Cycles of Elliptic Curves: Applications to Cryptography

A cycle of elliptic curves consists of elliptic curves defined over finite fields, such that the number of points on one curve is equal to the size of the field of definition of another, in a cyclic way. Such cycles are known to exist for arbitrary lengths, but their properties are not well studied. We discuss constructions of cycles with small embedding degrees, as well as their applications to cryptography. We also formulate open problems about these cycles.