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Is there an algorithm to find integers such that the following holds true : ${a_1}^2 + {a_2}^2 \dots + {a_k}^2 \equiv p $ $(mod $ $n)$, where $p$ is a prime and $n$ is any general integer.

By Lagrange 4 square theorem it is known that value of $k$ will be atmost 4. Also according to this post since, we are working in modular fields, the value of $k$ will be atmost 2, i.e. when it is not a quadratic residue. But I was interested in finding the integers rather than value of $k$. Please suggest an efficient algorithm for the above.

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    What do you mean "the maximum value"? Perhaps you meant "at most"? Since if it is true for $\;n=4\;$ then it is true for any $\;m>n\;$ , of course. – 2017-01-08
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    Updated the statement. – 2017-01-08
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    can someone please explain how the sum of squares in "modular fields" differ from the regular sum of squares a^2 + b^2 +... c^2 = n?. – 2017-01-08
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    For example, in modular field $4^2 + 1^2 \equiv 3 $ $(mod$ $ 7)$, which is not the case in normal integer system. – 2017-01-08

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