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How would you find $x$ in a modulo arithmetic expression $x^e \bmod p$ knowing only $e$ and $p$?

$e$ is an integer, $0 \leq e \lt p$, that is relatively prime to $p-1$; and $x$ is an integer, $0 \leq x < p$.

Gerry Myerson
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    Take a look at the Extended Euclidean algorithm at http://en.wikipedia.org/wiki/Euclidean_algorithm – Ross Millikan Mar 18 '11 at 21:51
  • possible duplicate of How to find the inverse modulo m? – Bill Dubuque Mar 18 '11 at 21:55
  • I don't think the question is a duplicate, it is a different question after all. But a good solution to this problem would be to find the inverse modulo $p$. – Eivind Mar 18 '11 at 22:07
  • @Eivind: They are equivalent: any method that solves $\rm\ e\ x \equiv d\ (mod\ p)\ $ can solve the special case $\rm\ d\equiv 1\ $ so it can compute $\rm 1/e:.\ $ Conversely any method that can compute $\rm\ 1/e\ $ can solve the equation by scaling the equation by $\rm\ 1/e:.$ – Bill Dubuque Mar 18 '11 at 22:35
  • @Ross, @Elvind, @Bill: I've edited the question in light of the comment Naan made to the answer below. – Arturo Magidin Mar 19 '11 at 01:51
  • Also posted here: http://math.stackexchange.com/questions/27867/modular-arithmetic/ – Zach Langley Mar 19 '11 at 01:53
  • Note: The question has changed from solving $\rm\ e\ x \equiv d\ (mod\ p)\ $ to solve $\rm\ x^e \equiv d\ (mod\ p):.\ $ The above comments and one of the answers were written before the question was changed. – Bill Dubuque Mar 20 '11 at 17:48

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Having fixed the problem: this is still a problem of using the extended Euclidean algorithm.

Because $e$ and $p-1$ are relatively prime, using the Euclidean algorithm you can find integers $r$ and $s$ such that $er + (p-1)s = 1$.

Now, from Fermat's Little Theorem, you should know that $x^{p-1} \equiv 1\pmod{p}$.

So, take $x^e$, and raise it to the $r$th power. We have: \begin{align*} (x^e)^r &\equiv x^{er} \pmod{p}\\ &= \equiv x^{1-(p-1)s} \pmod{p}\\ &\equiv x(x^{p-1})^{-s}\pmod{p}\\ &\equiv x(1)^{-s}\pmod{p}\\ &\equiv x\pmod{p}. \end{align*}

Arturo Magidin
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If I understand you correctly, your setup is $xe \equiv c (\text{mod} p)$ and you want to find $x$ given $e$ and $c$.

In general, I believe you need that $e$ and $c$ are relatively prime to $p$, so that these are elements of the group of units of $\mathbb{Z}/p\mathbb{Z}$. In this case, you can find a multiplicative inverse of $[e]^{-1}$ of $[e]$, the residue class of $e$. Using this we can see $[e]^{-1}[x][e] = [x] = [e]^{-1}[c]$ thus the residue class $[x]$ of $x$ is the same as the residue class $[e]^{-1}[c]$ and so we can take the representative of this class satisfying $0$ ≤ $x < p$ which necessarily exists and is unique.

Note: In case you aren't familiar with the terms above, the residue class of a number $x$ is in this case the set of numbers $y$ satisfying $y \equiv x (\text{mod} p)$, and the relation $x \equiv y (\text{mod} p)$ means that $x \text{mod} p = y \text{mod} p$.

Alex Becker
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