Cipolla's algorithm http://en.wikipedia.org/wiki/Cipolla's_algorithm is an efficient algorithm for finding a square root modulo a prime number. Is there an efficient algorithm for finding a square root modulo a prime power?

1$\begingroup$ Your link doesn't seem releveant; Cipolla's algorithm does have its own web page (maybe you meant to link to it?) You mean square root, I assume? "Finding a quadratic residue" makes it sound like you want something which is a square modulo $q$. $\endgroup$ – Sheikraisinrollbank Jan 14 '11 at 15:32

1$\begingroup$ no need to pile on... $\endgroup$ – Sheikraisinrollbank Jan 14 '11 at 15:44

$\begingroup$ I changed "quadratic residue" to "square root". $\endgroup$ – Craig Feinstein Jan 14 '11 at 15:44

14$\begingroup$ Cipolla + Hensel's Lemma should do it. $\endgroup$ – Franz Lemmermeyer Jan 14 '11 at 15:53

4$\begingroup$ Once you find a square root of $A$ modulo $p$, NewtonRapheson iteration applied to the polynomial $x^2  A$ will double the number of $p$adic digits of accuracy. (This is for $p\ge 3$. For $p=2$ you probably need to start with a square root modulo 8, and the convergence is slower.) The iteration is $x \to (x/2)+(A/2x)$. In other words, if $a^2 \equiv A \pmod{p^n}$ and you set $b=(a/2)+(A/2a)$, then $b^2\equiv A \pmod{p^{2n}}$. $\endgroup$ – Joe Silverman Jan 14 '11 at 21:20
Joe Silverman's comment gives the method. (if the square root of A mod p is 0 you have any easy first step.... let $\gcd(A\ ,p^n)=p^j.$ If $j$ is odd, give up, otherwise let $A=p^{2k}B$ and find the $\mod p \ $ square root of $B$ (if it is a quadratic residue.)
I ascertained this by looking at the modular square root code in Maple (a bit tricky to see the subprocedures..).
According to Wikipedia the TonelliShanks Algorithm is more efficient that Cipolla's for odd primes not of the form $64Q+1$: Let $m$ be the number of bits in the binary expansion of $p$ and $p1=Q2^S$ with $Q$ odd. Then it is asserted that Cipolla's method is better exactly when $S(S1)>8m+20$. Of course for even primes neither method is needed.
The designers of Maple seem to have determined or decided that trying $2,3,4,\cdots$ is best for primes under $80$ or so. I wasn't able to understand (in the limited time I put into it) which of the the modular square root methods Maple uses for the prime case for larger primes.

$\begingroup$ Excuse me, let's suppose that $\mathrm{gcd}(A,p^{n})=p^{j}$ and $j$ is odd. Then $A$ is not a square modulo $p^{n}$, right? $\endgroup$ – Srinivasa Granujan Mar 28 '16 at 9:47

$\begingroup$ Right. Suppose $B$ is a square root $\mod p^n$ and $B=p^iC \dots$ $\endgroup$ – Aaron Meyerowitz Mar 28 '16 at 20:53
An explicit formula is given in Tonelli's 1891 note referred to in the Wikipedia entry on the TonelliShanks algorithm: Given a prime $p>2$ and a quadratic residue $a \bmod p$, let $x$ be a square root of $a \bmod p$. Then for any power $q = p^k$ and $r:=q/p$, the square root of $a \bmod q$ is $x^r \cdot a^e$ where $e := (q  2r + 1)/2$. Tonelli's proof is straightforward: Square and apply the FermatEuler congruence with exponent $\varphi(q)=qr$ and the fact that $y \equiv z \bmod p$ implies $y^r \equiv z^r \bmod q$.
Dickson's History Of Numbers Vol 1 has formulas that find modular square roots for powers of prime modula. See p215 for Tonelli's algorithm and p218 for Cipolla's algorithm. Dickson's work can be found online at
https://ia800209.us.archive.org/12/items/historyoftheoryo01dick/historyoftheoryo01dick.pdf
Have you checked www.ma.utexas.edu/users/voloch/Preprints/roots.pdf, by Prof. Voloch and P Barreto? If I am not mistaken, in certain cases, their work improves on Cipolla's.

8$\begingroup$ The algorithm on that paper is not an algorithm for taking roots modulo prime powers, it's an algorithm for taking roots on finite fields whose order is a large power of a prime (which are different beasts). $\endgroup$ – Felipe Voloch Jan 14 '11 at 19:27