Constructing Permutation Rational Functions From Isogenies

A permutation rational function f∈𝔽_q(x) is a rational function that induces a bijection on 𝔽_q, that is, for all y∈𝔽_q there exists exactly one x∈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).