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Suppose I have a number $k$ such that $k < p$ but $k^2 > p$

Now I need to find smallest possible value of n such that

$n^2 \bmod p\equiv k^2 \bmod p$

I found a pattern $k^2 \bmod p \equiv (p - k)^2 \bmod p$.

Is there any other method? Are there any set of numbers if not all for which we can find such a number easily.

$P$ is not specifically a prime. It is any arbitrary positive number.

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    Is $p$ a prime, or just an arbitrary positive integer? – 2017-01-16
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    @DanielFischer It is any arbitrary positive integer – 2017-01-16

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