TY - JOUR T1 - Constructing elliptic curve isogenies in quantum subexponential time JF - Journal of Mathematical Cryptology Y1 - 2014 A1 - Andrew M. Childs A1 - David Jao A1 - Vladimir Soukharev AB - 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. VL - 8 U4 - 1 - 29 UR - http://arxiv.org/abs/1012.4019v2 CP - 1 J1 - J. Math. Cryptol. U5 - 10.1515/jmc-2012-0016 ER -