  A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function
that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$
there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation
rational functions are intimately related to exceptional rational functions,
and more generally exceptional covers of the projective line, of which they
form the first important example.
  In this paper, we show how to efficiently generate many permutation rational
functions over large finite fields using isogenies of elliptic curves, and
discuss some cryptographic applications. Our algorithm is based on Fried's
modular interpretation of certain dihedral exceptional covers of the projective
line (Cont. Math., 1994).
