Given two elliptic curves over a finite field having the same cardinality and
endomorphism ring, it is known that the curves admit an isogeny between them,
but finding such an isogeny is believed to be computationally difficult. The
fastest known classical algorithm takes exponential time, and prior to our work
no faster quantum algorithm was known. Recently, public-key cryptosystems based
on the presumed hardness of this problem have been proposed as candidates for
post-quantum cryptography. In this paper, we give a subexponential-time quantum
algorithm for constructing isogenies, assuming the Generalized Riemann
Hypothesis (but with no other assumptions). Our algorithm is based on a
reduction to a hidden shift problem, together with a new subexponential-time
algorithm for evaluating isogenies from kernel ideals (under only GRH), and
represents the first nontrivial application of Kuperberg's quantum algorithm
for the hidden shift problem. This result suggests that isogeny-based
cryptosystems may be uncompetitive with more mainstream quantum-resistant
cryptosystems such as lattice-based cryptosystems.
