the present invention comprises fast new methods for computing high - precision solutions of frobenius equations that arise in elliptic - curve cryptography . in particular , this invention may be used to accelerate the computation of the number of points on an elliptic curve over a finite field . the advantage over methods in prior art is that the invention is faster than previously known methods . the methods enable optimally fast canonical lifting of elliptic curves defined over finite fields , optimally fast pre - computations to determine an efficient representation of intermediate quantities , and optimally fast lifting of finite - field elements to compute multiplicative representatives . furthermore the invention enables rapid computation of norms and traces amongst other applications .