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 $q$. $\endgroup$– SheikraisinrollbankJan 14, 2011 at 15:32

1$\begingroup$ no need to pile on... $\endgroup$– SheikraisinrollbankJan 14, 2011 at 15:44

$\begingroup$ I changed "quadratic residue" to "square root". $\endgroup$– Craig FeinsteinJan 14, 2011 at 15:44

18$\begingroup$ Cipolla + Hensel's Lemma should do it. $\endgroup$– Franz LemmermeyerJan 14, 2011 at 15:53

5$\begingroup$ 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 SilvermanJan 14, 2011 at 21:20
4 Answers
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? $\endgroup$ Mar 28, 2016 at 9:47

$\begingroup$ Right. Suppose $B$ is a square root $\mod p^n$ and $B=p^iC \dots$ $\endgroup$ Mar 28, 2016 at 20:53

$\begingroup$ Can you please explain, how do I find a root modulo p^k if
A mod p = 0
? You said that in this caseyou have any easy first step...
 what does it mean? Does havingA mod p = 0
helps in any sense to find root modulo p^k? $\endgroup$– ArtyJul 16, 2022 at 18:27 
$\begingroup$ See the comment above. Square B, see what happens $\endgroup$ Jul 18, 2022 at 20:59
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 $y^r \equiv z^r \bmod q$.

$\begingroup$ Does this formula work always? For example if
a mod p = 0
(anda != 0
) does it still work? For examplep = 3, k = 4, q = p^k = 3^4 = 81, a = 7
, then roots ofa
modulop = 3
are1, 2
. Whilex^r * a^((q  2 * r + 1) / 2) mod q = 1^27 * 7^((81  2 * 27 + 1) / 2) mod q = 3
, but3^2 = 9
is not equal toa = 7
, so3
is not a square root ofa = 7
modulo81
. $\endgroup$– ArtyJul 16, 2022 at 18:39
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.

9$\begingroup$ 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). $\endgroup$ Jan 14, 2011 at 19:27