Information of this kind is critical to determining whether the equation necessarily has a solution, and whether such a solution will be unique.
