SISE: Supersingular Isogeny-Based Symmetric Encryption

In this paper, we propose the first symmetric encryption scheme based on traversals in the supersingular isogeny graph and point mapping under the 2ⁿ -isogeny, using the Legendre form of elliptic curves. In a supersingular isogeny graph, an elliptic curve can be transformed into an isogenous curve and then back to another curve with the same j-invariant by the means of a dual isogeny. However there are more than one elliptic curves with the same j-invariant and using just the dual isogenies won't lead to a correct curve mapping as well as point mapping that is required for symmetric encryption schemes. With Legendre form of elliptic curves, we define the orientation of six curves with the same j-invariant and precisely defined the rotation function as well as the standardized isogeny. The rotation as well as the standardized isogeny are used to define the accurate backtracking of elliptic curve, meaning we can move back and forth between isogenous curves. This capability naturally enabled us to design a symmetric encryption scheme which is quantum-safe due to the difficulty of the underlying problem. Our encryption scheme can be used to securely sharing short/frequently-changing secondary keys/secrets.