Is there a LaTeX symbol for 'order'?

I assume you're not looking for a certain symbol, but rather the operator ord, as found on the top of page 7 of the Mark Siggers' Cryptography Class notes. You can define your own operators using amsmath's \DeclareMathOperator:

\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator{\ord}{ord}

\begin{document}
\ldots the exponent, $e_i$, is called the order of $p_i$ in $a$, denoted $\ord_{p_i}(a)=e_i$.

\end{document}

From your comment it became clear that you were looking for the order operator.

In case somebody finds this question but is looking for the order (of magnitude):

\documentclass{article}
\DeclareRobustCommand{\orderof}{\ensuremath{\mathcal{O}}}
\begin{document}
$\orderof\left( 1234\right) =10^{3}$
\end{document}