The problem of computing an isogeny between two given elliptic curves has been studied by many authors and has several applications. A natural question that has not previously been considered is to construct isogenies between two given supersingular elliptic curves over F_p . Let p gt; 3 be a prime and let E , E' be supersingular elliptic curves over $ F_p . We want to construct an isogeny \\phi : E - gt; E' . The currently fastest algorithm for finding isogenies between supersingular elliptic curves solves this problem in the full supersingular isogeny graph over F_p^2. It takes an expected O(p^1/2 ) bit operations, and also O(p^1/2) space. we consider the structure of the isogeny graph of supersingular elliptic curves over F_p .
