<p>Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n?  More specifically, suppose I generate a random matrix M, and square it.  Can a probabilistic polynomial time algorithm output M with reasonable probability?  </p>

<p>If the matrix is diagonalizable, then does diagonalizing and taking the square roots work?  What if the matrix isn't diagonalizable, or if the diagonal has entries which aren't squares? What does this imply about square roots of the matrix?</p>

<p>Edits:</p>

<p>Also at:  <a href="https://math.stackexchange.com/questions/52474/taking-square-roots-of-matrices-over-z-nz">https://math.stackexchange.com/questions/52474/taking-square-roots-of-matrices-over-z-nz</a></p>

<p>Clearly the problem is hard in the worst case, since the identity matrix has exponentially many square roots.  So I guess the question is what to do in the average case.</p>

CC BY-SA 3.0
2011-07-20T02:30:55.340Z
SVD: http://en.wikipedia.org/wiki/Singular_value_decomposition. If you can take square root mod n, then just take the square roots of the singular values.
Sasho Nikolov
CC BY-SA 3.0
2011-07-20T02:58:13.197Z
@Sasha: an SVD over a finite field is undefined, see http://mathoverflow.net/questions/6987/singular-value-decomposition-over-finite-fields
Marcus Ritt
CC BY-SA 3.0
2011-07-20T03:29:22.173Z
Crosspost in Math.SE: http://math.stackexchange.com/q/52474/2767
Hsien-Chih Chang 張顯之
CC BY-SA 3.0
2011-07-20T10:43:14.190Z
Closing for simultaneous cross-posting without linking. Please wait a few days before posting the question on another site and always link to the other copies. Flag the post if you don't get an answer on Math.SE after a few days and we will reopen this question.
Kaveh
<ds.algorithms><linear-algebra><randomized-algorithms>
Taking Square Roots of Matrices over Z/nZ
WuTheFWasThat
# Taking Square Roots of Matrices over Z/nZ

(Asked by: username_0 on 2011-07-20)

Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n? More specifically, suppose I generate a random matrix M, and square it. Can a probabilistic polynomial time algorithm output M with reasonable probability?

If the matrix is diagonalizable, then does diagonalizing and taking the square roots work? What if the matrix isn't diagonalizable, or if the diagonal has entries which aren't squares? What does this imply about square roots of the matrix?

Edits:

Also at: <https://math.stackexchange.com/questions/52474/taking-square-roots-of-matrices-over-z-nz>

Clearly the problem is hard in the worst case, since the identity matrix has exponentially many square roots. So I guess the question is what to do in the average case.

**Tags:** ds.algorithms, linear-algebra, randomized-algorithms

> **Comments:**
> **username_1**: SVD: http://en.wikipedia.org/wiki/Singular_value_decomposition. If you can take square root mod n, then just take the square roots of the singular values.  
> **username_2**: @Sasha: an SVD over a finite field is undefined, see http://mathoverflow.net/questions/6987/singular-value-decomposition-over-finite-fields  
> **username_3**: Crosspost in Math.SE: http://math.stackexchange.com/q/52474/2767  
> **username_4**: Closing for simultaneous cross-posting without linking. Please wait a few days before posting the question on another site and always link to the other copies. Flag the post if you don't get an answer on Math.SE after a few days and we will reopen this question.  

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