Abstract: An exemplary method includes defining a CM field, representing coefficients of a Frobenius element of a hyperelliptic curve over a prime field as non-linear polynomials that are functions of an integer x and selecting a value for x whereby the product of the Frobenius element and its complex conjugate is a prime number. Such a method may further include determining the order of the Jacobian of the hyperelliptic curve, for example, where the order is an almost prime number. Various other methods, devices, systems, etc., are also disclosed, which may be optionally used for cryptography.