Abstract: In some aspects, a key establishment protocol is executed to generate a shared secret. A first entity calculates a first image curve EB representing an image of an elliptic curve E under a first isogeny ?B; calculates the shared secret based on the first image curve EB; receives a second image curve EA and a first pair of elliptic curve points {?A(PB), ?A(QB)} and from a second entity; obtains a basis {R, S}; calculates a third image curve EBA representing an image of the second image curve EA under a second isogeny ?B; calculates a third pair of elliptic curve points {?B(R), ?B(S)}; and sends the third image curve EBA and the third pair of elliptic curve points {?B(R), ?B(S)} to the second entity, wherein the third image curve EBA and the third pair of elliptic curve points {?B(R), ?B(S)} enable the second entity to compute the shared secret.

Abstract: In some aspects, a key establishment protocol is executed to generate a shared secret. A first entity calculates a first image curve EA representing an image of an elliptic curve E under a first isogeny ?A; calculates a first pair of elliptic curve points {?A(PB), ?A(QB)}; calculates secret integers {c, d}; sends the first image curve EA and the first pair of elliptic curve points {?A(PB), ?A(QB)} to a second entity; receives a second image curve EBA and a third pair of elliptic curve points {?B(R), ?B(S)} from the second entity; calculates a third image curve EÃBA representing an image of the second image curve EBA under a second isogeny {tilde over (?)}A, wherein the second isogeny {tilde over (?)}A is identified based on the secret integers {c, d} and the third pair of elliptic curve points {?B(R), ?B(S)}; and calculates the shared secret based on the third image curve EÃBA.