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Hello codeforces,
Today, I am regretably in math class with the subject of modular arithmetic. While thinking about a problem, I reminded myself of a question I asked myself some time ago.
Given $$$p$$$ and $$$mod$$$, find a $$$q$$$ such that $$$q^2$$$ = $$$p$$$ (modulo $$$mod$$$). $$$mod$$$ is prime and ≫ $$$10^8$$$
Does this admit an online logarithmic-complexity solution?
Yes, if you're fine with randomized approaches. See #8 here.
By the way, if $$$m \equiv 3 \pmod 4$$$, you can compute it as $$$q \equiv p^{\frac{m+1}{4}} \pmod{m}$$$ (see here).
Yes. The Tonelli-Shanks algorithm does it in logarithmic time. For an explanation, I think the Wikipedia article is straightforward enough.