Computing isogenies between supersingular elliptic curves over Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mat

Let $$p>3$$p>3 be a prime and let $$E$$E, $$E'$$E′ be supersingular elliptic curves over $${\mathbb {F}}_p$$Fp. We want to construct an isogeny $$\phi :E\rightarrow E'$$ϕ:E→E′. The currently fastest algorithm for finding isogenies between supersingular elliptic curves solves this problem in the full supersingular isogeny graph over $${\mathbb {F}}_{p^2}$$Fp2. It takes an expected $$\tilde{\mathcal {O}}(p^{1/2})$$O~(p1/2) bit operations, and also $$\tilde{\mathcal {O}}(p^{1/2})$$O~(p1/2) space, by performing a “meet-in-the-middle” breadth-first search in the isogeny graph. In this paper we consider the structure of the isogeny graph of supersingular elliptic curves over $${\mathbb {F}}_p$$Fp. We give an algorithm to construct isogenies between supersingular curves over $${\mathbb {F}}_p$$Fp that works in $$\tilde{\mathcal {O}}(p^{1/4})$$O~(p1/4) bit operations. We then discuss how this algorithm can be used to obtain an improved algorithm for the general supersingular isogeny problem.