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What efficient algorithms exist for the solving $x^N = a$ in GF(q)? What are their complexities?

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en.wikipedia.org/wiki/… –  Emil Jeřábek Feb 11 '13 at 18:57
    
Thank you! But it is too general method. Is there anything more specific? –  Alexey Feb 11 '13 at 19:47
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Which of $x$, $N$, $a$ are given and which are the variables to be solved for? –  Greg Martin Feb 11 '13 at 19:48
    
N and a are given. –  Alexey Feb 11 '13 at 19:59
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I will assume that $a,N,q$ are given and you want to solve for $x$. Even then, there is no simple single answer. The problem changes if $q$ is a big prime, versus the case $q$ a big power of a small prime, versus intermediate cases. It also changes if $N$ is big versus $N$ small. If are interested in a particular range, making that explicit will help getting a better answer. –  Felipe Voloch Feb 11 '13 at 20:02

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up vote 2 down vote accepted

Pasting the title of your question into Google gives references like http://www.math.leidenuniv.nl/~astolk/monday/notes/stolk-roots.pdf and http://www.ma.utexas.edu/users/voloch/Preprints/roots.pdf -- this should answer your question.

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