Suppose $$$p$$$ is an odd prime and $$$a$$$ is a quadratic residue modulo $$$p$$$. Cipolla's algorithm shows that $$$b:=x^{(p+1)/2}\bmod{(x^2-tx+a)}$$$ such that $$$b^2=a$$$ for some irreducible polynomial $$$x^2-tx+a\in\mathbb{F}_p[x]$$$.
I don't know how to prove this properly. If there are any mistakes or typos, please let me know, thanks! Let $$$\alpha$$$ be a zero of $$$x^2-tx+a$$$, $$$\mathbb{F}_p(\alpha):=\lbrace a_0+a_1\alpha :a_0,a_1\in\mathbb{F}_p\rbrace$$$, and let $$$\beta$$$ be a zero of $$$x^2-(t^2-4a)$$$, $$$\mathbb{F}_p(\beta):=\lbrace a_0+a_1\beta :a_0,a_1\in\mathbb{F}_p\rbrace$$$. We may easily find homomorphisms between $$$\mathbb{F}_p(\alpha)$$$ and $$$\mathbb{F}_p(\beta)$$$. So I think we just need to show that $$$((t+\beta )/2)^{p+1}=a$$$.
A lot of blogs show that $$$(t+\beta)^p=t-\beta$$$ according to binomial theorem and Fermat's little theorem, so
Bostan and Mori's paper shows that the computation of $$$x^n$$$ modulo a monic polynomial sometimes is equivalent to the computation of one term of a rational function $$$P(x)/Q(x)$$$ where $$$P,Q$$$ are both polynomials and $$$\deg(P(x))\lt \deg(Q(x))$$$.
We have
and
This may save some multiplications.
